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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make the six sided hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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what 's a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how do you make equilateral triangles ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
|
what is a `` mobius strip '' ?
|
so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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how to make a fexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ?
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if you made a hexaflexagon from paper with one side colored how many sides would end up colored when you finish it ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa !
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is it possible to have more than a six sided hexaflexagon ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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is there any relationship between a hexaflexagon and a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong .
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how do you make a 60 degree fold without a protractor ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes .
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what is a mobius strip ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square .
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i heard of something called a `` tri '' -hexaflexagon - is that like around the same thing as a regular hexaflexagon or like , totally different ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up .
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i did n't have the volume on so starting what is the point ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts .
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can you show me how to make a hexaflexagon in slow motion ?
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so say you just moved from england to the us and you 've got your old school supplies from england and your new school supplies from the us and it 's your first day of school and you get to class and find that your new american paper does n't fit in your old english binder . the paper is too wide , and hangs out . so you cut off the extra and end up with all these strips of paper . and to keep yourself amused during your math class you start playing with them . and by you , i mean arthur h. stone in 1939 . anyway , there 's lots of cool things you do with a strip of paper . you can fold it into shapes and more shapes . maybe spiral it around snugly like this . maybe make it into a square . maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts . in fact , there 's enough space here to keep wrapping the strip , and the your hexagon is pretty stable . and you 're like . `` i do n't know , hexagons are n't too exciting , but i guess it has symmetry or something . '' maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up . that 's symmetric , and it collapses down into these three triangles , which collapse down into one triangle , and collapsible hexagons are , you suppose , cool enough to at least amuse you a little but during your class . and then , since hexagons have six-way symmetry , you decide to try this three-way fold the other way , with flappy parts up , and are collapsing it down when suddenly the inside of your hexagon decides to open right up what , you close it back up and undo it . everything seems the same as before , the center is not open-uppable . but when you fold it that way again , it , like , flips inside-out . weird . this time , instead of going backwards , you try doing it again and again and again and again . and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side . yellow side , white side , yellow side , white side hmm . white side ? what ? where did the yellow side go ? so you go back and this time you colour the white side green , and find that your piece of paper has three sides . yellow , white and green . now this thing is definitely cool . therefore , you need to name it . and since it 's shaped like a hexagon and you flex it and flex rhymes with hex , hexaflexagon it is . that night , you ca n't sleep because you keep thinking about hexaflexagons . and the next day , as soon as you get to your math class you pull out your paper strips . you had made this sort of spirally folded paper that folds into again , the shape of a piece of paper , and you decide to take that and use it like a strip of paper to make a hexaflexagon . which would totally work , but it feels sturdier with the extra paper . and you color the three sides and are like , orange , yellow , pink . and you 're sort of trying to pay attention to class . math , yeah . orange , yellow , pink . orange , yellow , white ? wait a second . okay , so you colour that one green . and now it ; s orange , yellow , green , orange , yellow , green . who knows where the pink side went ? oh , there it is . now it 's back to orange , yellow , pink . orange , yellow , pink . hmm . blue . yellow , pink , blue . yellow , pink , blue . yellow , pink , huh . with the old flexagon , you could only flex it one way , flappy way up . but now there 's more flaps . so maybe you can fold it both ways . yes , one goes from pink to blue , but the other , from pink to orange . and now , one way goes from orange to yellow , but the other way goes from orange to neon yellow . during lunch you want to show this off to one of your new friends , bryant tuckerman . you start with the original , simple , three-faced hexaflexagon , which you call the trihexaflexagon . and he 's like , whoa ! and wants to learn how to make one . and you are like , it 's easy ! just start with a paper strip , fold it into equilateral traingles , and you 'll need nine of them , and you fold them around into this cycle and make sure it 's all symmetric . the flat parts are diamonds , and if they 're not , then you 're doing it wrong . and then you just tape the first triangle to the last along the edge , and you 're good . but tuckerman does n't have tape . after all , it was invented only 10 years ago . so he cuts out ten triangles instead of nine , and then glues the first to the last . then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and traingly , and then opening from the centre . you decide to start a flexagon committee together to explore the mysteries of flexagotion , but that will have to wait until next time .
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and you want to make one that 's a little less messy , so you try with another strip and tape it nicely into a twisty-foldy loop . you decide that it would be cool to colour the sides , so you get out a highlighter and make one yellow . now you can flip from yellow side to white side .
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how do you make a hexaflexagon with more than 3 sides ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 .
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what if the denominator is a prime number ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction .
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does anyone know how to divide a pie into 5 equal slices ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 .
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what do you call the line between the numerator and the denominator ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this .
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what does lcd stand for ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 .
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why did sal write a dot instead of a multiplication sign 3 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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so prime factorization is to make a composite number prime ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other .
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when you said the least common denominator is the product of the denominators 4 & 3 , in future problems can i just multiply the two denominators to find the least common denominator or do i need to make sure they do not share prime factors first ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 .
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and what exactly is a prime a number anyways ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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what is the point of prime numbers ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 .
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is 7/11 considered a simplified fraction ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 .
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what is the difference between the numerator and the denominator ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 .
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what 's a prime number ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it .
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how do you turn a fraction in to a percentage ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 .
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will a number line be a good way or a bar ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way .
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cant 21/28 be simplified more ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 .
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could someone explain what a prime number is please ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 .
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what would you do with negative fractions ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it .
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do you divide first or multiply first ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers .
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do i have to multiply the numbers ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 .
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what happens if the numerator and the denominator can not be devided by the same number ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 .
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how are you do you find that 21 and 28 is divisible by 7 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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what is the point of using prime numbers ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this .
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how do you do the sum ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this .
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what is short term division ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 .
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can a prime number be a multiple of any number except itself ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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what the hell is prime factor ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 .
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what is the simplified version of 12/6 and 27/9 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 .
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how can you make a pie chart out of fractions ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it .
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what is a equivalent fraction ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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when did you introduce factoring , prime numbers , etc ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 .
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how do i find th area of a base ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 .
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can u compare the fractions with a different denominator with decimals or only with fractions ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 .
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how do you know what to number to multiply ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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did we learn prime factorization before ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 .
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how do i show 4/6 & 5/7 on a number line ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 .
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what does the `` common '' thingy and the `` least common '' thingy mean ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first .
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sal is dividing by different numbers for example hes dividing 21/28 by 7 and hes dividing 6/9 by 3 but when i do that i get it wrong how is that possible ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 .
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if you have 2/8 and 3/89 could you cross multiply and then compare the products ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 .
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why did sal do all that complicated figuring , when he could just reduce the fractions ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward .
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what is 16 *8 equal to ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 .
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so my question is would n't the answer be 2/3 is greater than 3/4 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two .
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what are the prime factors ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 .
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how would we use comparing fractions in the real world ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators .
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do i have to do the harder way ( which i quite frankly ca n't grasp ) because i can not simplify the fractions before finding a common denominator ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 .
