Bluel0la/Set_1 · Datasets at Hugging Face

Instruction stringclasses 5 values	Input stringlengths 43 704	response stringlengths 1 1.77k
Provide a detailed solution to the given mathematical problem.	find the total number of prime factors in the expression ( 4 ) ^ 11 x ( 7 ) ^ 5 x ( 11 ) ^ 3 a ) 26 , b ) 22 , c ) 25 , d ) 30 , e ) 29	"( 4 ) ^ 11 x ( 7 ) ^ 5 x ( 11 ) ^ 3 = ( 2 x 2 ) ^ 11 x ( 7 ) ^ 5 x ( 11 ) ^ 3 = 2 ^ 11 x 2 ^ 11 x 7 ^ 5 x 11 ^ 3 = 2 ^ 22 x 7 ^ 5 x 11 ^ 3 total number of prime factors = ( 22 + 5 + 3 ) = 30 . answer is d ."
Explain the solution process for the given equation.	if the sum of 3 consecutive even numbers is 44 more than the average of these numbers , then the largest of these numbers is ? a ) 24 , b ) 66 , c ) 9 , d ) 7 , e ) 01	answer : 24 explanation : let the smallest of these number be x . the other two numbers are ( x + 2 ) and ( x + 4 ) . x + ( x + 2 ) + ( x + 4 ) = ( x + ( x + 2 ) + ( x + 4 ) ) / 3 + 44 3 x + 3 * ( x + 2 ) + 3 * ( x + 4 ) = x + ( x + 2 ) + ( x + 4 ) + 132 9 x + 18 = 3 x + 138 6 x = 120 x = 20 therefore , the largest number is 24 . answer : a
Provide a detailed solution to the given mathematical problem.	if [ [ x ] ] = x ^ 2 + 2 x + 4 , what is the value of [ [ 5 ] ] ? a ) 39 , b ) 9 , c ) 15 , d ) 19 , e ) 25	"these functions questions might look intimidating , but they just test your knowledge about how well you can substitute values [ [ x ] ] = x ^ 2 + 2 x + 4 [ [ 5 ] ] = 5 ^ 2 + 2 * 5 + 4 = 39 . option a"
Explain the solution process for the given equation.	a professional athlete was offered a 3 - year contract to play with team k that provided for an annual salary of $ 100000 in the first year , an increase in annual salary of 20 % over the previous year for the next two years , and a bonus of $ 50000 on signing . team l offered a 3 - year contract providing for an annual salary of $ 150000 in the first year , an increase in annual salary of 20 % over the previous year for the next two years , and no signing bonus . if he accepts the offer of team l and fulfills the 3 - year contract terms , the athlete will receive how much more money by choosing team l over team k ? a ) $ 132,000 , b ) $ 50,000 , c ) $ 82,500 , d ) $ 92,000 , e ) $ 100,000	team k' s contract = $ 100,000 + $ 100,000 * 1.2 + $ 100,000 * 1.2 * 1.2 + $ 50,000 = $ 414,000 team l' s contract = $ 150,000 + $ 150,000 * 1.2 + $ 150,000 * 1.2 * 1.2 = $ 546,000 the difference = $ 132,000 . answer : a .
Provide a detailed solution to the given mathematical problem.	a , b , k start from the same place and travel in the same direction at speeds of 30 km / hr , 40 km / hr , 60 km / hr respectively . b starts six hours after a . if b and k overtake a at the same instant , how many hours after a did k start ? a ) 3 , b ) 4.5 , c ) 4 , d ) d ) 12 , e ) e ) 5	"the table you made does n' t make sense to me . all three meet at the same point means the distance they cover is the same . we know their rates are 30 , 40 and 60 . say the time taken by b is t hrs . then a takes 6 + t hrs . and we need to find the time taken by k . distance covered by a = distance covered by b 30 * ( 6 + t ) = 40 * t t = 18 hrs distance covered by b = distance covered by k 40 * t = 60 * time taken by k time taken by k = 40 * 18 / 60 = 12 hrs time taken by a = 6 + t = 6 + 18 = 24 hrs time taken by k = 12 hrs so k starts 24 - 12 = 12 hrs after a . ( answer d )"
Solve the problem systematically, showing each step clearly.	if a 2 cm cube is cut into 1 cm cubes , then what is the percentage increase in the surface area of the resulting cubes ? a ) 50 % , b ) 100 % , c ) 150 % , d ) 200 % , e ) 250 %	"the area a of the large cube is 2 * 2 * 6 = 24 square cm . the area of the 8 small cubes is 8 * 6 = 48 = 2 a , an increase of 100 % . the answer is b ."
Solve the problem systematically, showing each step clearly.	carina has 115 ounces of coffee divided into 5 - and 10 - ounce packages . if she has 2 more 5 - ounce packages than 10 - ounce packages , how many 10 - ounce packages does she have ? a ) 5 , b ) 7 , c ) 6 , d ) 4 , e ) 3	"lets say 5 and 10 ounce packages be x and y respectively . given that , 5 x + 10 y = 115 and x = y + 2 . what is the value of y . substituting the x in first equation , 5 y + 10 + 10 y = 85 - > y = 105 / 15 . = 7 b"
Work through the equation methodically, ensuring clarity at each step.	the sides of the triangle are in the ratio 1 / 2 : 1 / 3 : 1 / 4 and its perimeter is 104 cm . the length of the longest side is ? ['a ) 32 cm','b ) 48 cm','c ) 50 cm','d ) 28 cm','e ) 30 cm']	ratio of sides = 1 / 2 : 1 / 3 : 1 / 4 = 6 : 4 : 3 largest side = 104 * 6 / 13 = 48 cm answer is b
Explain the solution process for the given equation.	find a two digit number , given that the sum of the digits is 13 and the difference of the digits is 5 . ? a ) 74 , b ) 82 , c ) 95 , d ) 76 , e ) 94	"using elimination method find which of the options fit the description of the number . . . from the option only 94 meets this description sum of digits - - - 9 + 4 = 13 difference of digits - - - 9 - 4 = 5 answer e ."
Provide a detailed solution to the given mathematical problem.	a group of 55 adults and 70 children go for trekking . if there is meal for either 70 adults or 90 children and if 14 adults have their meal , find the total number of children that can be catered with the remaining food . a ) 33 , b ) 54 , c ) 18 , d ) 17 , e ) 72	"explanation : as there is meal for 70 adults and 14 have their meal , the meal left can be catered to 56 adults . now , 70 adults = 90 children 7 adults = 9 children therefore , 56 adults = 72 children hence , the meal can be catered to 72 children . answer : e"
Solve the problem systematically, showing each step clearly.	in 1990 the budgets for projects q and v were $ 700,000 and $ 780,000 , respectively . in each of the next 10 years , the budget for q was increased by $ 30,000 and the budget for v was decreased by $ 10,000 . in which year was the budget for q equal to the budget for v ? a ) 1992 , b ) 1993 , c ) 1994 , d ) 1995 , e ) 1996	"let the no of years it takes is x . 700 + 30 x = 780 - 10 x - - > 40 x = 80 and x = 2 . thus , it happens in 1992 . a ."
Provide a detailed solution to the given mathematical problem.	a squirrel runs up a cylindrical post , in a perfect spiral path making one circuit for each rise of 4 feet . how many feet does the squirrel travels if the post is 16 feet tall and 2 feet in circumference ? a ) 10 feet , b ) 8 feet , c ) 13 feet , d ) 15 feet , e ) 18 feet	"total circuit = 16 / 4 = 4 total feet squirrel travels = 4 * 2 = 8 feet answer : b"
Work through the equation methodically, ensuring clarity at each step.	a man can do a job in 20 days . his father takes 20 days and his son finishes it in 25 days . how long will they take to complete the job if they all work together ? a ) 6.3 , b ) 6.9 , c ) 7.1 , d ) 6.1 , e ) 6.2	"1 day work of the three persons = ( 1 / 20 + 1 / 20 + 1 / 25 ) = 7 / 50 so , all three together will complete the work in 300 / 47 = 7.1 days . answer : c"
Break down and solve the equation step by step.	the h . c . f . of two numbers is 23 and the other two factors of their l . c . m . are 13 and 16 . the larger of the two numbers is : a ) 276 , b ) 299 , c ) 322 , d ) 345 , e ) 368	"clearly , the numbers are ( 23 x 13 ) and ( 23 x 16 ) . larger number = ( 23 x 16 ) = 368 . answer : option e"
Solve the problem systematically, showing each step clearly.	if k ^ 3 is divisible by 162 , what is the least possible value of integer k ? a ) 12 , b ) 15 , c ) 18 , d ) 24 , e ) 36	"k ^ 3 = 162 * x = 2 * 3 ^ 4 * x the factors of k must at minimum include 2 * 3 ^ 2 = 18 . the answer is c ."
Break down and solve the equation step by step.	if x < y < z and y - x > 5 , where x is an even integer and y and z are odd integers , what is the least possible value f of z - x ? a ) 6 , b ) 7 , c ) 8 , d ) 9 , e ) 10	"x < y < z to find the least possible value for z - x ; we need to find the values for z and x that can be closest to each other . if x is some even number , then what could be minimum possible odd z . if x is some even number y - x > 5 ; y > x + 5 ; minimum value for y = x + 5 + 2 = x + 7 [ note : x + 5 is as even + odd = odd and nearest odd greater than x + 5 is x + 5 + 2 ] minimum value for z = y + 2 = x + 7 + 2 = x + 9 [ note : z = y + 2 because both z and y are odd . difference between two odd numbers is 2 ] f = z - x = x + 9 - x = 9 ans : d"
Solve the problem systematically, showing each step clearly.	there are 20 players in a chess group , and each player plays each of the others once . given that each game is played by two players , how many total games will be played ? a ) 10 , b ) 30 , c ) 190 , d ) 60 , e ) 90	"20 players are there . two players play one game with one another . so 20 c 2 = 20 * 19 / 2 = 190 so option c is correct"
Solve the problem systematically, showing each step clearly.	john traveled 80 % of the way from yellow - town to green - fields by train at an average speed of 80 miles per hour . the rest of the way john traveled by car at an average speed of v miles per hour . if the average speed for the entire trip was 60 miles per hour , what is v in miles per hour ? a ) 30 , b ) 40 , c ) 50 , d ) 55 , e ) 70	"let distance be d . we can find the total time & equate it , which comes as : 0.8 d / 80 + 0.2 d / v = d / 60 = > v = 30 answer : a"
Provide a detailed solution to the given mathematical problem.	a 600 meter long train crosses a signal post in 40 seconds . how long will it take to cross a 1.8 kilometer long bridge , at the same speed ? a ) 4 min , b ) 2 min , c ) 8 min , d ) 9 min , e ) 3 min	s = 600 / 40 = 15 mps s = 1800 / 15 = 120 sec = 2 min . answer : b
Work through the equation methodically, ensuring clarity at each step.	at a particular graduation party with 300 guests , 70 % of the guests brought gifts , and 40 % of the female guests brought gifts . if 30 males did not bring gifts to the party , how many females did bring gifts ? a ) 18 , b ) 28 , c ) 40 , d ) 68 , e ) 70	"the correct method total = 300 . . 70 % of 300 = 210 got gifts . . 90 did not get gift , out of which 30 are males , so remaining 90 - 30 = 60 are females . . but 40 % females brought gift , so 60 % did not get it . . so 60 % = 60 , 100 % = 60 * 100 / 60 = 100 . ans 40 % of 100 = 40 c"
Provide a detailed solution to the given mathematical problem.	linda spent 3 / 4 of her savings on furniture and the rest on a tv . if the tv cost her $ 230 , what were her original savings ? a ) $ 900 , b ) $ 800 , c ) $ 1000 , d ) $ 920 , e ) $ 1500	"if linda spent 3 / 4 of her savings on furnitute , the rest 4 / 4 - 3 / 4 = 1 / 4 on a tv but the tv cost her $ 230 . so 1 / 4 of her savings is $ 230 . so her original savings are 4 times $ 230 = $ 920 correct answer d"
Provide a detailed solution to the given mathematical problem.	if rs . 7500 are borrowed at c . i at the rate of 4 % per annum , then after 2 years the amount to be paid is ? a ) rs . 8082 , b ) rs . 7800 , c ) rs . 8100 , d ) rs . 8112 , e ) rs . 9000	"explanation : a = 7500 ( 26 / 25 ) 2 = 8112 answer is d"
Explain the solution process for the given equation.	if the ratio of the sum of the first 6 terms of a g . p . to the sum of the first 3 terms of the g . p . is 28 , what is the common ratio of the g . p ? a ) 3 , b ) 1 / 3 , c ) 2 , d ) 9 , e ) 1 / 9	"28 = ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 ) / ( a 1 + a 2 + a 3 ) factorize the same terms 28 = 1 + ( a 4 + a 5 + a 6 ) / ( a 1 + a 2 + a 3 ) write every term with respect to r a 1 = a 1 a 2 = a 1 * r ^ 1 a 3 = a 1 * r ^ 2 . . . . . . . . . 28 = 1 + ( a 1 ( r ^ 3 + r ^ 4 + r ^ 5 ) ) / ( a 1 ( 1 + r ^ 1 + r ^ 2 ) ) 27 = ( r ^ 3 ( 1 + r ^ 1 + r ^ 2 ) ) / ( ( 1 + r ^ 1 + r ^ 2 ) ) 27 = r ^ 3 r = 3 a"
Break down and solve the equation step by step.	a watch was sold at a loss of 10 % . if it was sold for rs . 540 more , there would have been a gain of 8 % . what is the cost price ? a ) 1000 , b ) 2998 , c ) 3000 , d ) 2788 , e ) 2991	"90 % 108 % - - - - - - - - 18 % - - - - 540 100 % - - - - ? = > rs . 3000 answer : c"
Provide a detailed solution to the given mathematical problem.	calculate how much time could be saved if the train would run at its usual speed , given that it ran at 5 / 7 of its own speed and got to its destination in 15 hours ? a ) 1.29 hrs , b ) 2.29 hrs , c ) 4.29 hrs , d ) 5.29 hrs , e ) 3.29 hrs	"new speed = 5 / 7 of usual speed new time = 5 / 7 of usual time 5 / 7 of usual time = 15 hrs usual time = 15 * 5 / 7 = 10.71 hrs time saved = 15 - 10.71 = 4.29 hrs answer is c"