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by what % is 1/4 from 3/8 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 .
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so is 4/9 -- 7/12is 48/108 lessthan 63/108 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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is there any way to avoid introducing new concepts like prime factorization in the 4th grade fractions playlist ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two .
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so , if the numbers do n't share any prime factors , it 's the same of saying that they do not have a gcd ( greatest common divisor ) ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 .
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you know how if you multiply or divide the numerator or denominator of a fraction ( if you 're comparing it with another fraction ) you have to do the same to the same to both the top and bottom ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 .
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how do u get the common denominaors when you have to find out which one is smaller ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 .
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during the eclipse sal taks about t why is the moon the exact size of the sun , offering complete 1 for 1 coverage ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 .
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what was the point of doing a prime factorization tree when you just ended up multiplying the denominators to find the lcm ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 .
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when in the real world is comparing fractions useful ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 .
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what would happen if one or both fractions were improper fractions ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first .
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today you dusted and vacuumed your home if your dusted your house every 6 days and run the vacuumed every 9 days when is the next time you will do both chores again ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet .
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can you use smaller numbers for the first question , like 4x7 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 .
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what is 600 divided by 22 ?
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use less than , greater than , or equal to compare the two fractions 21/28 , or 21 over 28 , and 6/9 , or 6 over 9 . so there 's a bunch of ways to do this . the easiest way is if they had the same denominator , you could just compare the numerators . unlucky for us , we do not have the same denominator . so what we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators . or even more simply , we could simplify them first and then try to do it . so let me do that last one , because i have a feeling that 'll be the fastest way to do it . so 21/28 -- you can see that they are both divisible by 7 . so let 's divide both the numerator and the denominator by 7 . so we could divide 21 by 7 . and we can divide -- so let me make the numerator -- and we can divide the denominator by 7 . we 're doing the same thing to the numerator and the denominator , so we 're not going to change the value of the fraction . so 21 divided by 7 is 3 , and 28 divided by 7 is 4 . so 21/28 is the exact same fraction as 3/4 . 3/4 is the simplified version of it . let 's do the same thing for 6/9 . 6 and 9 are both divisible by 3 . so let 's divide them both by 3 so we can simplify this fraction . so let 's divide both of them by 3 . 6 divided by 3 is 2 , and 9 divided by 3 is 3 . so 21/28 is 3/4 . they 're the exact same fraction , just written a different way . this is the more simplified version . and 6/9 is the exact same fraction as 2/3 . so we really can compare 3/4 and 2/3 . so this is really comparing 3/4 and 2/3 . and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 . then we would have to multiply big numbers . here we could do fairly small numbers . the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 . and 4 and 3 do n't share any prime factors with each other . so their least common multiple is really just going to be the product of the two . so we can write 3/4 as something over 12 . and we can write 2/3 as something over 12 . and i got the 12 by multiplying 3 times 4 . they have no common factors . another way you could think about it is 4 , if you do a prime factorization , is 2 times 2 . and 3 -- it 's already a prime number , so you ca n't prime factorize it any more . so what you want to do is think of a number that has all of the prime factors of 4 and 3 . so it needs one 2 , another 2 , and a 3 . well , 2 times 2 times 3 is 12 . and either way you think about it , that 's how you would get the least common multiple or the common denominator for 4 and 3 . well , to get from 4 to 12 , you 've got to multiply by 3 . so we 're multiplying the denominator by 3 to get to 12 . so we also have to multiply the numerator by 3 . so 3 times 3 is 9 . over here , to get from 3 to 12 , we have to multiply the denominator by 4 . so we also have to multiply the numerator by 4 . so we get 8 . and so now when we compare the fractions , it 's pretty straightforward . 21/28 is the exact same thing as 9/12 , and 6/9 is the exact same thing as 8/12 . so which of these is a greater quantity ? well , clearly , we have the same denominator right now . we have 9/12 is clearly greater than 8/12 . so 9/12 is clearly greater than 8/12 . or if you go back and you realize that 9/12 is the exact same thing as 21/28 , we could say 21/28 is definitely greater than -- and 8/12 is the same thing as 6/9 -- is definitely greater than 6/9 . and we are done . another way we could have done it -- we did n't necessarily have to simplify that . and let me show you that just for fun . so if we were doing it with -- if we did n't think to simplify our two numbers first . i 'm trying to find a color i have n't used yet . so 21/28 and 6/9 . so we could just find a least common multiple in the traditional way without simplifying first . so what 's the prime factorization of 28 ? it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization . prime factorization of 9 is 3 times 3 . so the least common multiple of 28 and 9 have to contain a 2 , a 2 , a 7 , a 3 and a 3 . or essentially , it 's going to be 28 times 9 . so let 's over here multiply 28 times 9 . there 's a couple of ways you could do it . you could multiply in your head 28 times 10 , which would be 280 , and then subtract 28 from that , which would be what ? 252 . or we could just multiply it out if that confuses you . so let 's just do the second way . 9 times 8 is 72 . 9 times 2 is 18 . 18 plus 7 is 25 . so we get 252 . so i 'm saying the common denominator here is going to be 252 . least common multiple of 28 and 9 . well , to go from 28 to 252 , we had to multiply it by 9 . we had to multiply 28 times 9 . so we 're multiplying 28 times 9 . so we also have to multiply the numerator times 9 . so what is 21 times 9 ? that 's easier to do in your head . 20 times 9 is 180 . and then 1 times 9 is 9 . so this is going to be 189 . to go from 9 to 252 , we had to multiply by 28 . so we also have to multiply the numerator by 28 if we do n't want to change the value of the fraction . so 6 times 28 -- 6 times 20 is 120 . 6 times 8 is 48 . so we get 168 . let me write that out just to make sure i did n't make a mistake . so 28 times 6 -- 8 times 6 is 48 . 2 times 6 is 12 , plus 4 is 16 . so right , 168 . so now we have a common denominator here . and so we can really just compare the numerators . and 189 is clearly greater than 168 . so 189/252 is clearly greater than 168/252 . or that 's the same thing as saying 21/28 , because that 's what this is over here . the left-hand side is 21/28 , is clearly greater than the right-hand side , which is really 6/9 .
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it 's 2 times 14 . and 14 is 2 times 7 . that 's its prime factorization .
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how do we order fractions because for example 1/2 7/2 4/5 it is 1/2 first 4/5 next and finally 7/2 but the thing says no it is incorrect ?