Provide a detailed solution to the given mathematical problem.	convert 0.35 in to a vulgar fraction ? a ) 18 / 50 , b ) 16 / 50 , c ) 7 / 20 , d ) 19 / 50 , e ) none	"answer 0.35 = 35 / 100 = 7 / 20 correct option : c"
Provide a detailed solution to the given mathematical problem.	5 n + 2 > 17 and 7 n - 5 < 44 ; n must be between which numbers ? a ) 1 and 8 , b ) 2 and 6 , c ) 0 and 9 , d ) 3 and 7 , e ) 2 and 9	"5 n > 15 - - > n > 3 7 n < 49 - - > n < 7 3 < n < 7 answer : d"
Work through the equation methodically, ensuring clarity at each step.	the grade point average of one third of the classroom is 54 ; the grade point average of the rest is 45 . what is the grade point average of the whole class ? a ) 52 , b ) 54 , c ) 48 , d ) 58 , e ) 60	"let n = total students in class total points for 1 / 3 class = 54 n / 3 = 18 n total points for 2 / 3 class = 45 * 2 n / 3 = 30 n total points for whole class = 18 n + 30 n = 48 n 48 n total class points / n total students = 48 grade point average for total class answer : c"
Provide a detailed solution to the given mathematical problem.	in the fifth grade at parkway elementary school there are 420 students . 312 students are boys and 250 students are playing soccer . 86 % of the students that play soccer are boys . how many girl student are in parkway that is not playing soccer ? a ) 69 . , b ) 73 . , c ) 81 , d ) 91 , e ) 108	total students = 420 boys = 312 , girls = 108 total playing soccer = 250 86 % of 250 = 215 are boys who play soccer . girls who play soccer = 35 . total girls who do not play soccer = 108 - 35 = 73 . correct option : b
Work through the equation methodically, ensuring clarity at each step.	from january 1 , 1991 , to january 1 , 1993 , the number of people enrolled in health maintenance organizations increased by 15 percent . the enrollment on january 1 , 1993 , was 60 million . how many million people , to the nearest million , were enrolled in health maintenance organizations on january 1 , 1991 ? a ) 38 , b ) 39 , c ) 40 , d ) 41 , e ) 52	"1.15 x = 60 - - > 23 / 20 * x = 60 - - > x = 60 * 20 / 23 = 1200 / 23 = ~ 52 . answer : e ."
Explain the solution process for the given equation.	a box contains 6 pairs of shoes ( 12 shoes in total ) . if two shoes are selected at random , what it is the probability that they are matching shoes ? a ) 1 / 11 , b ) 1 / 20 , c ) 1 / 19 , d ) 1 / 10 , e ) 1 / 9	the problem with your solution is that we do n' t choose 1 shoe from 12 , but rather choose the needed one after we just took one and need the second to be the pair of it . so , the probability would simply be : 1 / 1 * 1 / 11 as after taking one at random there are 11 shoes left and only one is the pair of the first one ) = 1 / 11 answer : a .
Solve the problem systematically, showing each step clearly.	the average of first 25 prime numbers is ? a ) 9.9 , b ) 8.8 , c ) 5.6 , d ) 11.9 , e ) 11.11	"explanation : average = ( 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 / 9 = 100 / 9 = 11.11 ( approx ) answer is e"
Work through the equation methodically, ensuring clarity at each step.	of 60 children , 30 are happy , 10 are sad , and 20 are neither happy nor sad . there are 17 boys and 43 girls . if there are 6 happy boys and 4 sad girls , how many boys are neither happy nor sad ? a ) 5 , b ) 4 , c ) 6 , d ) 8 , e ) 10	"venn diagrams are useful for multiple values of a single variable e . g . state of mind - happy / sad / neither . when you have two or more variables such as here where you have gender - boy / girl too , it becomes unwieldy . in this case , either use the table or logic . table method is shown above ; here is how you will use logic : there are 6 happy boys . there are 4 sad girls but total 10 sad children . so rest 6 sad children must be sad boys . we have 6 happy boys and 6 sad boys . total we have 17 boys . so 17 - 6 - 6 = 5 boys must be neither happy nor sad . answer ( a )"
Work through the equation methodically, ensuring clarity at each step.	a man sells an article at a profit of 25 % . if he had bought it at 20 % less and sold it for rs . 10.50 less , he would have gained 30 % . find the cost of the article . a ) 30 , b ) 40 , c ) 50 , d ) 60 , e ) 70	let c . p = 100 gain = 25 % s . p = 125 supposed c . p = 80 gain = 30 % s . p = ( 130 * 80 ) / 100 = 104 diff = ( 125 - 104 ) = 21 diff 21 when c . p = 100 then diff 10.50 when c . p = ( 100 * 10.50 ) / 21 = 50 answer : c
Break down and solve the equation step by step.	what is the least possible value of expression e = ( x - 1 ) ( x - 3 ) ( x - 4 ) ( x - 6 ) + 10 for real values of x ? a ) 1 , b ) 10 , c ) 9 , d ) 0 , e ) 8	explanation : e = ( x - 1 ) ( x - 6 ) ( x - 3 ) ( x - 4 ) + 10 e = ( x 2 - 7 x + 6 ) ( x 2 - 7 x + 12 ) + 10 let x 2 - 7 x + 6 = y e = y 2 + 6 y + 10 e = ( y + 3 ) 2 + 1 minimum value = 1 , when y = - 3 answer : a
Provide a detailed solution to the given mathematical problem.	in a rectangular coordinate system , what is the area of a rhombus whose vertices have the coordinates ( 0 , 5.5 ) , ( 8 , 0 ) , ( 0 , - 5.5 ) , ( - 8 , 0 ) ? ['a ) 56','b ) 88','c ) 112','d ) 116','e ) 120']	ares of rhombus = 1 / 2 * d 1 * d 2 length of 1 st diagonal = 8 + 8 = 16 length of 2 nd diagonal = 5.5 + 5.5 = 11 area = 1 / 2 * 16 * 11 = 88 b is the answer
Solve the problem systematically, showing each step clearly.	a train moves at average speed of 40 kmph reaches its destination on time . what its average speed becomes 35 kmph , then it reaches its destination 15 late . find the length of the journey ? a ) 70 km , b ) 85 km , c ) 38 km , d ) 14 km , e ) 1 km	difference between timings = 15 mis = 1 / 4 hr let the length of the journey be x km . then x / 35 - x / 40 = 1 / 4 8 x - 7 x = 70 x = 70 km answer ( a )
Break down and solve the equation step by step.	1370 , 1320 , 1070 , - 180 ? a ) - 6430 , b ) - 6530 , c ) - 6630 , d ) - 6730 , e ) - 6830	1370 , 13201070 , - 180 , ? 1370 - 50 * ( 5 ^ 0 ) = 1320 1320 - 50 * ( 5 ^ 1 ) = 1070 1070 - 50 * ( 5 ^ 2 ) = - 180 - 180 - 50 * ( 5 ^ 3 ) = - 6430 answer : a .
Work through the equation methodically, ensuring clarity at each step.	if x is less than y by 45 % then y exceed x by : a ) 33.33 % , b ) 25 % , c ) 81.81 % , d ) 66.66 % , e ) none of these	"using formula ( x / ( 100 - x ) * 100 ) where x is percentage decrease ( here it is 25 % ) = > 45 / ( 100 - 45 ) * 100 = 81.81 % answer : c"
Work through the equation methodically, ensuring clarity at each step.	the ratio between the present ages of a and b is 4.5 : 3 respectively . the ratio between a' s age 4 years ago and b' s age 4 years hence is 1 : 1 . what is the ratio between a' s age 4 years hence and b' s age 4 years ago ? a ) 3 : 4 , b ) 3 : 0 , c ) 3 : 1 , d ) 7 : 3 , e ) 3 : 7	"let the present ages of a and b be 4.5 x and 3 x years respectively . then , ( 4.5 x - 4 ) / ( 3 x + 4 ) = 1 / 1 1.5 x = 8 = > x = 5.33 required ratio = ( 5 x + 4 ) : ( 3 x - 4 ) = 28 : 12 = 7 : 3 . answer : d"
Break down and solve the equation step by step.	two pipes a and b can fill a tank in 30 and 60 minutes respectively . if both the pipes are used together , then how long will it take to fill the tank ? a ) 10 , b ) 20 , c ) 30 , d ) 24 , e ) 25	"part filled by a in 1 min . = 1 / 30 part filled by b in 1 min . = 1 / 60 part filled by ( a + b ) in 1 min . = 1 / 30 + 1 / 60 = 1 / 20 . both the pipes can fill the tank in 20 minutes . answer : b"
Solve the problem systematically, showing each step clearly.	there are 8 stores in town that had a total of 24 visitors on a particular day . however , only 12 people went shopping that day ; some people visited more than one store . if 9 people visited exactly two stores each , and everyone visited at least one store , what is the largest number of stores anyone could have visited ? a ) 2 , b ) 3 , c ) 4 , d ) 5 , e ) 6	"9 people visited 2 stores each for 18 visits . to maximize the number of stores that one person visited , let' s assume that 2 people visited 1 store each . the number of remaining visits is 24 - 18 - 2 = 4 , which is the maximum that one person could have visited . the answer is c ."
Explain the solution process for the given equation.	how many positive integers less than 6,000 are there in which the sum of the digits equals 5 ? a ) 65 , b ) 62 , c ) 61 , d ) 56 , e ) 58	"basically , the question asks how many 4 digit numbers ( including those in the form 0 xxx , 00 xx , and 000 x ) have digits which add up to 5 . think about the question this way : we know that there is a total of 5 to be spread among the 4 digits , we just have to determine the number of ways it can be spread . let x represent a sum of 1 , and \| represent a seperator between two digits . as a result , we will have 5 x' s ( digits add up to the 5 ) , and 3 \|' s ( 3 digit seperators ) . so , for example : xx \| x \| x \| x = 2111 \| \| xxx \| xx = 0032 etc . there are 8 c 3 ways to determine where to place the separators . hence , the answer is 8 c 3 = 56 . d"
Provide a detailed solution to the given mathematical problem.	if the charge of staying in a student youth hostel $ 18.00 / day for the first week , and $ 11.00 / day for each additional week , how much does it cost to stay for 23 days ? a ) $ 160 , b ) $ 176 , c ) $ 282 , d ) $ 302 , e ) $ 286	"total number of days of stay = 23 charge of staying in first week = 18 * 7 = 126 $ charge of staying for additional days = ( 23 - 7 ) * 11 = 16 * 11 = 176 $ total charge = 126 + 176 = 302 $ answer d"
Work through the equation methodically, ensuring clarity at each step.	a certain number of men can finish a piece of work in 10 days . however , if there were 10 less men it will take 10 days more for the work to be finished . how many men were there originally ? a ) 10 men , b ) 30 men , c ) 40 men , d ) 20 men , e ) none of these	"let x be the number of men originally thus , 10 x = ( 10 + 10 ) ( x - 10 ) 10 x = 20 ( x - 10 ) x = 2 ( x - 10 ) x = 20 answer : d"
Break down and solve the equation step by step.	in a family 15 people eat only vegetarian , 8 people eat only non veg . , 11 people eat both veg and non veg . . how many people eat veg in the family ? a ) 20 , b ) 11 , c ) 23 , d ) 26 , e ) 21	"total people eat veg = only veg + both veg and non veg total = 15 + 11 = 26 answer = d"
Solve the problem systematically, showing each step clearly.	a math teacher has 24 cards , each of which is in the shape of a geometric figure . half of the cards are rectangles , and a third of the cards are rhombuses . if 8 cards are squares , what is the maximum possible number of cards that re circles . a ) 7 , b ) 10 , c ) 11 , d ) 12 , e ) 13	"a square is a special kind of rhombus ( sides are perpendicular ) a square is a special kind of rectangles ( sides with same length ) among the 24 cards with have : 15 rectangles 10 rhombus 8 squares among the 15 rectangles , there could be 8 special ones ( with sides of same length ) that are squares . that lets at least 7 rectangles that are not square . among the 10 rectangles , there could be 8 special ones ( with sides perpendicular ) that are squares . that lets at least 2 rhombus that are not square . we have 8 squares . so the minimum different cards that represent a square , a rhombus or a rectangle is 2 + 7 + 8 = 17 which means that the maximum number of circles that you could have is 24 - 17 = 7 answer ( a )"
Explain the solution process for the given equation.	excluding stoppages , the speed of a bus is 54 kmph and including stoppages , it is 36 kmph . for how many minutes does the bus stop per hour ? a ) 20 min , b ) 8 min , c ) 9 min , d ) 10 min , e ) 11 min	"due to stoppages , it covers 18 km less . time taken to cover 18 km = ( 18 / 54 ) x 60 = 20 min answer : a"
Explain the solution process for the given equation.	what is the greatest 6 - digit number when divided by 8 , 9 , and 10 leaves a remainder of 6 , 7 , and 8 respectively ? a ) 456780 , b ) 678918 , c ) 997479 , d ) 997916 , e ) 997920	when you divide a positive integer by 10 , the remainder will just be the units digit . we know the remainder is 8 when we divide by 10 , so b is the only possible answer .
Provide a detailed solution to the given mathematical problem.	if x is equal to the sum of the integers from 30 to 50 , inclusive , and y is the number of even integers from 30 to 50 , inclusive , what is the value of x + y ? a ) 810 , b ) 811 , c ) 830 , d ) 850 , e ) 851	since the integers from 30 - 50 are consecutive we can use the ` ` median x # of terms formula'' to determine x . there are 21 terms and the median is 40 . this gives a total of 21 x 40 = 840 . there are 11 even integers from 30 - 50 = y . therefore , x + y = 851 . answer : e
Work through the equation methodically, ensuring clarity at each step.	a man has some hens and cows . if the number of heads be 50 and the number of feet equals 144 , then the number of hens will be a ) 22 , b ) 24 , c ) 26 , d ) 20 , e ) 28	"explanation : let number of hens = h and number of cows = c number of heads = 50 = > h + c = 48 - - - ( equation 1 ) number of feet = 144 = > 2 h + 4 c = 144 = > h + 2 c = 72 - - - ( equation 2 ) ( equation 2 ) - ( equation 1 ) gives 2 c - c = 72 - 50 = > c = 22 substituting the value of c in equation 1 , we get h + 22 = 50 = > h = 50 - 22 = 28 i . e . , number of hens = 28 answer : e"
Work through the equation methodically, ensuring clarity at each step.	for how many values of q , is \| \| \| q - 5 \| - 10 \| - 5 \| = 2 ? ( those ls are mods ) a ) 0 , b ) 2 , c ) 4 , d ) 8 , e ) more than 8	i think its 8 \| \| \| q - 5 \| - 10 \| - 5 \| = 2 let \| q - 5 \| = a which makes above \| \| a - 10 \| - 5 \| = 2 let \| a - 10 \| = b which makes \| b - 5 \| = 2 now for the above b can take 3 , 7 for every b = 3 a can have 13 , 7 and for b = 7 a can have 17 and 3 so' a' has four solutions 13 , 7 , 17 and 3 for a = 13 ; q has 18 or - 8 thus has 2 for every combination hence 4 x 2 = 8 answer d
Work through the equation methodically, ensuring clarity at each step.	the size of a television screen is given as the length of the screen' s diagonal . if the screens were flat , then the area of a square 21 - inch screen would be how many square inches greater than the area of a square 17 - inch screen ? a ) 2 , b ) 4 , c ) 16 , d ) 38 , e ) 76	"pythogoras will help here ! let the sides be x and diagonal be d then d ^ 2 = 2 x ^ 2 and area = x ^ 2 now plug in the given diagonal values to find x values and then subtract the areas ans will be 21 ^ 2 / 2 - 17 ^ 2 / 2 = 152 / 2 = 76 ans e ."
Provide a detailed solution to the given mathematical problem.	if a car went the first third of the distance at 80 kmh , the second third at 24 kmh , and the last third at 54 kmh , what was the average speed of the car for the entire trip ? a ) 36 kmh , b ) 40 kmh , c ) 41 kmh , d ) 44 kmh , e ) 50 kmh	"assume d / 3 = 2160 ( this number is convenient because it is divisible by 80 , 24 and 54 ) so : 2160 = 80 * t 1 = 27 hrs 2160 = 24 * t 2 = 90 hrs 2160 = 54 * t 3 = 40 hrs t = t 1 + t 2 + t 3 = 157 hrs d = rt ( 240 * 3 ) = r * 157 r = 41.27 answer : c"
Provide a detailed solution to the given mathematical problem.	if \| 20 x - 10 \| = 90 , then find the product of the values of x ? a ) - 45 , b ) 50 , c ) - 62 , d ) 35 , e ) - 20	"\| 20 x - 10 \| = 90 20 x - 10 = 90 or 20 x - 10 = - 90 20 x = 100 or 20 x = - 80 x = 5 or x = - 4 product = - 4 * 5 = - 20 answer is e"
Provide a detailed solution to the given mathematical problem.	how many even integers n , such that 20 < = n < = 180 are of the form 3 k + 4 , where k is any natural number ? a ) 24 , b ) 27 , c ) 30 , d ) 33 , e ) 36	the first number is 22 = 16 + 6 ( 1 ) . we can continue adding 6 to make a list : 22 , 28 , 34 , . . . the last number is 278 = 16 + 6 ( 27 ) there are 27 numbers in the list . the answer is b .
Explain the solution process for the given equation.	find the area of trapezium whose parallel sides are 20 cm and 18 cm long , and the distance between them is 13 cm . a ) 335 cm 2 , b ) 885 cm 2 , c ) 285 cm 2 , d ) 247 cm 2 , e ) 725 cm 2	"area of a trapezium = 1 / 2 ( sum of parallel sides ) * ( perpendicular distance between them ) = 1 / 2 ( 20 + 18 ) * ( 13 ) = 247 cm 2 answer : d"
Break down and solve the equation step by step.	the length of a bridge in meters , which a train 80 - meters long and traveling at 45 km / hr can cross in 30 seconds is ? a ) 215 , b ) 235 , c ) 255 , d ) 275 , e ) 295	"45 km / h = 45000 m / 3600 s = 12.5 m / s in 30 seconds , the train can go 30 ( 12.5 ) = 375 meters let x be the length of the bridge . x + 80 = 375 meters x = 295 meters the answer is e ."
Explain the solution process for the given equation.	an empty wooden vessel weighs 14 % of its total weight when filled with paint . if the weight of a partially filled vessel is one half that of a completely filled vessel , what fraction of the vessel is filled . a ) 6 / 13 , b ) 5 / 9 , c ) 1 / 24 , d ) 4 / 9 , e ) 2 / 5	"an empty wooden vessel weighs 14 % of its total weight when filled with paint : vessel = 0.14 ( vessel + paint ) ; 14 v = v + p ( so the weight of completely filled vessel is 14 v ) p = 13 v ( so the weight of the paint when the vessels is completely filled is 13 v ) . the weight of a partially filled vessel is one half that of a completely filled vessel : v + p' = 1 / 2 * 14 v ; p' = 6 v ( so the weight of the paint when the vessels is partially filled is 6 v ) . what fraction of the vessel is filled ? so , we need to find the ratio of the weight of the paint when the vessel iscompletely filledto the weight of the paint when the vessel ispartially filled : p' / p = 6 v / 13 v = 6 / 13 . answer : a ."
Break down and solve the equation step by step.	tabby is training for a triathlon . she swims at a speed of 1 mile per hour . she runs at a speed of 8 miles per hour . she wants to figure out her average speed for these two events . what is the correct answer for her ? a ) 8 mph , b ) 5.25 mph , c ) 3.5 mph , d ) 4.5 mph , e ) 0.5 mph	"( 1 mph + 8 mph ) / 2 = 4.5 mph correct option is : d"
Explain the solution process for the given equation.	how many 3 - digits number are palindromic numbers ? a palindromic number reads the same forward and backward , example 121 . a ) 90 , b ) 610 , c ) 729 , d ) 900 , e ) 1000	"take the task of building palindromes and break it intostages . stage 1 : select the hundreds digit we can choose 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , or 9 so , we can complete stage 1 in 9 ways stage 2 : select the tens digit we can choose 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , or 9 so , we can complete stage 2 in 10 ways important : at this point , the remaining digits are alreadylocked in . stage 3 : select the units digit this digit must be the same as the thousands digit ( which we already chose in stage 2 ) so , we can complete this stage in 1 way . by thefundamental counting principle ( fcp ) , we can complete all 5 stages ( and thus build a 3 - digit palindrome ) in ( 9 ) ( 10 ) ( 1 ) ways ( = 900 ways ) answer : a"
Work through the equation methodically, ensuring clarity at each step.	a cistern can be filled by a tap in 4 hours while it can be emptied by another tap in 10 hours . if both the taps are opened simultaneously , then after how much time will the cistern get filled ? a ) 2.9 hrs , b ) 8.9 hrs , c ) 2.9 hrs , d ) 6.7 hrs , e ) 8.6 hrs	"net part filled in 1 hour = ( 1 / 4 - 1 / 10 ) = 3 / 20 the cistern will be filled in 20 / 3 hrs i . e . , 6.7 hrs . answer : d"
Work through the equation methodically, ensuring clarity at each step.	the ratio of money with ram and gopal is 7 : 17 and that with gopal and krishan is 7 : 17 . if ram has rs . 637 , krishan has ? a ) s . 2890 , b ) s . 2330 , c ) s . 3757 , d ) s . 1620 , e ) s . 2680	"ram : gopal = 7 : 17 = 49 : 119 gopal : krishan = 7 : 17 = 119 : 289 ram : gopal : krishan = 49 : 119 : 289 ram : krishan = 49 : 289 thus , 49 : 289 = 637 : n & there n = 289 x 637 / 49 = rs . 3757 answer : c"
Solve the problem systematically, showing each step clearly.	a theater charges $ 12 for seats in the orchestra and $ 8 for seats in the balcony . on a certain night , a total of 350 tickets were sold for a total cost of $ 3,320 . how many more tickets were sold that night for seats in the balcony than for seats in the orchestra ? a ) 90 , b ) 110 , c ) 120 , d ) 130 , e ) 220	"orchestra seats - a balcony seats - b a + b = 350 and 12 a + 8 b = 3320 solving equations simultaneously ( multiply equation 1 with 8 and subtract from second equation ) 4 a = 3320 - 8 * 350 = 3320 - 2800 = 520 i . e . a = 130 and b = 350 - 130 = 220 more seats in balcony than orchestra = b - a = 220 - 130 = 90 answer : option a"
Work through the equation methodically, ensuring clarity at each step.	the area of an isosceles trapezoid with sides of length 5 and bases of length 8 and 14 is ? a ) 22 , b ) 32 , c ) 44 , d ) 55 , e ) 56	"we can divide the trapezoid into two triangles and a rectangle sides of triangle , t 1 - - > h 1 = 5 , b 1 = 3 , s 1 = root ( 25 - 9 ) = 4 = s 2 of triangle t 2 area of triangle = ( 1 / 2 ) * 4 * 3 = 6 side of the rectangle = 8 * 4 = 32 total area of the trapezoid = 32 + 6 + 6 = 44 answer : c"
Provide a detailed solution to the given mathematical problem.	if 20 % of a number = 1000 , then 120 % of that number will be ? a ) 20 , b ) 120 , c ) 360 , d ) 6000 , e ) 820	"let the number x . then , 20 % of x = 1000 x = ( 1000 * 100 ) / 20 = 5000 120 % of x = ( 120 / 100 * 5000 ) = 6000 . answer : d"
Solve the problem systematically, showing each step clearly.	two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 26 seconds . the ratio of their speeds is ? a ) 3 / 1 , b ) 9 / 1 , c ) 3 / 3 , d ) 3 / 5 , e ) 5 / 2	"let the speeds of the two trains be x m / sec and y m / sec respectively . then , length of the first train = 27 x meters , and length of the second train = 17 y meters . ( 27 x + 17 y ) / ( x + y ) = 26 = = > 27 x + 17 y = 26 x + 26 y = = > 1 x = 9 y = = > x / y = 9 / 1 . answer : b"
Break down and solve the equation step by step.	a train covers a distance of 9 km in 6 min . if it takes 5 sec to pass a telegraph post , then the length of the train is ? a ) 100 , b ) 105 , c ) 110 , d ) 120 , e ) 125	"speed = ( 9 / 6 * 60 ) km / hr = ( 90 * 5 / 18 ) m / sec = 25 m / sec . length of the train = 25 * 5 = 125 m . answer : option e"
Solve the problem systematically, showing each step clearly.	a watch was sold at a loss of 12 % . if it was sold for rs . 140 more , there would have been a gain of 4 % . what is the cost price ? a ) s . 1000 , b ) s . 875 , c ) s . 1007 , d ) s . 1006 , e ) s . 1002	"explanation : 88 % 104 % - - - - - - - - 16 % - - - - 140 100 % - - - - ? = > rs . 875 answer : b"
Provide a detailed solution to the given mathematical problem.	a batsman makes a score of 83 runs in the 17 th inning and thus increases his averages by 3 . find his average after 17 th inning ? a ) 19 , b ) 29 , c ) 35 , d ) 49 , e ) 59	"let the average after 17 th inning = x then average after 16 th inning = ( x - 3 ) therefore 16 ( x - 3 ) + 83 = 17 x therefore x = 35 answer : c"
Break down and solve the equation step by step.	a train traveling at 100 kmph overtakes a motorbike traveling at 64 kmph in 85 seconds . what is the length of the train in meters ? a ) 400 meters , b ) 1111 meters , c ) 160 meters , d ) 850 meters , e ) none of these	train overtakes a bike means that we are talking about total length of the train . ( train' s head is close to bike when it started and its tail crosses the bike when it overtakes the bike ) relative speed = 100 - 64 = 36 km / h = 36000 m / h time = 85 seconds distance = speed * time 36000 * 85 / 3600 = 850 meters . d is the answer .
Explain the solution process for the given equation.	25.25 / 1500 is equal to : a ) 1.012526333 , b ) 0.012625333 , c ) 0.125263333 , d ) 0.126253333 , e ) 0.016833333	"25.25 / 2000 = 2525 / 200000 = 0.016833333 answer : e"
Explain the solution process for the given equation.	the ratio , by volume , of bleach to detergent to water in a certain solution is 2 : 40 : 100 . the solution will be altered so that the ratio of bleach to detergent is tripled while the ratio of detergent to water is halved . if the altered solution will contain 200 liters of water , how many liters of detergent will it contain ? a ) 60 , b ) 55 , c ) 50 , d ) 45 , e ) 40	"b : d : w = 2 : 40 : 100 bnew / dnew = ( 1 / 3 ) * ( 2 / 40 ) = ( 1 / 60 ) dnew / wnew = ( 1 / 2 ) * ( 40 / 100 ) = ( 1 / 5 ) wnew = 200 dnew = wnew / 5 = 200 / 5 = 40 so , answer will be e"
Work through the equation methodically, ensuring clarity at each step.	the difference of two numbers is 11 . one third of their sum is 7 . what are the two numbers ? a ) 21,10 , b ) 16,5 , c ) 3,14 , d ) 11,10 , e ) 7,14	"let the two numbers be x and y . equation ( i ) : x - y = 11 equation ( ii ) : ( x + y ) / 3 = 7 solve system of equations : x + y = 21 x - y = 11 ( add equations together ) - - - 2 x = 32 - - - x = 16 16 - y = 11 - - - y = 5 since x = 16 and y = 5 , answer b ( 16,5 ) is correct ."