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we are here in the portland art museum in the jubilee center for modern and contemporary art in front of marcel duchamp 's landmark boite-en-valise the red box , series f when i hear the word `` valise '' i think of could carry around like a suitcase . yeah . like a suitcase . i think my grandmother used `` valise '' for a suitcase exactly , and we use backpacks . it 's a new nomenclature . duchamp at the end of world war one had seen in the desturction in europe . he was a painter , he saw his brother injured in battle and die he came back to paris and he said `` my work could be lost for the ages '' and he decided to reproduce everything and make his own history his own musuem in a suitcase . he packaged himself absolutely . in a new an completely unbranded way . he reproduced a new descending staircase all of his cubist and vaguely surrealist paintings on reproductions it 's like a retrospective in a box . absolutely and you know the interesting thing about this box is that he updates it twice so that the box here at the portland art museum is a box that he created as an addition for schwartz in italy in 1960 how many are there ? 100 there are 100 boxes in the red version there are red , green , leather and tan versions the earliest version is actually a valise a leather valise , because these are mechanically reproduced objects he had a little cottage industry he prints a bunch of them but they never get assembled and so when schwartz comes to him in the '60s and says i want to do the boite-de-valise as a real addition duchamp says okay i happened to have a stack of them right here . i never put them together and so his wife teeny sat in paris apartment gluing the reproductions onto these black cardboard backgrounds and little labels they had made and they created this retrospective in a box and there is something to me that 's sort of about the idea of the artist packaging himself selling himself , almost like a traveling salesman going around to galleries and try to get their work there is something about that artist and the commercial environment . as a curator i always immidiately pullback when someone introduces themselves as an artist because the modern artist carried slides and now they carry an iphone with their entire work on it sort of like duchamp but in the technology of today as opposed to the reproductive technology of his age what i find interesting and beautiful about this is the variety because he was both a painter and a sculptor in the sense that the ready made was a sculpture we have the tiny verison of a urel the entire thing was a ready made he gave us his lifes work to that date as a ready made surrogate for the experience . none of it is an original print in the sense of an artist pulled , plate numbered and signed but in fact these are all mechanically reproduced from images that may or may not have been an original work of art in the first place what 's great to me about it is this kind of embracing of mechanical reproduction which is sort of a thread through all of duchamp 's work and a sort of loss of the aura of the original which was always a sort of issue that duchamp confronted you 'll notive that this object is sitting not in 1929 when he first concieves it but it sits here in the jubit centre halfway up in the 1960-70 when america discovers his ideas when duchamp becomes the grandfather of pop and of cindy sherman and richard prince and all of the artists who practice appropriation and sherry levine and all of that absolutely and sherry levine who would not exist without duchamp and this little miniturization of them too it 's like little souveniers a little duchamp souvenir it 's like warhol , your little vail of the air of paris right here
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we are here in the portland art museum in the jubilee center for modern and contemporary art in front of marcel duchamp 's landmark boite-en-valise the red box , series f when i hear the word `` valise '' i think of could carry around like a suitcase . yeah .
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what is mona lisa doing in his collection ?
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- you know , there 's that saying that it takes a village to raise a kid . well , i guess you could say that it takes an entire body , and i mean every single organ of a body , to make a baby . it 's like the best example of teamwork that there is , all the organs in the body working together to support the growth and the development of the fetus , but also just as important , to make sure that mom 's body is n't sacrificing its own needs to support the pregnancy . so in order to accomplish that , pretty much every organ system undergoes a significant amount of change . and i wan na go through what those changes are . so let 's start with the cardiovascular system , so the heart and the blood vessels throughout the body . and , i guess you could say , at the very basic , most essential , level , the body needs more blood to carry oxygen and those nutrients to the fetus . so the blood volume increases by something like 40 to 50 percent throughout the pregnancy . and it 's not just that there 's more blood . the heart is also working harder to more efficiently supply that blood to the fetus . so , for example , the heart beats more quickly . it beats , like , 10 to 15 beats more each minute than usual . and the stroke volume , so the amount of blood that 's pumped out with each heartbeat , increases , which means that the cardiac output , or the amount of blood that 's pumped out of the heart each minute , also increases . so in summary , more blood is being pumped out of the heart to meet the demands of the fetus . so if you had to take a guess , what do you think happens to blood pressure during pregnancy ? i 'm gon na guess that you guys guessed that it increases , because that 's certainly what i thought happened . but it actually decreases . and it decreases for a couple of different reasons . firstly , there 's a lot of progesterone floating around in the blood during pregnancy . and , if there 's something that progesterone does really well , it relaxes smooth muscle . and that includes the smooth muscle that surrounds all of the blood vessels . so that relaxation causes dilation of the blood vessels , which then lowers the blood pressure . so that 's a first cause , sort of , of the decrease in blood pressure through pregnancy . the second thing that contributes to the lower blood pressure is the placenta , which is an addition of an entirely brand-new blood vessel circuit to the circulatory system . it 's like when you add a resistor in parallel , reducing the resistance of the entire circuit . i 'm just joking . that does n't help anyone understand it any better . i guess you can think of it kind of , kind of as a tall apartment building and what would happen to the pressure in the shower heads if you added a whole additional floor of apartments , with shower heads that are really leaky and let out a lot of water . the placenta is kind of like that . it 's a really low-resistance circuit . and also , while we 're talking about the cardiovascular system , there 's a syndrome that 's called supine , it 's called supine .. `` supine '' means when you 're lying on your back , supine postural , so it 's supine postural , `` postural '' meaning related to posture , supine postural hypotensive syndrome , so supine postural hypotensive syndrome . and it 's weird that i 'm talk about a syndrome in a video that 's talking about physiology during pregnancy . but i guess you could say , it 's a syndrome , or something that goes wrong , due to normal pregnancy physiology . so anyways , what it refers to is when , late in the course of a pregnancy , the uterus becomes larger , right ? and when the uterus becomes larger , it can compress the inferior vena cava . so that kind of looks like this . and since the inferior vena cava gathers the blood from the veins of the lower body and returns that blood to the heart , the compression of the inferior vena cava means that less blood is pushed back to the heart , meaning that less blood is pumped out with each heartbeat . and that leads to low blood pressure , or hypotension . right ? so that 's where the `` hypotensive '' in this name comes from . so the woman starts to feel light headed and like she 's about to faint , especially when she 's on her back , because that 's the position , when she 's on her back , that 's the position in which the uterus is exerting the most pressure on the inferior vena cava . a really quick way to resolve that issue is for the woman to turn to her left side , and that tilts the uterus to the left and off of the inferior vena cava , allowing more blood to return to the heart . so that 's all or most of the functional changes that occur with the cardiovascular system in pregnancy . and the growing uterus also shifts the heart to the left a little , so there are also some anatomical changes , too . now , i know that 's a lot of information , but the