Work through the equation methodically, ensuring clarity at each step.	rahim bought 65 books for rs . 1150 from one shop and 50 books for rs . 920 from another . what is the average price he paid per book ? a ) rs . 17 , b ) rs . 18 , c ) rs . 12 , d ) rs . 11 , e ) rs . 10	"average price per book = ( 1150 + 920 ) / ( 65 + 50 ) = 2070 / 115 = rs . 18 answer : b"
Work through the equation methodically, ensuring clarity at each step.	if the area of a square with sides of length 4 centimeters is equal to the area of a rectangle with a width of 8 centimeters , what is the length of the rectangle , in centimeters ? a ) 4 , b ) 8 , c ) 2 , d ) 16 , e ) 18	"let length of rectangle = l 4 ^ 2 = l * 8 = > l = 16 / 8 = 2 answer c"
Provide a detailed solution to the given mathematical problem.	if 20 men can build a wall 112 metres long in 6 days , what length of a similar wall can be built by 10 men in 3 days ? a ) 65 mtr . , b ) 28 mtr , c ) 70 mtr . , d ) 78 mtr . , e ) 17 mtr .	"20 men is 6 days can build 112 metres 10 men in 3 days can build = 112 * ( 10 / 20 ) x ( 3 / 6 ) = 28 meters answer : b ."
Provide a detailed solution to the given mathematical problem.	the average age of 39 students in a group is 14 years . when teacher' s age is included to it , the average increases by one . what is the teacher' s age in years ? a ) 35 years , b ) 45 years , c ) 51 years , d ) 54 years , e ) none of these	"explanation : age of the teacher = ( 40 * 15 - 39 * 14 ) years = 54 years . answer : d"
Provide a detailed solution to the given mathematical problem.	the cost price of 12 articles is equal to the selling price of 10 articles . what is the profit percent ? a ) 10 % , b ) 20 % , c ) 30 % , d ) 40 % , e ) 50 %	"10 * sp = 12 * cp sp = 1.2 * cp the profit percent is 20 % . the answer is b ."
Solve the problem systematically, showing each step clearly.	on the first of the year , james invested x dollars at proudstar bank in an account that yields 2.2 % in interest every quarter year . at the end of the year , during which he made no additional deposits or withdrawals , he had y dollars in the account . if james had invested the same amount in an account which pays interest on a yearly basis , what must the interest rate be for james to have y dollars at the end of the year ? a ) 2.04 % , b ) 6.12 % , c ) 8 % , d ) 8.25 % , e ) 9 %	"if the interest were compounded annually instead of quarterly then in one year the interest would be 2.2 * 4 = 8.8 % . now , since the interest is compounded quarterly then there would be interest earned on interest ( very small amount ) thus the actual interest would be a little bit more than 8.8 % answer : e ."
Provide a detailed solution to the given mathematical problem.	suppose you work for a manufacturing plant that pays you $ 12.50 an hour and $ . 16 for each widget you make . how many widgets must you produce in a 40 hour week to earn $ 800 ( before payroll deductions ) ? a ) 1800 , b ) 1825 , c ) 1850 , d ) 1855 , e ) 1875	"hourly = $ 12.50 / hour totalhours = 40 piecework = $ 0.16 / widget totalpieces = ? ? ? ? total pay = ( hourly * totalhours ) + ( piecework * totalpieces ) so , put it all together : $ 800 = ( $ 12.50 * 40 ) + ( $ 0.16 * total pieces ) totalpieces = ( $ 800 - ( $ 12.50 * 40 ) ) / $ 0.16 = 1875 widgets e . 1875"
Work through the equation methodically, ensuring clarity at each step.	gold is 19 times as heavy as water and copper is 9 times as heavy as water . in what ratio should these be mixed to get an alloy 16 times as heavy as water ? a ) 1 : 2 , b ) 7 : 3 , c ) 4 : 1 , d ) 5 : 2 , e ) 6 : 5	"g = 19 w c = 9 w let 1 gm of gold mixed with x gm of copper to get 1 + x gm of the alloy 1 gm gold + x gm copper = x + 1 gm of alloy 19 w + 9 wx = x + 1 * 16 w 19 + 9 x = 16 ( x + 1 ) x = 3 / 7 ratio of gold with copper = 1 : 3 / 7 = 7 : 3 answer is b"
Break down and solve the equation step by step.	if a randomly selected non - negative single digit integer is added to { 2 , 3 , 5 , 8 } . what is the probability that the median of the set will increase but the range still remains the same ? a ) 0.2 , b ) 0.3 , c ) 0.4 , d ) 0.5 , e ) 0.6	we are selecting from non - negative single digit integers , so from { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . these 10 digits represent the total number of outcomes . hence , the total number of outcomes is 10 . we need to find the probability that the median of the set will increase but the range still remains the same . the median of the set is ( 3 + 5 ) / 2 = 4 , thus the number selected must be 5 or greater . for the range to remain the same , the number must be between 2 and 8 inclusive . to satisfy both conditions , the number selected must be 5 , 6 , 7 , or 8 . the probability is 4 / 10 = 0.4 the answer is c .
Solve the problem systematically, showing each step clearly.	two employees m and n are paid a total of $ 616 per week by their employer . if m is paid 120 percent of the salary paid to n , how much is n paid per week ? a ) $ 270 , b ) $ 280 , c ) $ 290 , d ) $ 300 , e ) $ 310	"1.2 n + n = 616 2.2 n = 616 n = 280 the answer is b ."
Work through the equation methodically, ensuring clarity at each step.	with a uniform speed a car covers the distance in 8 hours . had the speed been increased by 4 km / hr , the same distance could have been covered in 7 1 / 2 hours . what is the distance covered ? a ) 347 , b ) 480 , c ) 288 , d ) 278 , e ) 131	"let the distance be x km . then , x / ( 7 1 / 2 ) - x / 8 = 4 2 x / 15 - x / 8 = 4 = > x = 480 km . answer : b"
Break down and solve the equation step by step.	if 9 a - b = 10 b + 70 = - 12 b - 2 a , what is the value of 11 a + 11 b ? a ) - 4 , b ) - 2 , c ) 0 , d ) 2 , e ) 4	"( i ) 9 a - 11 b = 70 ( ii ) 2 a + 22 b = - 70 adding ( i ) and ( ii ) : 11 a + 11 b = 0 the answer is c ."
Solve the problem systematically, showing each step clearly.	a man can row downstream at the rate of 16 km / hr and upstream at 8 km / hr . find man' s rate in still water and the rate of current ? a ) 12,4 , b ) 6,3 , c ) 9,3 , d ) 6,6 , e ) none of these	"explanation : rate of still water = 1 / 2 ( 16 + 8 ) = 12 km / hr rate of current = 1 / 2 ( 16 - 8 ) = 4 km / hr answer : option a"
Work through the equation methodically, ensuring clarity at each step.	find the ones digit of 73 ^ 352 a ) 1 , b ) 5 , c ) 6 , d ) 7 , e ) 9	"the units digit of 73 ^ 352 will be the same as the units digit of 3 ^ 352 . 3 ^ 1 = 3 - - > the units digit is 3 ; 3 ^ 2 = 9 - - > the units digit is 9 ; 3 ^ 3 = 27 - - > the units digit is 7 ; 3 ^ 4 = 81 - - > the units digit is 1 ; 3 ^ 5 = 243 - - > the units digit is 3 again ; . . . so , as you can see the units digit repeats in blocks of 4 : { 3 , 9 , 7 , 1 } , { 3 , 9 , 7 , 1 } , . . . now , since 352 = ( multiple of 4 ) + 0 , since 3 is an odd number , the units digit of 3 ^ 352 will be 1 . answer : a ."
Explain the solution process for the given equation.	- 64 x 29 + 165 = ? a ) 2436 , b ) 2801 , c ) - 2801 , d ) - 1601 , e ) none of them	"given exp . = - 64 x ( 30 - 1 ) + 165 = - ( 64 x 30 ) + 64 + 165 = - 1830 + 229 = - 1601 answer is d"
Explain the solution process for the given equation.	a watch was sold at a loss of 10 % . if it was sold for rs . 168 more , there would have been a gain of 4 % . what is the cost price ? a ) s : 1000 , b ) s : 1067 , c ) s : 1200 , d ) s : 1028 , e ) s : 1027	"90 % 104 % - - - - - - - - 14 % - - - - 168 100 % - - - - ? = > rs : 1200 answer : c"
Explain the solution process for the given equation.	how many integers are divisible by 5 between 10 ! and 10 ! + 20 inclusive ? a ) 5 , b ) 7 , c ) 8 , d ) 9 , e ) 10	a - 7 10 ! is divisible by 5 there are 4 numbers between 10 ! and 10 ! + 20 that are divisible by 5 . hence 5
Explain the solution process for the given equation.	in a flight of 600 km , an aircraft was slowed down due to bad weather . its average speed for the trip was reduced by 200 km / hr and the time of flight increased by 30 minutes . the duration of the flight is a ) 3 hr . , b ) 5 hr . , c ) 2 hr . , d ) 7 hr . , e ) 1 hr .	"e 1 hr . let the duration of the flight be x hours . then , 600 / x - 600 / 0.5 x = 200 - - > 600 / x - 1200 / ( 2 x + 1 ) = 200 - - > x ( 2 x + 1 ) = 3 - - > 2 x ^ 2 + x - 3 = 0 - - > ( 2 x + 3 ) ( x - 1 ) = 0 - - > x = 1 hr ."
Provide a detailed solution to the given mathematical problem.	martin bought 10 concert tickets , some at the full price of $ 2.00 per ticket , and some at a discounted price of $ 1.60 per ticket . if he spent a total of $ 17.60 , how many discounted tickets did he buy ? a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7	"let x be the number of tickets he bought at $ 2 per ticket . then 2 x + ( 10 - x ) 1.6 = 17.6 0.4 x = 1.6 = > x = 4 discounted tickets = 10 - x = 6 ans : d"
Solve the problem systematically, showing each step clearly.	a man can do a piece of work in 5 days , but with the help of his son , he can do it in 3 days . in what time can the son do it alone ? a ) 13 / 2 , b ) 7 , c ) 15 / 2 , d ) 8 , e ) 9	"son' s 1 day' s work = ( 1 / 3 ) - ( 1 / 5 ) = 2 / 5 the son alone can do the work in 15 / 2 days answer is c"
Break down and solve the equation step by step.	a cycle is bought for rs . 670 and sold for rs . 780 , find the gain percent ? a ) 70 % , b ) 20 % , c ) 16 % , d ) 19 % , e ) 80 %	"explanation : 670 - - - - 110 100 - - - - ? = > 16 % answer : c"
Solve the problem systematically, showing each step clearly.	in a certain corporation , there are 300 male employees and 150 female employees . it is known that 10 % of the male employees have advanced degrees and 40 % of the females have advanced degrees . if one of the 450 employees is chosen at random , what is the probability this employee has an advanced degree or is female ? a ) 1 / 5 , b ) 5 / 15 , c ) 6 / 15 , d ) 3 / 10 , e ) 4 / 15	p ( female ) = 150 / 450 = 1 / 3 p ( male with advanced degree ) = 0.1 * 300 / 450 = 30 / 450 = 1 / 15 the sum of the probabilities is 6 / 15 = 2 / 5 the answer is c .
Explain the solution process for the given equation.	if ( 2 to the x ) - ( 2 to the ( x - 2 ) ) = 3 ( 2 to the 10 ) , what is the value of x ? a ) 9 , b ) 11 , c ) 13 , d ) 12 , e ) 17	"( 2 to the power x ) - ( 2 to the power ( x - 2 ) ) = 3 ( 2 to the power 10 ) 2 ^ x - 2 ^ ( x - 2 ) = 3 . 2 ^ 10 hence x = 12 . answer is d"
Work through the equation methodically, ensuring clarity at each step.	a worker' s daily wage is increased by 50 % and the new wage is $ 51 per day . what was the worker' s daily wage before the increase ? a ) $ 28 , b ) $ 32 , c ) $ 34 , d ) $ 35 , e ) $ 40	"let x be the daily wage before the increase . 1.5 x = $ 51 x = $ 34 the answer is c ."
Break down and solve the equation step by step.	the manager of a produce market purchased a quantity of tomatoes for $ 0.80 per pound . due to improper handling , 20 percent of the tomatoes , by weight , were ruined and discarded . at what price per pound should the manager sell the remaining tomatoes if she wishes to make a profit on the sale of the tomatoes equal to 8 percent of the cost of the tomatoes . a ) $ 0.94 , b ) $ 0.96 , c ) $ 0.98 , d ) $ 1.00 , e ) $ 1.09	"assume the manager bought 100 tomatoes . cost price = 80 given : 20 % are damaged - - > available tomatoes to sell = 80 80 * x - 80 = 0.08 * 80 80 x - 80 = 6.4 80 x = 86.64 x = 86.64 / 80 = 87 / 80 ( approx ) = 1.0875 x is slightly under 1.0875 = 1.09 answer : e"
Explain the solution process for the given equation.	consider the word rotor . whichever way you read it , from left to right or from right to left , you get the same word . such a word is known as palindrome . find the maximum possible number of 5 - letter palindromes ? a ) 56678 , b ) 16789 , c ) 17576 , d ) 35142 , e ) none of these	"explanation : the first letter from the right can be chosen in 26 ways because there are 26 alphabets . having chosen this , the second letter can be chosen in 26 ways . = > the first two letters can be chosen in 26 * 26 = 676 wayshaving chosen the first two letters , the third letter can be chosen in 26 ways . = > all the three letters can be chosen in 676 * 26 = 17576 ways . it implies that the maximum possible number of five letter palindromes is 17576 because the fourth letter is the same as the second letter and the fifth letter is the same as the first letter . answer : c"

Provide a detailed solution to the given mathematical problem.

find the total number of prime factors in the expression ( 4 ) ^ 11 x ( 7 ) ^ 5 x ( 11 ) ^ 3 a ) 26 , b ) 22 , c ) 25 , d ) 30 , e ) 29

"( 4 ) ^ 11 x ( 7 ) ^ 5 x ( 11 ) ^ 3 = ( 2 x 2 ) ^ 11 x ( 7 ) ^ 5 x ( 11 ) ^ 3 = 2 ^ 11 x 2 ^ 11 x 7 ^ 5 x 11 ^ 3 = 2 ^ 22 x 7 ^ 5 x 11 ^ 3 total number of prime factors = ( 22 + 5 + 3 ) = 30 . answer is d ."

Explain the solution process for the given equation.

if the sum of 3 consecutive even numbers is 44 more than the average of these numbers , then the largest of these numbers is ? a ) 24 , b ) 66 , c ) 9 , d ) 7 , e ) 01

answer : 24 explanation : let the smallest of these number be x . the other two numbers are ( x + 2 ) and ( x + 4 ) . x + ( x + 2 ) + ( x + 4 ) = ( x + ( x + 2 ) + ( x + 4 ) ) / 3 + 44 3 x + 3 * ( x + 2 ) + 3 * ( x + 4 ) = x + ( x + 2 ) + ( x + 4 ) + 132 9 x + 18 = 3 x + 138 6 x = 120 x = 20 therefore , the largest number is 24 . answer : a

Provide a detailed solution to the given mathematical problem.

if [ [ x ] ] = x ^ 2 + 2 x + 4 , what is the value of [ [ 5 ] ] ? a ) 39 , b ) 9 , c ) 15 , d ) 19 , e ) 25

"these functions questions might look intimidating , but they just test your knowledge about how well you can substitute values [ [ x ] ] = x ^ 2 + 2 x + 4 [ [ 5 ] ] = 5 ^ 2 + 2 * 5 + 4 = 39 . option a"

Explain the solution process for the given equation.

a professional athlete was offered a 3 - year contract to play with team k that provided for an annual salary of $ 100000 in the first year , an increase in annual salary of 20 % over the previous year for the next two years , and a bonus of $ 50000 on signing . team l offered a 3 - year contract providing for an annual salary of $ 150000 in the first year , an increase in annual salary of 20 % over the previous year for the next two years , and no signing bonus . if he accepts the offer of team l and fulfills the 3 - year contract terms , the athlete will receive how much more money by choosing team l over team k ? a ) $ 132,000 , b ) $ 50,000 , c ) $ 82,500 , d ) $ 92,000 , e ) $ 100,000

team k' s contract = $ 100,000 + $ 100,000 * 1.2 + $ 100,000 * 1.2 * 1.2 + $ 50,000 = $ 414,000 team l' s contract = $ 150,000 + $ 150,000 * 1.2 + $ 150,000 * 1.2 * 1.2 = $ 546,000 the difference = $ 132,000 . answer : a .

Provide a detailed solution to the given mathematical problem.

a , b , k start from the same place and travel in the same direction at speeds of 30 km / hr , 40 km / hr , 60 km / hr respectively . b starts six hours after a . if b and k overtake a at the same instant , how many hours after a did k start ? a ) 3 , b ) 4.5 , c ) 4 , d ) d ) 12 , e ) e ) 5

"the table you made does n' t make sense to me . all three meet at the same point means the distance they cover is the same . we know their rates are 30 , 40 and 60 . say the time taken by b is t hrs . then a takes 6 + t hrs . and we need to find the time taken by k . distance covered by a = distance covered by b 30 * ( 6 + t ) = 40 * t t = 18 hrs distance covered by b = distance covered by k 40 * t = 60 * time taken by k time taken by k = 40 * 18 / 60 = 12 hrs time taken by a = 6 + t = 6 + 18 = 24 hrs time taken by k = 12 hrs so k starts 24 - 12 = 12 hrs after a . ( answer d )"

Solve the problem systematically, showing each step clearly.

if a 2 cm cube is cut into 1 cm cubes , then what is the percentage increase in the surface area of the resulting cubes ? a ) 50 % , b ) 100 % , c ) 150 % , d ) 200 % , e ) 250 %

"the area a of the large cube is 2 * 2 * 6 = 24 square cm . the area of the 8 small cubes is 8 * 6 = 48 = 2 a , an increase of 100 % . the answer is b ."

Solve the problem systematically, showing each step clearly.

carina has 115 ounces of coffee divided into 5 - and 10 - ounce packages . if she has 2 more 5 - ounce packages than 10 - ounce packages , how many 10 - ounce packages does she have ? a ) 5 , b ) 7 , c ) 6 , d ) 4 , e ) 3

"lets say 5 and 10 ounce packages be x and y respectively . given that , 5 x + 10 y = 115 and x = y + 2 . what is the value of y . substituting the x in first equation , 5 y + 10 + 10 y = 85 - > y = 105 / 15 . = 7 b"

Work through the equation methodically, ensuring clarity at each step.

the sides of the triangle are in the ratio 1 / 2 : 1 / 3 : 1 / 4 and its perimeter is 104 cm . the length of the longest side is ? ['a ) 32 cm','b ) 48 cm','c ) 50 cm','d ) 28 cm','e ) 30 cm']

ratio of sides = 1 / 2 : 1 / 3 : 1 / 4 = 6 : 4 : 3 largest side = 104 * 6 / 13 = 48 cm answer is b

Explain the solution process for the given equation.

find a two digit number , given that the sum of the digits is 13 and the difference of the digits is 5 . ? a ) 74 , b ) 82 , c ) 95 , d ) 76 , e ) 94

"using elimination method find which of the options fit the description of the number . . . from the option only 94 meets this description sum of digits - - - 9 + 4 = 13 difference of digits - - - 9 - 4 = 5 answer e ."

Provide a detailed solution to the given mathematical problem.

a group of 55 adults and 70 children go for trekking . if there is meal for either 70 adults or 90 children and if 14 adults have their meal , find the total number of children that can be catered with the remaining food . a ) 33 , b ) 54 , c ) 18 , d ) 17 , e ) 72

"explanation : as there is meal for 70 adults and 14 have their meal , the meal left can be catered to 56 adults . now , 70 adults = 90 children 7 adults = 9 children therefore , 56 adults = 72 children hence , the meal can be catered to 72 children . answer : e"

Solve the problem systematically, showing each step clearly.

in 1990 the budgets for projects q and v were $ 700,000 and $ 780,000 , respectively . in each of the next 10 years , the budget for q was increased by $ 30,000 and the budget for v was decreased by $ 10,000 . in which year was the budget for q equal to the budget for v ? a ) 1992 , b ) 1993 , c ) 1994 , d ) 1995 , e ) 1996

"let the no of years it takes is x . 700 + 30 x = 780 - 10 x - - > 40 x = 80 and x = 2 . thus , it happens in 1992 . a ."

Provide a detailed solution to the given mathematical problem.

a squirrel runs up a cylindrical post , in a perfect spiral path making one circuit for each rise of 4 feet . how many feet does the squirrel travels if the post is 16 feet tall and 2 feet in circumference ? a ) 10 feet , b ) 8 feet , c ) 13 feet , d ) 15 feet , e ) 18 feet

"total circuit = 16 / 4 = 4 total feet squirrel travels = 4 * 2 = 8 feet answer : b"

Work through the equation methodically, ensuring clarity at each step.

a man can do a job in 20 days . his father takes 20 days and his son finishes it in 25 days . how long will they take to complete the job if they all work together ? a ) 6.3 , b ) 6.9 , c ) 7.1 , d ) 6.1 , e ) 6.2

"1 day work of the three persons = ( 1 / 20 + 1 / 20 + 1 / 25 ) = 7 / 50 so , all three together will complete the work in 300 / 47 = 7.1 days . answer : c"

Break down and solve the equation step by step.

the h . c . f . of two numbers is 23 and the other two factors of their l . c . m . are 13 and 16 . the larger of the two numbers is : a ) 276 , b ) 299 , c ) 322 , d ) 345 , e ) 368

"clearly , the numbers are ( 23 x 13 ) and ( 23 x 16 ) . larger number = ( 23 x 16 ) = 368 . answer : option e"

Solve the problem systematically, showing each step clearly.

if k ^ 3 is divisible by 162 , what is the least possible value of integer k ? a ) 12 , b ) 15 , c ) 18 , d ) 24 , e ) 36

"k ^ 3 = 162 * x = 2 * 3 ^ 4 * x the factors of k must at minimum include 2 * 3 ^ 2 = 18 . the answer is c ."