cardiovascular system undergoes lots of changes to support the pregnancy . so now let 's move on to what changes occur in the respiratory system . so oxygen . okay . oxygen consumption increases in pregnancy . right ? the fetus uses oxygen , the mom is using more oxygen to support all the changes in the body . so that means that mom 's body needs to bring in more oxygen into the blood . and that 's mostly done by increasing minute ventilation . it 's mostly done by increasing minute ventilation , or the volume of air that 's taken in each minute . and it 's not that pregnant ladies intentionally take deeper breaths , because that would get really uncomfortable very quickly . it 's all that progesterone once again . so it 's all that progesterone in the blood . and what that progesterone does is , it acts on the central respiratory centers in the brain to instruct the lungs to take in more air with each breath . so that 's how you end up with more air being taken in with each breath during pregnancy . and a quick thing that needs to be mentioned is that , when you have more air being inhaled with each breath , more carbon dioxide is being exhaled with each breath . right ? does that make sense ? and carbon dioxide is an acid . so in order to keep the ph of the blood balanced , the body responds to that decrease in carbon dioxide , so that decrease in an acid , by increasing the secretion of bicarbonate , which is a base , from the kidneys . so you have increased secretion of bicarbonate from the kidneys . so what you end up with is either normal or a very slightly alkalotic , so a slightly basic , blood ph . okay . and lastly , there are a couple of anatomical changes , too . so , the enlarging uterus pushes the diaphragm upwards , almost four centimeters through the course of the pregnancy . and that would really make it difficult to breathe . but the chest wall during pregnancy is also more mobile , it 's more flexible . and your chest wall circumference is larger . so that works to make up for that upward shift of the diaphragm . okay , so let 's finish off down here by discussing the changes that occur with the kidney . so two things . first thing , we said that there 's an increase in blood volume during pregnancy , right ? and secondly , all of the arteries in the body are dilated during pregnancy , including the ones that supply the kidney . so if you add those two things up , you end up with having more blood flow to the kidney . and what that means is , you end up with an increase in the rate of filtration of blood through the kidney . it 's kind of like , you know those water filters that you can attach directly to your faucet ? right ? imagine if you had one of those . and if your pipes and your faucet got much larger , they got much wider , and there 's more water running through the pipes , the rate at which the water was being filtered through the water filter would increase drastically . well , this is the exact same thing . the kidney is just like your water filter in that it filters all of your blood . now , with regards to the bladder , there is a contentious topic of whether the bladder holds more or less urine in a pregnant woman . there 's some thought that progesterone , which , remember , causes relaxation of smooth muscle , relaxes and increases the capacity of the bladder . and then there 's other thought that the pressure of the uterus , the large uterus on the bladder , decreases the capacity of the bladder . so we 're not entirely sure . but one thing is certain , and that is that pregnant women definitely urinate more frequently than normal . and that has to do with increased urine production , as well as that pressure on the bladder from the uterus . and that pressure from the uterus also leads to dilatation of the ureter . so the ureters become dilated . and that 's really important . and it kind of looks like this , where the pressure from the larger uterus causes the ureters to become wider , to become dilated . and that uterus putting pressure here sort of acts as a road block . and urine builds up in the ureters behind that road block . that built-up stagnant urine acts as a medium for bacterial growth , right ? because we know urinary stasis is a risk factor for bacterial growth . and that 's perhaps why pregnant women are more susceptible to developing pyelonephritis , or infection of the kidney , than are non-pregnant women . it 's because of that urinary stasis that occurs as a result of the large uterus putting pressure on the ureters . all right . so those are some of the physiologic changes that occur in pregnancy , with the cardiovascular system , the respiratory system , and the renal system .
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okay . oxygen consumption increases in pregnancy . right ?
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what is the physiologic explanation for the unusual , insistent food cravings that some women experience during pregnancy ?
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- you know , there 's that saying that it takes a village to raise a kid . well , i guess you could say that it takes an entire body , and i mean every single organ of a body , to make a baby . it 's like the best example of teamwork that there is , all the organs in the body working together to support the growth and the development of the fetus , but also just as important , to make sure that mom 's body is n't sacrificing its own needs to support the pregnancy . so in order to accomplish that , pretty much every organ system undergoes a significant amount of change . and i wan na go through what those changes are . so let 's start with the cardiovascular system , so the heart and the blood vessels throughout the body . and , i guess you could say , at the very basic , most essential , level , the body needs more blood to carry oxygen and those nutrients to the fetus . so the blood volume increases by something like 40 to 50 percent throughout the pregnancy . and it 's not just that there 's more blood . the heart is also working harder to more efficiently supply that blood to the fetus . so , for example , the heart beats more quickly . it beats , like , 10 to 15 beats more each minute than usual . and the stroke volume , so the amount of blood that 's pumped out with each heartbeat , increases , which means that the cardiac output , or the amount of blood that 's pumped out of the heart each minute , also increases . so in summary , more blood is being pumped out of the heart to meet the demands of the fetus . so if you had to take a guess , what do you think happens to blood pressure during pregnancy ? i 'm gon na guess that you guys guessed that it increases , because that 's certainly what i thought happened . but it actually decreases . and it decreases for a couple of different reasons . firstly , there 's a lot of progesterone floating around in the blood during pregnancy . and , if there 's something that progesterone does really well , it relaxes smooth muscle . and that includes the smooth muscle that surrounds all of the blood vessels . so that relaxation causes dilation of the blood vessels , which then lowers the blood pressure . so that 's a first cause , sort of , of the decrease in blood pressure through pregnancy . the second thing that contributes to the lower blood pressure is the placenta , which is an addition of an entirely brand-new blood vessel circuit to the circulatory system . it 's like when you add a resistor in parallel , reducing the resistance of the entire circuit . i 'm just joking . that does n't help anyone understand it any better . i guess you can think of it kind of , kind of as a tall apartment building and what would happen to the pressure in the shower heads if you added a whole additional floor of apartments , with shower heads that are really leaky and let out a lot of water . the placenta is kind of like that . it 's a really low-resistance circuit . and also , while we 're talking about the cardiovascular system , there 's a syndrome that 's called supine , it 's called supine .. `` supine '' means when you 're lying on your back , supine postural , so it 's supine postural , `` postural '' meaning related to posture , supine postural hypotensive syndrome , so supine postural hypotensive syndrome . and it 's weird that i 'm talk about a syndrome in a video that 's talking about physiology during pregnancy . but i guess you could say , it 's a syndrome , or something that goes wrong , due to normal pregnancy physiology . so anyways , what it refers to is when , late in the course of a pregnancy , the uterus becomes larger , right ? and when the uterus becomes larger , it can compress the inferior vena cava . so that kind of looks like this . and since the inferior vena cava gathers the blood from the veins of the lower body and returns that blood to the heart , the compression of the inferior vena cava means that less blood is pushed back to the heart , meaning that less blood is pumped out with each heartbeat . and that leads to low blood pressure , or hypotension . right ? so that 's where the `` hypotensive '' in this name comes from . so the woman starts to feel light headed and like she 's about to faint , especially when she 's on her back , because that 's the position , when she 's on her back , that 's the position in which the uterus is exerting the most pressure on the inferior vena cava . a really quick way to resolve that issue is for the woman to turn to her left side , and that tilts the uterus to the left and off of the inferior vena cava , allowing more blood to return to the heart . so that 's all or most of the functional changes that occur with the cardiovascular system in pregnancy . and the growing uterus also shifts the heart to the left a little , so there are also some anatomical changes , too . now , i know that 's a lot of information , but the cardiovascular system undergoes lots of changes to support the pregnancy . so now let 's move on to what changes occur in the respiratory system . so oxygen . okay . oxygen consumption increases in pregnancy . right ? the fetus uses oxygen , the mom is using more oxygen to support all the changes in the body . so that means that mom 's body needs to bring in more oxygen into the blood . and that 's mostly done by increasing minute ventilation . it 's mostly done by increasing minute ventilation , or the volume of air that 's taken in each minute . and it 's not that pregnant ladies intentionally take deeper breaths , because that would get really uncomfortable very quickly . it 's all that progesterone once again . so it 's all that progesterone in the blood . and what that progesterone does is , it acts on the central respiratory centers in the brain to instruct the lungs to take in more air with each breath . so that 's how you end up with more air being taken in with each breath during pregnancy . and a quick thing that needs to be mentioned is that , when you have more air being inhaled with each breath , more carbon dioxide is being exhaled with each breath . right ? does that make sense ? and carbon dioxide is an acid . so in order to keep the ph of the blood balanced , the body responds to that decrease in carbon dioxide , so that decrease in an acid , by increasing the secretion of bicarbonate , which is a base , from the kidneys . so you have increased secretion of bicarbonate from the kidneys . so what you end up with is either normal or a very slightly alkalotic , so a slightly basic , blood ph . okay . and lastly , there are a couple of anatomical changes , too . so , the enlarging uterus pushes the diaphragm upwards , almost four centimeters through the course of the pregnancy . and that would really make it difficult to breathe . but the chest wall during pregnancy is also more mobile , it 's more flexible . and your chest wall circumference is larger . so that works to make up for that upward shift of the diaphragm . okay , so let 's finish off down here by discussing the changes that occur with the kidney . so two things . first thing , we said that there 's an increase in blood volume during pregnancy , right ? and secondly , all of the arteries in the body are dilated during pregnancy , including the ones that supply the kidney . so if you add those two things up , you end up with having more blood flow to the kidney . and what that means is , you end up with an increase in the rate of filtration of blood through the kidney . it 's kind of like , you know those water filters that you can attach directly to your faucet ? right ? imagine if you had one of those . and if your pipes and your faucet got much larger , they got much wider , and there 's more water running through the pipes , the rate at which the water was being filtered through the water filter would increase drastically . well , this is the exact same thing . the kidney is just like your water filter in that it filters all of your blood . now , with regards to the bladder , there is a contentious topic of whether the bladder holds more or less urine in a pregnant woman . there 's some thought that progesterone , which , remember , causes relaxation of smooth muscle , relaxes and increases the capacity of the bladder . and then there 's other thought that the pressure of the uterus , the large uterus on the bladder , decreases the capacity of the bladder . so we 're not entirely sure . but one thing is certain , and that is that pregnant women definitely urinate more frequently than normal . and that has to do with increased urine production , as well as that pressure on the bladder from the uterus . and that pressure from the uterus also leads to dilatation of the ureter . so the ureters become dilated . and that 's really important . and it kind of looks like this , where the pressure from the larger uterus causes the ureters to become wider , to become dilated . and that uterus putting pressure here sort of acts as a road block . and urine builds up in the ureters behind that road block . that built-up stagnant urine acts as a medium for bacterial growth , right ? because we know urinary stasis is a risk factor for bacterial growth . and that 's perhaps why pregnant women are more susceptible to developing pyelonephritis , or infection of the kidney , than are non-pregnant women . it 's because of that urinary stasis that occurs as a result of the large uterus putting pressure on the ureters . all right . so those are some of the physiologic changes that occur in pregnancy , with the cardiovascular system , the respiratory system , and the renal system .
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because we know urinary stasis is a risk factor for bacterial growth . and that 's perhaps why pregnant women are more susceptible to developing pyelonephritis , or infection of the kidney , than are non-pregnant women . it 's because of that urinary stasis that occurs as a result of the large uterus putting pressure on the ureters .
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as pregnant women are more likely to get kidney infections , what postures or exercises are recommended to drain out the stagnant urine into the bladder ?
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- you know , there 's that saying that it takes a village to raise a kid . well , i guess you could say that it takes an entire body , and i mean every single organ of a body , to make a baby . it 's like the best example of teamwork that there is , all the organs in the body working together to support the growth and the development of the fetus , but also just as important , to make sure that mom 's body is n't sacrificing its own needs to support the pregnancy . so in order to accomplish that , pretty much every organ system undergoes a significant amount of change . and i wan na go through what those changes are . so let 's start with the cardiovascular system , so the heart and the blood vessels throughout the body . and , i guess you could say , at the very basic , most essential , level , the body needs more blood to carry oxygen and those nutrients to the fetus . so the blood volume increases by something like 40 to 50 percent throughout the pregnancy . and it 's not just that there 's more blood . the heart is also working harder to more efficiently supply that blood to the fetus . so , for example , the heart beats more quickly . it beats , like , 10 to 15 beats more each minute than usual . and the stroke volume , so the amount of blood that 's pumped out with each heartbeat , increases , which means that the cardiac output , or the amount of blood that 's pumped out of the heart each minute , also increases . so in summary , more blood is being pumped out of the heart to meet the demands of the fetus . so if you had to take a guess , what do you think happens to blood pressure during pregnancy ? i 'm gon na guess that you guys guessed that it increases , because that 's certainly what i thought happened . but it actually decreases . and it decreases for a couple of different reasons . firstly , there 's a lot of progesterone floating around in the blood during pregnancy . and , if there 's something that progesterone does really well , it relaxes smooth muscle . and that includes the smooth muscle that surrounds all of the blood vessels . so that relaxation causes dilation of the blood vessels , which then lowers the blood pressure . so that 's a first cause , sort of , of the decrease in blood pressure through pregnancy . the second thing that contributes to the lower blood pressure is the placenta , which is an addition of an entirely brand-new blood vessel circuit to the circulatory system . it 's like when you add a resistor in parallel , reducing the resistance of the entire circuit . i 'm just joking . that does n't help anyone understand it any better . i guess you can think of it kind of , kind of as a tall apartment building and what would happen to the pressure in the shower heads if you added a whole additional floor of apartments , with shower heads that are really leaky and let out a lot of water . the placenta is kind of like that . it 's a really low-resistance circuit . and also , while we 're talking about the cardiovascular system , there 's a syndrome that 's called supine , it 's called supine .. `` supine '' means when you 're lying on your back , supine postural , so it 's supine postural , `` postural '' meaning related to posture , supine postural hypotensive syndrome , so supine postural hypotensive syndrome . and it 's weird that i 'm talk about a syndrome in a video that 's talking about physiology during pregnancy . but i guess you could say , it 's a syndrome , or something that goes wrong , due to normal pregnancy physiology . so anyways , what it refers to is when , late in the course of a pregnancy , the uterus becomes larger , right ? and when the uterus becomes larger , it can compress the inferior vena cava . so that kind of looks like this . and since the inferior vena cava gathers the blood from the veins of the lower body and returns that blood to the heart , the compression of the inferior vena cava means that less blood is pushed back to the heart , meaning that less blood is pumped out with each heartbeat . and that leads to low blood pressure , or hypotension . right ? so that 's where the `` hypotensive '' in this name comes from . so the woman starts to feel light headed and like she 's about to faint , especially when she 's on her back , because that 's the position , when she 's on her back , that 's the position in which the uterus is exerting the most pressure on the inferior vena cava . a really quick way to resolve that issue is for the woman to turn to her left side , and that tilts the uterus to the left and off of the inferior vena cava , allowing more blood to return to the heart . so that 's all or most of the functional changes that occur with the cardiovascular system in pregnancy . and the growing uterus also shifts the heart to the left a little , so there are also some anatomical changes , too . now , i know that 's a lot of information , but the cardiovascular system undergoes lots of changes to support the pregnancy . so now let 's move on to what changes occur in the respiratory system . so oxygen . okay . oxygen consumption increases in pregnancy . right ? the fetus uses oxygen , the mom is using more oxygen to support all the changes in the body . so that means that mom 's body needs to bring in more oxygen into the blood . and that 's mostly done by increasing minute ventilation . it 's mostly done by increasing minute ventilation , or the volume of air that 's taken in each minute . and it 's not that pregnant ladies intentionally take deeper breaths , because that would get really uncomfortable very quickly . it 's all that progesterone once again . so it 's all that progesterone in the blood . and what that progesterone does is , it acts on the central respiratory centers in the brain to instruct the lungs to take in more air with each breath . so that 's how you end up with more air being taken in with each breath during pregnancy . and a quick thing that needs to be mentioned is that , when you have more air being inhaled with each breath , more carbon dioxide is being exhaled with each breath . right ? does that make sense ? and carbon dioxide is an acid . so in order to keep the ph of the blood balanced , the body responds to that decrease in carbon dioxide , so that decrease in an acid , by increasing the secretion of bicarbonate , which is a base , from the kidneys . so you have increased secretion of bicarbonate from the kidneys . so what you end up with is either normal or a very slightly alkalotic , so a slightly basic , blood ph . okay . and lastly , there are a couple of anatomical changes , too . so , the enlarging uterus pushes the diaphragm upwards , almost four centimeters through the course of the pregnancy . and that would really make it difficult to breathe . but the chest wall during pregnancy is also more mobile , it 's more flexible . and your chest wall circumference is larger . so that works to make up for that upward shift of the diaphragm . okay , so let 's finish off down here by discussing the changes that occur with the kidney . so two things . first thing , we said that there 's an increase in blood volume during pregnancy , right ? and secondly , all of the arteries in the body are dilated during pregnancy , including the ones that supply the kidney . so if you add those two things up , you end up with having more blood flow to the kidney . and what that means is , you end up with an increase in the rate of filtration of blood through the kidney . it 's kind of like , you know those water filters that you can attach directly to your faucet ? right ? imagine if you had one of those . and if your pipes and your faucet got much larger , they got much wider , and there 's more water running through the pipes , the rate at which the water was being filtered through the water filter would increase drastically . well , this is the exact same thing . the kidney is just like your water filter in that it filters all of your blood . now , with regards to the bladder , there is a contentious topic of whether the bladder holds more or less urine in a pregnant woman . there 's some thought that progesterone , which , remember , causes relaxation of smooth muscle , relaxes and increases the capacity of the bladder . and then there 's other thought that the pressure of the uterus , the large uterus on the bladder , decreases the capacity of the bladder . so we 're not entirely sure . but one thing is certain , and that is that pregnant women definitely urinate more frequently than normal . and that has to do with increased urine production , as well as that pressure on the bladder from the uterus . and that pressure from the uterus also leads to dilatation of the ureter . so the ureters become dilated . and that 's really important . and it kind of looks like this , where the pressure from the larger uterus causes the ureters to become wider , to become dilated . and that uterus putting pressure here sort of acts as a road block . and urine builds up in the ureters behind that road block . that built-up stagnant urine acts as a medium for bacterial growth , right ? because we know urinary stasis is a risk factor for bacterial growth . and that 's perhaps why pregnant women are more susceptible to developing pyelonephritis , or infection of the kidney , than are non-pregnant women . it 's because of that urinary stasis that occurs as a result of the large uterus putting pressure on the ureters . all right . so those are some of the physiologic changes that occur in pregnancy , with the cardiovascular system , the respiratory system , and the renal system .
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and that includes the smooth muscle that surrounds all of the blood vessels . so that relaxation causes dilation of the blood vessels , which then lowers the blood pressure . so that 's a first cause , sort of , of the decrease in blood pressure through pregnancy .
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heart beat rate is inversely proportional to blood pressure , is this law not applicable to all normal situations ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative .
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would n't the derivative of a^x be just x ( a^ ( x-1 ) ) according to the product rule ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function .
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how come when using the chain rule and taking the derivative ( ( ln a ) * x ) = ln a ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative .
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where does the x go ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a .
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0 would n't derivative of 8 equal to 0 ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative .
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now , what power does 2 need to be to raised to get 2 ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative .
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should n't we use the power rule ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative .
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why did n't he use the product rule to get the result of [ 8 * 3^x ] = -ln ( x ) * 3^x * 8/3^2x ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function .
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how come when using the chain rule and taking the derivative ( ln a ) * x we get ln ( a ) ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative .
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just like x. would n't we use the product rule here , or at least convert ln ( a ) to 1/a ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing .
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how can ln ( a ) be treated as a constant ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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so really think about this . do n't just accept this as a leap of faith . it should make sense to you .
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is n't differentiation of ln ( a ) =1/ ?