Break down and solve the equation step by step.

if x < y < z and y - x > 5 , where x is an even integer and y and z are odd integers , what is the least possible value f of z - x ? a ) 6 , b ) 7 , c ) 8 , d ) 9 , e ) 10

"x < y < z to find the least possible value for z - x ; we need to find the values for z and x that can be closest to each other . if x is some even number , then what could be minimum possible odd z . if x is some even number y - x > 5 ; y > x + 5 ; minimum value for y = x + 5 + 2 = x + 7 [ note : x + 5 is as even + odd = odd and nearest odd greater than x + 5 is x + 5 + 2 ] minimum value for z = y + 2 = x + 7 + 2 = x + 9 [ note : z = y + 2 because both z and y are odd . difference between two odd numbers is 2 ] f = z - x = x + 9 - x = 9 ans : d"

Solve the problem systematically, showing each step clearly.

there are 20 players in a chess group , and each player plays each of the others once . given that each game is played by two players , how many total games will be played ? a ) 10 , b ) 30 , c ) 190 , d ) 60 , e ) 90

"20 players are there . two players play one game with one another . so 20 c 2 = 20 * 19 / 2 = 190 so option c is correct"

Solve the problem systematically, showing each step clearly.

john traveled 80 % of the way from yellow - town to green - fields by train at an average speed of 80 miles per hour . the rest of the way john traveled by car at an average speed of v miles per hour . if the average speed for the entire trip was 60 miles per hour , what is v in miles per hour ? a ) 30 , b ) 40 , c ) 50 , d ) 55 , e ) 70

"let distance be d . we can find the total time & equate it , which comes as : 0.8 d / 80 + 0.2 d / v = d / 60 = > v = 30 answer : a"

Provide a detailed solution to the given mathematical problem.

a 600 meter long train crosses a signal post in 40 seconds . how long will it take to cross a 1.8 kilometer long bridge , at the same speed ? a ) 4 min , b ) 2 min , c ) 8 min , d ) 9 min , e ) 3 min

s = 600 / 40 = 15 mps s = 1800 / 15 = 120 sec = 2 min . answer : b

Work through the equation methodically, ensuring clarity at each step.

at a particular graduation party with 300 guests , 70 % of the guests brought gifts , and 40 % of the female guests brought gifts . if 30 males did not bring gifts to the party , how many females did bring gifts ? a ) 18 , b ) 28 , c ) 40 , d ) 68 , e ) 70

"the correct method total = 300 . . 70 % of 300 = 210 got gifts . . 90 did not get gift , out of which 30 are males , so remaining 90 - 30 = 60 are females . . but 40 % females brought gift , so 60 % did not get it . . so 60 % = 60 , 100 % = 60 * 100 / 60 = 100 . ans 40 % of 100 = 40 c"

Provide a detailed solution to the given mathematical problem.

linda spent 3 / 4 of her savings on furniture and the rest on a tv . if the tv cost her $ 230 , what were her original savings ? a ) $ 900 , b ) $ 800 , c ) $ 1000 , d ) $ 920 , e ) $ 1500

"if linda spent 3 / 4 of her savings on furnitute , the rest 4 / 4 - 3 / 4 = 1 / 4 on a tv but the tv cost her $ 230 . so 1 / 4 of her savings is $ 230 . so her original savings are 4 times $ 230 = $ 920 correct answer d"

Provide a detailed solution to the given mathematical problem.

if rs . 7500 are borrowed at c . i at the rate of 4 % per annum , then after 2 years the amount to be paid is ? a ) rs . 8082 , b ) rs . 7800 , c ) rs . 8100 , d ) rs . 8112 , e ) rs . 9000

"explanation : a = 7500 ( 26 / 25 ) 2 = 8112 answer is d"

Explain the solution process for the given equation.

if the ratio of the sum of the first 6 terms of a g . p . to the sum of the first 3 terms of the g . p . is 28 , what is the common ratio of the g . p ? a ) 3 , b ) 1 / 3 , c ) 2 , d ) 9 , e ) 1 / 9

"28 = ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 ) / ( a 1 + a 2 + a 3 ) factorize the same terms 28 = 1 + ( a 4 + a 5 + a 6 ) / ( a 1 + a 2 + a 3 ) write every term with respect to r a 1 = a 1 a 2 = a 1 * r ^ 1 a 3 = a 1 * r ^ 2 . . . . . . . . . 28 = 1 + ( a 1 ( r ^ 3 + r ^ 4 + r ^ 5 ) ) / ( a 1 ( 1 + r ^ 1 + r ^ 2 ) ) 27 = ( r ^ 3 ( 1 + r ^ 1 + r ^ 2 ) ) / ( ( 1 + r ^ 1 + r ^ 2 ) ) 27 = r ^ 3 r = 3 a"

Break down and solve the equation step by step.

a watch was sold at a loss of 10 % . if it was sold for rs . 540 more , there would have been a gain of 8 % . what is the cost price ? a ) 1000 , b ) 2998 , c ) 3000 , d ) 2788 , e ) 2991

"90 % 108 % - - - - - - - - 18 % - - - - 540 100 % - - - - ? = > rs . 3000 answer : c"

Provide a detailed solution to the given mathematical problem.

calculate how much time could be saved if the train would run at its usual speed , given that it ran at 5 / 7 of its own speed and got to its destination in 15 hours ? a ) 1.29 hrs , b ) 2.29 hrs , c ) 4.29 hrs , d ) 5.29 hrs , e ) 3.29 hrs

"new speed = 5 / 7 of usual speed new time = 5 / 7 of usual time 5 / 7 of usual time = 15 hrs usual time = 15 * 5 / 7 = 10.71 hrs time saved = 15 - 10.71 = 4.29 hrs answer is c"

Provide a detailed solution to the given mathematical problem.

convert 0.35 in to a vulgar fraction ? a ) 18 / 50 , b ) 16 / 50 , c ) 7 / 20 , d ) 19 / 50 , e ) none

"answer 0.35 = 35 / 100 = 7 / 20 correct option : c"

Provide a detailed solution to the given mathematical problem.

5 n + 2 > 17 and 7 n - 5 < 44 ; n must be between which numbers ? a ) 1 and 8 , b ) 2 and 6 , c ) 0 and 9 , d ) 3 and 7 , e ) 2 and 9

"5 n > 15 - - > n > 3 7 n < 49 - - > n < 7 3 < n < 7 answer : d"

Work through the equation methodically, ensuring clarity at each step.

the grade point average of one third of the classroom is 54 ; the grade point average of the rest is 45 . what is the grade point average of the whole class ? a ) 52 , b ) 54 , c ) 48 , d ) 58 , e ) 60

"let n = total students in class total points for 1 / 3 class = 54 n / 3 = 18 n total points for 2 / 3 class = 45 * 2 n / 3 = 30 n total points for whole class = 18 n + 30 n = 48 n 48 n total class points / n total students = 48 grade point average for total class answer : c"

Provide a detailed solution to the given mathematical problem.

in the fifth grade at parkway elementary school there are 420 students . 312 students are boys and 250 students are playing soccer . 86 % of the students that play soccer are boys . how many girl student are in parkway that is not playing soccer ? a ) 69 . , b ) 73 . , c ) 81 , d ) 91 , e ) 108

total students = 420 boys = 312 , girls = 108 total playing soccer = 250 86 % of 250 = 215 are boys who play soccer . girls who play soccer = 35 . total girls who do not play soccer = 108 - 35 = 73 . correct option : b

Work through the equation methodically, ensuring clarity at each step.

from january 1 , 1991 , to january 1 , 1993 , the number of people enrolled in health maintenance organizations increased by 15 percent . the enrollment on january 1 , 1993 , was 60 million . how many million people , to the nearest million , were enrolled in health maintenance organizations on january 1 , 1991 ? a ) 38 , b ) 39 , c ) 40 , d ) 41 , e ) 52

"1.15 x = 60 - - > 23 / 20 * x = 60 - - > x = 60 * 20 / 23 = 1200 / 23 = ~ 52 . answer : e ."

Explain the solution process for the given equation.

a box contains 6 pairs of shoes ( 12 shoes in total ) . if two shoes are selected at random , what it is the probability that they are matching shoes ? a ) 1 / 11 , b ) 1 / 20 , c ) 1 / 19 , d ) 1 / 10 , e ) 1 / 9

the problem with your solution is that we do n' t choose 1 shoe from 12 , but rather choose the needed one after we just took one and need the second to be the pair of it . so , the probability would simply be : 1 / 1 * 1 / 11 as after taking one at random there are 11 shoes left and only one is the pair of the first one ) = 1 / 11 answer : a .

Solve the problem systematically, showing each step clearly.

the average of first 25 prime numbers is ? a ) 9.9 , b ) 8.8 , c ) 5.6 , d ) 11.9 , e ) 11.11

"explanation : average = ( 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 / 9 = 100 / 9 = 11.11 ( approx ) answer is e"

Work through the equation methodically, ensuring clarity at each step.

of 60 children , 30 are happy , 10 are sad , and 20 are neither happy nor sad . there are 17 boys and 43 girls . if there are 6 happy boys and 4 sad girls , how many boys are neither happy nor sad ? a ) 5 , b ) 4 , c ) 6 , d ) 8 , e ) 10

"venn diagrams are useful for multiple values of a single variable e . g . state of mind - happy / sad / neither . when you have two or more variables such as here where you have gender - boy / girl too , it becomes unwieldy . in this case , either use the table or logic . table method is shown above ; here is how you will use logic : there are 6 happy boys . there are 4 sad girls but total 10 sad children . so rest 6 sad children must be sad boys . we have 6 happy boys and 6 sad boys . total we have 17 boys . so 17 - 6 - 6 = 5 boys must be neither happy nor sad . answer ( a )"

Work through the equation methodically, ensuring clarity at each step.

a man sells an article at a profit of 25 % . if he had bought it at 20 % less and sold it for rs . 10.50 less , he would have gained 30 % . find the cost of the article . a ) 30 , b ) 40 , c ) 50 , d ) 60 , e ) 70

let c . p = 100 gain = 25 % s . p = 125 supposed c . p = 80 gain = 30 % s . p = ( 130 * 80 ) / 100 = 104 diff = ( 125 - 104 ) = 21 diff 21 when c . p = 100 then diff 10.50 when c . p = ( 100 * 10.50 ) / 21 = 50 answer : c

Break down and solve the equation step by step.

what is the least possible value of expression e = ( x - 1 ) ( x - 3 ) ( x - 4 ) ( x - 6 ) + 10 for real values of x ? a ) 1 , b ) 10 , c ) 9 , d ) 0 , e ) 8

explanation : e = ( x - 1 ) ( x - 6 ) ( x - 3 ) ( x - 4 ) + 10 e = ( x 2 - 7 x + 6 ) ( x 2 - 7 x + 12 ) + 10 let x 2 - 7 x + 6 = y e = y 2 + 6 y + 10 e = ( y + 3 ) 2 + 1 minimum value = 1 , when y = - 3 answer : a

Provide a detailed solution to the given mathematical problem.

in a rectangular coordinate system , what is the area of a rhombus whose vertices have the coordinates ( 0 , 5.5 ) , ( 8 , 0 ) , ( 0 , - 5.5 ) , ( - 8 , 0 ) ? ['a ) 56','b ) 88','c ) 112','d ) 116','e ) 120']

ares of rhombus = 1 / 2 * d 1 * d 2 length of 1 st diagonal = 8 + 8 = 16 length of 2 nd diagonal = 5.5 + 5.5 = 11 area = 1 / 2 * 16 * 11 = 88 b is the answer

Solve the problem systematically, showing each step clearly.

a train moves at average speed of 40 kmph reaches its destination on time . what its average speed becomes 35 kmph , then it reaches its destination 15 late . find the length of the journey ? a ) 70 km , b ) 85 km , c ) 38 km , d ) 14 km , e ) 1 km

difference between timings = 15 mis = 1 / 4 hr let the length of the journey be x km . then x / 35 - x / 40 = 1 / 4 8 x - 7 x = 70 x = 70 km answer ( a )

Break down and solve the equation step by step.

1370 , 1320 , 1070 , - 180 ? a ) - 6430 , b ) - 6530 , c ) - 6630 , d ) - 6730 , e ) - 6830

1370 , 13201070 , - 180 , ? 1370 - 50 * ( 5 ^ 0 ) = 1320 1320 - 50 * ( 5 ^ 1 ) = 1070 1070 - 50 * ( 5 ^ 2 ) = - 180 - 180 - 50 * ( 5 ^ 3 ) = - 6430 answer : a .

Work through the equation methodically, ensuring clarity at each step.

if x is less than y by 45 % then y exceed x by : a ) 33.33 % , b ) 25 % , c ) 81.81 % , d ) 66.66 % , e ) none of these

"using formula ( x / ( 100 - x ) * 100 ) where x is percentage decrease ( here it is 25 % ) = > 45 / ( 100 - 45 ) * 100 = 81.81 % answer : c"

Work through the equation methodically, ensuring clarity at each step.

the ratio between the present ages of a and b is 4.5 : 3 respectively . the ratio between a' s age 4 years ago and b' s age 4 years hence is 1 : 1 . what is the ratio between a' s age 4 years hence and b' s age 4 years ago ? a ) 3 : 4 , b ) 3 : 0 , c ) 3 : 1 , d ) 7 : 3 , e ) 3 : 7

"let the present ages of a and b be 4.5 x and 3 x years respectively . then , ( 4.5 x - 4 ) / ( 3 x + 4 ) = 1 / 1 1.5 x = 8 = > x = 5.33 required ratio = ( 5 x + 4 ) : ( 3 x - 4 ) = 28 : 12 = 7 : 3 . answer : d"

Break down and solve the equation step by step.

two pipes a and b can fill a tank in 30 and 60 minutes respectively . if both the pipes are used together , then how long will it take to fill the tank ? a ) 10 , b ) 20 , c ) 30 , d ) 24 , e ) 25

"part filled by a in 1 min . = 1 / 30 part filled by b in 1 min . = 1 / 60 part filled by ( a + b ) in 1 min . = 1 / 30 + 1 / 60 = 1 / 20 . both the pipes can fill the tank in 20 minutes . answer : b"

Solve the problem systematically, showing each step clearly.

there are 8 stores in town that had a total of 24 visitors on a particular day . however , only 12 people went shopping that day ; some people visited more than one store . if 9 people visited exactly two stores each , and everyone visited at least one store , what is the largest number of stores anyone could have visited ? a ) 2 , b ) 3 , c ) 4 , d ) 5 , e ) 6

"9 people visited 2 stores each for 18 visits . to maximize the number of stores that one person visited , let' s assume that 2 people visited 1 store each . the number of remaining visits is 24 - 18 - 2 = 4 , which is the maximum that one person could have visited . the answer is c ."

Explain the solution process for the given equation.

how many positive integers less than 6,000 are there in which the sum of the digits equals 5 ? a ) 65 , b ) 62 , c ) 61 , d ) 56 , e ) 58

"basically , the question asks how many 4 digit numbers ( including those in the form 0 xxx , 00 xx , and 000 x ) have digits which add up to 5 . think about the question this way : we know that there is a total of 5 to be spread among the 4 digits , we just have to determine the number of ways it can be spread . let x represent a sum of 1 , and | represent a seperator between two digits . as a result , we will have 5 x' s ( digits add up to the 5 ) , and 3 |' s ( 3 digit seperators ) . so , for example : xx | x | x | x = 2111 | | xxx | xx = 0032 etc . there are 8 c 3 ways to determine where to place the separators . hence , the answer is 8 c 3 = 56 . d"

Provide a detailed solution to the given mathematical problem.

if the charge of staying in a student youth hostel $ 18.00 / day for the first week , and $ 11.00 / day for each additional week , how much does it cost to stay for 23 days ? a ) $ 160 , b ) $ 176 , c ) $ 282 , d ) $ 302 , e ) $ 286

"total number of days of stay = 23 charge of staying in first week = 18 * 7 = 126 $ charge of staying for additional days = ( 23 - 7 ) * 11 = 16 * 11 = 176 $ total charge = 126 + 176 = 302 $ answer d"

Work through the equation methodically, ensuring clarity at each step.

a certain number of men can finish a piece of work in 10 days . however , if there were 10 less men it will take 10 days more for the work to be finished . how many men were there originally ? a ) 10 men , b ) 30 men , c ) 40 men , d ) 20 men , e ) none of these

"let x be the number of men originally thus , 10 x = ( 10 + 10 ) ( x - 10 ) 10 x = 20 ( x - 10 ) x = 2 ( x - 10 ) x = 20 answer : d"

Break down and solve the equation step by step.

in a family 15 people eat only vegetarian , 8 people eat only non veg . , 11 people eat both veg and non veg . . how many people eat veg in the family ? a ) 20 , b ) 11 , c ) 23 , d ) 26 , e ) 21

"total people eat veg = only veg + both veg and non veg total = 15 + 11 = 26 answer = d"

Solve the problem systematically, showing each step clearly.

a math teacher has 24 cards , each of which is in the shape of a geometric figure . half of the cards are rectangles , and a third of the cards are rhombuses . if 8 cards are squares , what is the maximum possible number of cards that re circles . a ) 7 , b ) 10 , c ) 11 , d ) 12 , e ) 13

"a square is a special kind of rhombus ( sides are perpendicular ) a square is a special kind of rectangles ( sides with same length ) among the 24 cards with have : 15 rectangles 10 rhombus 8 squares among the 15 rectangles , there could be 8 special ones ( with sides of same length ) that are squares . that lets at least 7 rectangles that are not square . among the 10 rectangles , there could be 8 special ones ( with sides perpendicular ) that are squares . that lets at least 2 rhombus that are not square . we have 8 squares . so the minimum different cards that represent a square , a rhombus or a rectangle is 2 + 7 + 8 = 17 which means that the maximum number of circles that you could have is 24 - 17 = 7 answer ( a )"

Explain the solution process for the given equation.

excluding stoppages , the speed of a bus is 54 kmph and including stoppages , it is 36 kmph . for how many minutes does the bus stop per hour ? a ) 20 min , b ) 8 min , c ) 9 min , d ) 10 min , e ) 11 min

"due to stoppages , it covers 18 km less . time taken to cover 18 km = ( 18 / 54 ) x 60 = 20 min answer : a"

Explain the solution process for the given equation.

what is the greatest 6 - digit number when divided by 8 , 9 , and 10 leaves a remainder of 6 , 7 , and 8 respectively ? a ) 456780 , b ) 678918 , c ) 997479 , d ) 997916 , e ) 997920

when you divide a positive integer by 10 , the remainder will just be the units digit . we know the remainder is 8 when we divide by 10 , so b is the only possible answer .

Provide a detailed solution to the given mathematical problem.

if x is equal to the sum of the integers from 30 to 50 , inclusive , and y is the number of even integers from 30 to 50 , inclusive , what is the value of x + y ? a ) 810 , b ) 811 , c ) 830 , d ) 850 , e ) 851

since the integers from 30 - 50 are consecutive we can use the ` ` median x # of terms formula'' to determine x . there are 21 terms and the median is 40 . this gives a total of 21 x 40 = 840 . there are 11 even integers from 30 - 50 = y . therefore , x + y = 851 . answer : e

Work through the equation methodically, ensuring clarity at each step.

a man has some hens and cows . if the number of heads be 50 and the number of feet equals 144 , then the number of hens will be a ) 22 , b ) 24 , c ) 26 , d ) 20 , e ) 28

"explanation : let number of hens = h and number of cows = c number of heads = 50 = > h + c = 48 - - - ( equation 1 ) number of feet = 144 = > 2 h + 4 c = 144 = > h + 2 c = 72 - - - ( equation 2 ) ( equation 2 ) - ( equation 1 ) gives 2 c - c = 72 - 50 = > c = 22 substituting the value of c in equation 1 , we get h + 22 = 50 = > h = 50 - 22 = 28 i . e . , number of hens = 28 answer : e"

Work through the equation methodically, ensuring clarity at each step.

for how many values of q , is | | | q - 5 | - 10 | - 5 | = 2 ? ( those ls are mods ) a ) 0 , b ) 2 , c ) 4 , d ) 8 , e ) more than 8

i think its 8 | | | q - 5 | - 10 | - 5 | = 2 let | q - 5 | = a which makes above | | a - 10 | - 5 | = 2 let | a - 10 | = b which makes | b - 5 | = 2 now for the above b can take 3 , 7 for every b = 3 a can have 13 , 7 and for b = 7 a can have 17 and 3 so' a' has four solutions 13 , 7 , 17 and 3 for a = 13 ; q has 18 or - 8 thus has 2 for every combination hence 4 x 2 = 8 answer d

Work through the equation methodically, ensuring clarity at each step.

the size of a television screen is given as the length of the screen' s diagonal . if the screens were flat , then the area of a square 21 - inch screen would be how many square inches greater than the area of a square 17 - inch screen ? a ) 2 , b ) 4 , c ) 16 , d ) 38 , e ) 76

"pythogoras will help here ! let the sides be x and diagonal be d then d ^ 2 = 2 x ^ 2 and area = x ^ 2 now plug in the given diagonal values to find x values and then subtract the areas ans will be 21 ^ 2 / 2 - 17 ^ 2 / 2 = 152 / 2 = 76 ans e ."

Provide a detailed solution to the given mathematical problem.

if a car went the first third of the distance at 80 kmh , the second third at 24 kmh , and the last third at 54 kmh , what was the average speed of the car for the entire trip ? a ) 36 kmh , b ) 40 kmh , c ) 41 kmh , d ) 44 kmh , e ) 50 kmh

"assume d / 3 = 2160 ( this number is convenient because it is divisible by 80 , 24 and 54 ) so : 2160 = 80 * t 1 = 27 hrs 2160 = 24 * t 2 = 90 hrs 2160 = 54 * t 3 = 40 hrs t = t 1 + t 2 + t 3 = 157 hrs d = rt ( 240 * 3 ) = r * 157 r = 41.27 answer : c"

Provide a detailed solution to the given mathematical problem.

if | 20 x - 10 | = 90 , then find the product of the values of x ? a ) - 45 , b ) 50 , c ) - 62 , d ) 35 , e ) - 20

"| 20 x - 10 | = 90 20 x - 10 = 90 or 20 x - 10 = - 90 20 x = 100 or 20 x = - 80 x = 5 or x = - 4 product = - 4 * 5 = - 20 answer is e"

Provide a detailed solution to the given mathematical problem.

how many even integers n , such that 20 < = n < = 180 are of the form 3 k + 4 , where k is any natural number ? a ) 24 , b ) 27 , c ) 30 , d ) 33 , e ) 36

the first number is 22 = 16 + 6 ( 1 ) . we can continue adding 6 to make a list : 22 , 28 , 34 , . . . the last number is 278 = 16 + 6 ( 27 ) there are 27 numbers in the list . the answer is b .

Explain the solution process for the given equation.

find the area of trapezium whose parallel sides are 20 cm and 18 cm long , and the distance between them is 13 cm . a ) 335 cm 2 , b ) 885 cm 2 , c ) 285 cm 2 , d ) 247 cm 2 , e ) 725 cm 2

"area of a trapezium = 1 / 2 ( sum of parallel sides ) * ( perpendicular distance between them ) = 1 / 2 ( 20 + 18 ) * ( 13 ) = 247 cm 2 answer : d"

Break down and solve the equation step by step.

the length of a bridge in meters , which a train 80 - meters long and traveling at 45 km / hr can cross in 30 seconds is ? a ) 215 , b ) 235 , c ) 255 , d ) 275 , e ) 295

"45 km / h = 45000 m / 3600 s = 12.5 m / s in 30 seconds , the train can go 30 ( 12.5 ) = 375 meters let x be the length of the bridge . x + 80 = 375 meters x = 295 meters the answer is e ."

Explain the solution process for the given equation.

an empty wooden vessel weighs 14 % of its total weight when filled with paint . if the weight of a partially filled vessel is one half that of a completely filled vessel , what fraction of the vessel is filled . a ) 6 / 13 , b ) 5 / 9 , c ) 1 / 24 , d ) 4 / 9 , e ) 2 / 5

"an empty wooden vessel weighs 14 % of its total weight when filled with paint : vessel = 0.14 ( vessel + paint ) ; 14 v = v + p ( so the weight of completely filled vessel is 14 v ) p = 13 v ( so the weight of the paint when the vessels is completely filled is 13 v ) . the weight of a partially filled vessel is one half that of a completely filled vessel : v + p' = 1 / 2 * 14 v ; p' = 6 v ( so the weight of the paint when the vessels is partially filled is 6 v ) . what fraction of the vessel is filled ? so , we need to find the ratio of the weight of the paint when the vessel iscompletely filledto the weight of the paint when the vessel ispartially filled : p' / p = 6 v / 13 v = 6 / 13 . answer : a ."