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what i want to do in this video is explore taking the derivatives of exponential functions . so we 've already seen that the derivative with respect to x of e to the x is equal to e to x , which is a pretty amazing thing . one of the many things that makes e somewhat special . though when you have an exponential with your base right over here as e , the derivative of it , the slope at any point , is equal to the value of that actual function . but now let 's start exploring when we have other bases . can we somehow figure out what is the derivative , what is the derivative with respect to x when we have a to the x , where a could be any number ? is there some way to figure this out ? and maybe using our knowledge that the derivative of e to the x , is e to the x . well can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base ? well , you could view a , you could view a as being equal to e. let me write it this way . well all right , a as being equal to e to the natural log of a . now if this is n't obvious to you , i really want you to think about it . what is the natural log of a ? the natural log of a is the power you need to raise e to , to get to a . so if you actually raise e to that power , if you raise e to the power you need to raise e too to get to a . well then you 're just going to get to a . so really think about this . do n't just accept this as a leap of faith . it should make sense to you . and it just comes out of really what a logarithm is . and so we can replace a with this whole expression here . if a is the same thing as e to the natural log of a , well then this is going to be , then this is going to be equal to the derivative with respect to x of e to the natural log , i keep writing la ( laughs ) , to the natural log of a and then we 're going to raise that to the xth power . we 're going to raise that to the x power . and now this , just using our exponent properties , this is going to be equal to the derivative with respect to x of , and i 'll just keep color-coding it . if i raise something to an exponent and then raise that to an exponent , that 's the same thing as raising our original base to the product of those exponents . that 's just a basic exponent property . so that 's going to be the same thing as e to the natural log of a , natural log of a times x power . times x power . and now we can use the chain rule to evaluate this derivative . so what we will do is we will first take the derivative of the outside function . so e to the natural log of a times x with respect to the inside function , with respect to natural log of a times x . and so , this is going to be equal to e to the natural log of a times x . and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative . if it was the derivative of three x , it would just be three . if it 's the derivative of natural log a times x , it 's just going to be natural log of a . and so this is going to give us the natural log of a times e to the natural log of a . and i 'm going to write it like this . natural log of a to the x power . well we 've already seen this . this right over here is just a . so it all simplifies . it all simplifies to the natural log of a times a to the x , which is a pretty neat result . so if you 're taking the derivative of e to the x , it 's just going to be e to the x . if you 're taking the derivative of a to the x , it 's just going to be the natural log of a times a to the x . and so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. so if i want to find the derivative with respect to x of eight times three to the x power , well what 's that going to be ? well that 's just going to be eight times and then the derivative of this right over here is going to be , based on what we just saw , it 's going to be the natural log of our base , natural log of three times three to the x . times three to the x . so it 's equal to eight natural log of three times three to the x . times three to the x power .
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and then we take the derivative of that inside function with respect to x . well natural log of a , it might not immediately jump out to you , but that 's just going to be a number . so that 's just going to be , so times the derivative .
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is `` a '' should be an real number ?
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture . so , i 'm gon na choose six . this is fun , lem me keep going . how many wheels do i see in the picture ? one , two , three , four , five , six , seven , eight . wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels . these four cars have 16 wheels . so , 16 wheels , let 's keep going . how many people are in the boat ? i see one , two , and three . there 's the guy rowing it , there 's i guess the mother , and then there 's the baby . there 's three people in this boat . three people in the boat . this is too much fun , how many faces do i see ? one , two , three , four five , six , seven , eight , nine , 10 faces . keep going . how many bottles do i see ? one , two , three , four bottles . and we 're done .
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one , two , three , four , five , six , seven , eight . wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels .
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what will the solar system be like if we did n't have a jupiter ?
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture . so , i 'm gon na choose six . this is fun , lem me keep going . how many wheels do i see in the picture ? one , two , three , four , five , six , seven , eight . wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels . these four cars have 16 wheels . so , 16 wheels , let 's keep going . how many people are in the boat ? i see one , two , and three . there 's the guy rowing it , there 's i guess the mother , and then there 's the baby . there 's three people in this boat . three people in the boat . this is too much fun , how many faces do i see ? one , two , three , four five , six , seven , eight , nine , 10 faces . keep going . how many bottles do i see ? one , two , three , four bottles . and we 're done .
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture .
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how do we pronounce numbers more than 1000 ?
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture . so , i 'm gon na choose six . this is fun , lem me keep going . how many wheels do i see in the picture ? one , two , three , four , five , six , seven , eight . wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels . these four cars have 16 wheels . so , 16 wheels , let 's keep going . how many people are in the boat ? i see one , two , and three . there 's the guy rowing it , there 's i guess the mother , and then there 's the baby . there 's three people in this boat . three people in the boat . this is too much fun , how many faces do i see ? one , two , three , four five , six , seven , eight , nine , 10 faces . keep going . how many bottles do i see ? one , two , three , four bottles . and we 're done .
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture .
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what is an easy way of carrying decimals in my head ?
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture . so , i 'm gon na choose six . this is fun , lem me keep going . how many wheels do i see in the picture ? one , two , three , four , five , six , seven , eight . wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels . these four cars have 16 wheels . so , 16 wheels , let 's keep going . how many people are in the boat ? i see one , two , and three . there 's the guy rowing it , there 's i guess the mother , and then there 's the baby . there 's three people in this boat . three people in the boat . this is too much fun , how many faces do i see ? one , two , three , four five , six , seven , eight , nine , 10 faces . keep going . how many bottles do i see ? one , two , three , four bottles . and we 're done .
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wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels . these four cars have 16 wheels .
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why did steering wheels not count as wheels ?
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how many people do you see in this picture ? let 's see , i see one , two , three , four , five , six people in this picture . so , i 'm gon na choose six . this is fun , lem me keep going . how many wheels do i see in the picture ? one , two , three , four , five , six , seven , eight . wan na scroll down and make sure i do n't miss these over here , nine , 10 , 11 , 12 , 13 , 14 , 15 , 16 wheels . there are 16 wheels . these four cars have 16 wheels . so , 16 wheels , let 's keep going . how many people are in the boat ? i see one , two , and three . there 's the guy rowing it , there 's i guess the mother , and then there 's the baby . there 's three people in this boat . three people in the boat . this is too much fun , how many faces do i see ? one , two , three , four five , six , seven , eight , nine , 10 faces . keep going . how many bottles do i see ? one , two , three , four bottles . and we 're done .
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this is fun , lem me keep going . how many wheels do i see in the picture ? one , two , three , four , five , six , seven , eight .
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what happens if the picture is blurry and you can not count ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't .
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at the very end , when the compound is finally named , should the prefix be 2s instead of s , because we must specify the chiral center ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four .
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how does the molecule go from sticking out of the page to going into the page , while the others just move from the plane of the page to sticking out ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon .
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what 's the actual difference between chiral molecules or chirality and enantiomers ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done .
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how do you decide on the r/s configuration when given a flat skeletal or lewis structure that does n't have the substituents pre-oriented in 3d like the example in this video ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on .
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when substituents are in a tie while we are ranking them , you look at the highest atom attached to each , but how do double or triple bonds get counted in the ranking process ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system .
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is it possible for a molecule to have multiple chiral centres ?
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let 's see if we can name this molecule using the -- sometimes called the r-s system , or the cahn-ingold-prelog system . and the first thing to do is just to see if there are any chiral centers in this molecule . if there are n't , then we do n't even have to use the r-s system . we can just use our standard nomenclature rules and we 'd be done . so if we look here , this carbon is attached to three hydrogens , so it 's definitely not attached to four different groups . same thing about this carbon right here . this carbon right here is attached to a fluorine , but then it 's attached to two methyl groups . so it 's the same group , so this is also not a chiral carbon or an asymmetric carbon . this carbon right here is attached to a hydrogen and three other carbons , but each of these three carbons look like different groups . this carbon is attached to two methyls and a fluorine . this carbon is attached to two hydrogens and a bromine . this carbon is just a methyl group . so this right here does look like a chiral center . it does look like a chiral carbon , and the other ones do n't . this is just a methyl group . it has three hydrogens , so definitely not attached to four different groups . and this is attached to two hydrogens , and those are obviously the same group , so this is also not a chiral center . so we have one chiral center , so the r-s naming system will apply . but a good starting point will just be naming it using our standard nomenclature rules . and to do that we look for the longest carbon chain here . let 's see , if we start over here , and i do n't know what direction i 'm going to name it from yet , but i just want to identify the longest chain . if we went from here , we have one , two , three . we can either go to four or to four there , so we definitely have four carbons , four carbon , longest chain . so that tells us that we will be using the prefix but- , or it will be a butane , because they 're all single bonds here , so it is a butane . but to decide whether we branch off , it does n't matter whether we use this ch3 or this ch3 , they 're the same group . but to decide whether we use this part of the longest chain or we use that , we think about the rule that the core chain to use should have as many simple groups attached to it as possible , as opposed to as few complex groups . so if we used this carbon as part of our longest chain , then this will be a group that 's attached to it , which would be a bromomethyl group , which is not as simple as maybe it could be . but if we use this carbon in our longest chain , we 'll have two groups . we 'll have a bromo attached , and we 'll also have a methyl group . and that 's what we want . we want more simple groups attached to the longest chain . so what we 're going to do is we 're going to use this carbon , this carbon , this carbon , and that carbon as our longest chain . and we want to start from the end that is closest to something being attached to it , and that bromine is right there . so there 's going to be our number one carbon , our number two carbon , our number three carbon , and our number four carbon . and then we can label the different groups and then figure out what order they should be listed in . so this is a 1-bromo and then this will be a 2-methyl right here . and then just a hydrogen . then three we have a fluoro , so on a carbon three , we have a fluoro , and then on carbon three , we also have a methyl group right here , so we also have a 3-methyl . so when we name it , we put in alphabetical order . bromo comes first , so this thing right here is 1-bromo . then alphabetically , fluoro comes next , 1-bromo-3-fluoro . we have two methyls , so it 's going to be 2 comma 3-dimethyl . and remember , the d does n't count in alphabetical order . dimethylbutane , because we have the longest chain is four carbons . dimethylbutane . so that 's just the standard nomenclature rules . we still have n't used the r-s system . now we can do that . now to think about that , we already said that this is our chiral center , so we just have to essentially rank the groups attached to it in order of atomic number and then use the cahn-ingold-prelog rules , and we 'll do all that in this example . so let 's look at the different groups attached to it . so when you look at it , this guy has three carbons and a hydrogen . carbon is definitely higher in atomic number on the periodic table . it has an atomic number 6 . hydrogen is 1 . you probably know that already . so hydrogen is definitely going to be number four . so let me put number four there next to the hydrogen . and let me find a nice color . i 'll do it in white . so hydrogen is definitely the number four group . we have to differentiate between this carbon group , that carbon group , and that carbon group . and the way you do it , if there 's a tie on the three carbons , you then look at what is attached to those carbons , and you compare the highest thing attached to each of those carbons to the highest things attached to the other carbons , and then you do the same ranking . and if that 's a tie , then you keep going on and on and on . so on this carbon right here , we have a bromine . bromine has an atomic number of 35 , which is higher than carbon . so this guy has a bromine attached to it . this guy only has hydrogen attached to it . this guy has a fluorine attached to it . that 's the highest thing . so this is going to be the third lowest , or i should say the second to lowest , because it only has hydrogens attached to it , so that is number three . the one has the bromine attached to it is going to be number one , and the one that has the fluorine attached to it is number two . and just a reminder , we were tied with the carbon , so we have to look at the next highest constituent , and even if this had three fluorines attached to it , the bromine would still trump it . you compare the highest to the highest . so now that we 've done that , let me redraw this molecule so it 's a little bit easier to visualize . so i 'll draw our chiral carbon in the middle . and i 'm just doing this for visualization purposes . and right here we have our number one group . i 'll literally just call that our number one group . so right there that is our number one group . it 's in the plane of the screen . so i 'll just call that our number one group . over here , also in the plane of the screen , i have our number two group . so let me do it like that . so then you have your number two group , just like that . and then you have your number three group behind the molecule right now the way it 's drawn . i 'll do that in magenta . so then you have your number three group . it 's behind the molecule , so i 'll draw it like this . this is our number three group . and then we have our number four group , which is the hydrogen pointing out right now . and i 'll just do that in a yellow . we have our number four group pointing out in front right now . so that is number four , just like that . actually , let me draw it a little bit clearer , so it looks a little bit more like the tripod structure that it 's supposed to be . so let me redraw the number three group . the number three group should look like -- so this is our number three group . let me draw it a little bit more like this . the number three group is behind us . and then finally , you have your number four group in yellow , which is just a hydrogen that 's coming straight out . so that is coming straight out of -- well , not straight out , but at an angle out of the page . so that 's our number four group , i 'll just label it number four . it really is just a hydrogen , so i really did n't have to simplify it much there . now by the r-s system , or by the cahn-ingold-prelog system , we want our number four group to be the one furthest back . so we really want it where the number three position is . and so the easiest way i can think of doing that is you can imagine this is a tripod that 's leaning upside down . or another way to view it is you can view it as an umbrella , where this is the handle of the umbrella and that 's the top of the umbrella that would block the rain , i guess . but the easiest way to get the number four group that 's actually a hydrogen in the number three position would be to rotate it . you could imagine , rotate it around the axis defined by the number one group . so the number one group is just going to stay where it is . the number four is going to rotate to the number three group . number three is going to rotate around to the number two group , and then the number two group is going to rotate to where the number four group is right now . so if we were to redraw that , let 's redraw our chiral carbon . so let me scroll over a little bit . so we have our chiral carbon . i put the little asterisk there to say that that 's our chiral carbon . the number four group is now behind . i 'll do it with the circles . it makes it look a little bit more like atoms . so the number four group is now behind where the number three group used to be , so number four is now there . number one has n't changed . that 's kind of the axis that we rotated around . so the number one group has not changed . number one is still there . number two is now where number four used to be , so number two is now jutting out of the page . and then we have number three is now where number two was . so number three is there . and now that we 've put our fourth group behind the molecule , we literally just figure out whether we have to go clockwise or counterclockwise to go from one , two to three . and that 's pretty straightforward . to go from one to two to three , we have to go counterclockwise . or another way to think of it , we 're going to the left , counterclockwise . at least on the top of the clock , we 're going to the left . and so , since we 're going to left , this is s or sinister . this is s , which stands for sinister , which is latin for left . so we 're done . we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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we 've named it using the r-s system . this molecule is ( s ) -- sinister -- 1-bromo-3-fluoro-2,3-di --
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once you have determined that the chiral center for c2 is s , to fully name the molecule could it be correct to say ( 2s ) 1-bromo-3-fluoro-2,3-dimethylbutane ?
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