Break down and solve the equation step by step.

tabby is training for a triathlon . she swims at a speed of 1 mile per hour . she runs at a speed of 8 miles per hour . she wants to figure out her average speed for these two events . what is the correct answer for her ? a ) 8 mph , b ) 5.25 mph , c ) 3.5 mph , d ) 4.5 mph , e ) 0.5 mph

"( 1 mph + 8 mph ) / 2 = 4.5 mph correct option is : d"

Explain the solution process for the given equation.

how many 3 - digits number are palindromic numbers ? a palindromic number reads the same forward and backward , example 121 . a ) 90 , b ) 610 , c ) 729 , d ) 900 , e ) 1000

"take the task of building palindromes and break it intostages . stage 1 : select the hundreds digit we can choose 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , or 9 so , we can complete stage 1 in 9 ways stage 2 : select the tens digit we can choose 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , or 9 so , we can complete stage 2 in 10 ways important : at this point , the remaining digits are alreadylocked in . stage 3 : select the units digit this digit must be the same as the thousands digit ( which we already chose in stage 2 ) so , we can complete this stage in 1 way . by thefundamental counting principle ( fcp ) , we can complete all 5 stages ( and thus build a 3 - digit palindrome ) in ( 9 ) ( 10 ) ( 1 ) ways ( = 900 ways ) answer : a"

Work through the equation methodically, ensuring clarity at each step.

a cistern can be filled by a tap in 4 hours while it can be emptied by another tap in 10 hours . if both the taps are opened simultaneously , then after how much time will the cistern get filled ? a ) 2.9 hrs , b ) 8.9 hrs , c ) 2.9 hrs , d ) 6.7 hrs , e ) 8.6 hrs

"net part filled in 1 hour = ( 1 / 4 - 1 / 10 ) = 3 / 20 the cistern will be filled in 20 / 3 hrs i . e . , 6.7 hrs . answer : d"

Work through the equation methodically, ensuring clarity at each step.

the ratio of money with ram and gopal is 7 : 17 and that with gopal and krishan is 7 : 17 . if ram has rs . 637 , krishan has ? a ) s . 2890 , b ) s . 2330 , c ) s . 3757 , d ) s . 1620 , e ) s . 2680

"ram : gopal = 7 : 17 = 49 : 119 gopal : krishan = 7 : 17 = 119 : 289 ram : gopal : krishan = 49 : 119 : 289 ram : krishan = 49 : 289 thus , 49 : 289 = 637 : n & there n = 289 x 637 / 49 = rs . 3757 answer : c"

Solve the problem systematically, showing each step clearly.

a theater charges $ 12 for seats in the orchestra and $ 8 for seats in the balcony . on a certain night , a total of 350 tickets were sold for a total cost of $ 3,320 . how many more tickets were sold that night for seats in the balcony than for seats in the orchestra ? a ) 90 , b ) 110 , c ) 120 , d ) 130 , e ) 220

"orchestra seats - a balcony seats - b a + b = 350 and 12 a + 8 b = 3320 solving equations simultaneously ( multiply equation 1 with 8 and subtract from second equation ) 4 a = 3320 - 8 * 350 = 3320 - 2800 = 520 i . e . a = 130 and b = 350 - 130 = 220 more seats in balcony than orchestra = b - a = 220 - 130 = 90 answer : option a"

Work through the equation methodically, ensuring clarity at each step.

the area of an isosceles trapezoid with sides of length 5 and bases of length 8 and 14 is ? a ) 22 , b ) 32 , c ) 44 , d ) 55 , e ) 56

"we can divide the trapezoid into two triangles and a rectangle sides of triangle , t 1 - - > h 1 = 5 , b 1 = 3 , s 1 = root ( 25 - 9 ) = 4 = s 2 of triangle t 2 area of triangle = ( 1 / 2 ) * 4 * 3 = 6 side of the rectangle = 8 * 4 = 32 total area of the trapezoid = 32 + 6 + 6 = 44 answer : c"

Provide a detailed solution to the given mathematical problem.

if 20 % of a number = 1000 , then 120 % of that number will be ? a ) 20 , b ) 120 , c ) 360 , d ) 6000 , e ) 820

"let the number x . then , 20 % of x = 1000 x = ( 1000 * 100 ) / 20 = 5000 120 % of x = ( 120 / 100 * 5000 ) = 6000 . answer : d"

Solve the problem systematically, showing each step clearly.

two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 26 seconds . the ratio of their speeds is ? a ) 3 / 1 , b ) 9 / 1 , c ) 3 / 3 , d ) 3 / 5 , e ) 5 / 2

"let the speeds of the two trains be x m / sec and y m / sec respectively . then , length of the first train = 27 x meters , and length of the second train = 17 y meters . ( 27 x + 17 y ) / ( x + y ) = 26 = = > 27 x + 17 y = 26 x + 26 y = = > 1 x = 9 y = = > x / y = 9 / 1 . answer : b"

Break down and solve the equation step by step.

a train covers a distance of 9 km in 6 min . if it takes 5 sec to pass a telegraph post , then the length of the train is ? a ) 100 , b ) 105 , c ) 110 , d ) 120 , e ) 125

"speed = ( 9 / 6 * 60 ) km / hr = ( 90 * 5 / 18 ) m / sec = 25 m / sec . length of the train = 25 * 5 = 125 m . answer : option e"

Solve the problem systematically, showing each step clearly.

a watch was sold at a loss of 12 % . if it was sold for rs . 140 more , there would have been a gain of 4 % . what is the cost price ? a ) s . 1000 , b ) s . 875 , c ) s . 1007 , d ) s . 1006 , e ) s . 1002

"explanation : 88 % 104 % - - - - - - - - 16 % - - - - 140 100 % - - - - ? = > rs . 875 answer : b"

Provide a detailed solution to the given mathematical problem.

a batsman makes a score of 83 runs in the 17 th inning and thus increases his averages by 3 . find his average after 17 th inning ? a ) 19 , b ) 29 , c ) 35 , d ) 49 , e ) 59

"let the average after 17 th inning = x then average after 16 th inning = ( x - 3 ) therefore 16 ( x - 3 ) + 83 = 17 x therefore x = 35 answer : c"

Break down and solve the equation step by step.

a train traveling at 100 kmph overtakes a motorbike traveling at 64 kmph in 85 seconds . what is the length of the train in meters ? a ) 400 meters , b ) 1111 meters , c ) 160 meters , d ) 850 meters , e ) none of these

train overtakes a bike means that we are talking about total length of the train . ( train' s head is close to bike when it started and its tail crosses the bike when it overtakes the bike ) relative speed = 100 - 64 = 36 km / h = 36000 m / h time = 85 seconds distance = speed * time 36000 * 85 / 3600 = 850 meters . d is the answer .

Explain the solution process for the given equation.

25.25 / 1500 is equal to : a ) 1.012526333 , b ) 0.012625333 , c ) 0.125263333 , d ) 0.126253333 , e ) 0.016833333

"25.25 / 2000 = 2525 / 200000 = 0.016833333 answer : e"

Explain the solution process for the given equation.

the ratio , by volume , of bleach to detergent to water in a certain solution is 2 : 40 : 100 . the solution will be altered so that the ratio of bleach to detergent is tripled while the ratio of detergent to water is halved . if the altered solution will contain 200 liters of water , how many liters of detergent will it contain ? a ) 60 , b ) 55 , c ) 50 , d ) 45 , e ) 40

"b : d : w = 2 : 40 : 100 bnew / dnew = ( 1 / 3 ) * ( 2 / 40 ) = ( 1 / 60 ) dnew / wnew = ( 1 / 2 ) * ( 40 / 100 ) = ( 1 / 5 ) wnew = 200 dnew = wnew / 5 = 200 / 5 = 40 so , answer will be e"

Work through the equation methodically, ensuring clarity at each step.

the difference of two numbers is 11 . one third of their sum is 7 . what are the two numbers ? a ) 21,10 , b ) 16,5 , c ) 3,14 , d ) 11,10 , e ) 7,14

"let the two numbers be x and y . equation ( i ) : x - y = 11 equation ( ii ) : ( x + y ) / 3 = 7 solve system of equations : x + y = 21 x - y = 11 ( add equations together ) - - - 2 x = 32 - - - x = 16 16 - y = 11 - - - y = 5 since x = 16 and y = 5 , answer b ( 16,5 ) is correct ."

Work through the equation methodically, ensuring clarity at each step.

rahim bought 65 books for rs . 1150 from one shop and 50 books for rs . 920 from another . what is the average price he paid per book ? a ) rs . 17 , b ) rs . 18 , c ) rs . 12 , d ) rs . 11 , e ) rs . 10

"average price per book = ( 1150 + 920 ) / ( 65 + 50 ) = 2070 / 115 = rs . 18 answer : b"

Work through the equation methodically, ensuring clarity at each step.

if the area of a square with sides of length 4 centimeters is equal to the area of a rectangle with a width of 8 centimeters , what is the length of the rectangle , in centimeters ? a ) 4 , b ) 8 , c ) 2 , d ) 16 , e ) 18

"let length of rectangle = l 4 ^ 2 = l * 8 = > l = 16 / 8 = 2 answer c"

Provide a detailed solution to the given mathematical problem.

if 20 men can build a wall 112 metres long in 6 days , what length of a similar wall can be built by 10 men in 3 days ? a ) 65 mtr . , b ) 28 mtr , c ) 70 mtr . , d ) 78 mtr . , e ) 17 mtr .

"20 men is 6 days can build 112 metres 10 men in 3 days can build = 112 * ( 10 / 20 ) x ( 3 / 6 ) = 28 meters answer : b ."

Provide a detailed solution to the given mathematical problem.

the average age of 39 students in a group is 14 years . when teacher' s age is included to it , the average increases by one . what is the teacher' s age in years ? a ) 35 years , b ) 45 years , c ) 51 years , d ) 54 years , e ) none of these

"explanation : age of the teacher = ( 40 * 15 - 39 * 14 ) years = 54 years . answer : d"

Provide a detailed solution to the given mathematical problem.

the cost price of 12 articles is equal to the selling price of 10 articles . what is the profit percent ? a ) 10 % , b ) 20 % , c ) 30 % , d ) 40 % , e ) 50 %

"10 * sp = 12 * cp sp = 1.2 * cp the profit percent is 20 % . the answer is b ."

Solve the problem systematically, showing each step clearly.

on the first of the year , james invested x dollars at proudstar bank in an account that yields 2.2 % in interest every quarter year . at the end of the year , during which he made no additional deposits or withdrawals , he had y dollars in the account . if james had invested the same amount in an account which pays interest on a yearly basis , what must the interest rate be for james to have y dollars at the end of the year ? a ) 2.04 % , b ) 6.12 % , c ) 8 % , d ) 8.25 % , e ) 9 %

"if the interest were compounded annually instead of quarterly then in one year the interest would be 2.2 * 4 = 8.8 % . now , since the interest is compounded quarterly then there would be interest earned on interest ( very small amount ) thus the actual interest would be a little bit more than 8.8 % answer : e ."

Provide a detailed solution to the given mathematical problem.

suppose you work for a manufacturing plant that pays you $ 12.50 an hour and $ . 16 for each widget you make . how many widgets must you produce in a 40 hour week to earn $ 800 ( before payroll deductions ) ? a ) 1800 , b ) 1825 , c ) 1850 , d ) 1855 , e ) 1875

"hourly = $ 12.50 / hour totalhours = 40 piecework = $ 0.16 / widget totalpieces = ? ? ? ? total pay = ( hourly * totalhours ) + ( piecework * totalpieces ) so , put it all together : $ 800 = ( $ 12.50 * 40 ) + ( $ 0.16 * total pieces ) totalpieces = ( $ 800 - ( $ 12.50 * 40 ) ) / $ 0.16 = 1875 widgets e . 1875"

Work through the equation methodically, ensuring clarity at each step.

gold is 19 times as heavy as water and copper is 9 times as heavy as water . in what ratio should these be mixed to get an alloy 16 times as heavy as water ? a ) 1 : 2 , b ) 7 : 3 , c ) 4 : 1 , d ) 5 : 2 , e ) 6 : 5

"g = 19 w c = 9 w let 1 gm of gold mixed with x gm of copper to get 1 + x gm of the alloy 1 gm gold + x gm copper = x + 1 gm of alloy 19 w + 9 wx = x + 1 * 16 w 19 + 9 x = 16 ( x + 1 ) x = 3 / 7 ratio of gold with copper = 1 : 3 / 7 = 7 : 3 answer is b"

Break down and solve the equation step by step.

if a randomly selected non - negative single digit integer is added to { 2 , 3 , 5 , 8 } . what is the probability that the median of the set will increase but the range still remains the same ? a ) 0.2 , b ) 0.3 , c ) 0.4 , d ) 0.5 , e ) 0.6

we are selecting from non - negative single digit integers , so from { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . these 10 digits represent the total number of outcomes . hence , the total number of outcomes is 10 . we need to find the probability that the median of the set will increase but the range still remains the same . the median of the set is ( 3 + 5 ) / 2 = 4 , thus the number selected must be 5 or greater . for the range to remain the same , the number must be between 2 and 8 inclusive . to satisfy both conditions , the number selected must be 5 , 6 , 7 , or 8 . the probability is 4 / 10 = 0.4 the answer is c .

Solve the problem systematically, showing each step clearly.

two employees m and n are paid a total of $ 616 per week by their employer . if m is paid 120 percent of the salary paid to n , how much is n paid per week ? a ) $ 270 , b ) $ 280 , c ) $ 290 , d ) $ 300 , e ) $ 310

"1.2 n + n = 616 2.2 n = 616 n = 280 the answer is b ."

Work through the equation methodically, ensuring clarity at each step.

with a uniform speed a car covers the distance in 8 hours . had the speed been increased by 4 km / hr , the same distance could have been covered in 7 1 / 2 hours . what is the distance covered ? a ) 347 , b ) 480 , c ) 288 , d ) 278 , e ) 131

"let the distance be x km . then , x / ( 7 1 / 2 ) - x / 8 = 4 2 x / 15 - x / 8 = 4 = > x = 480 km . answer : b"

Break down and solve the equation step by step.

if 9 a - b = 10 b + 70 = - 12 b - 2 a , what is the value of 11 a + 11 b ? a ) - 4 , b ) - 2 , c ) 0 , d ) 2 , e ) 4

"( i ) 9 a - 11 b = 70 ( ii ) 2 a + 22 b = - 70 adding ( i ) and ( ii ) : 11 a + 11 b = 0 the answer is c ."

Solve the problem systematically, showing each step clearly.

a man can row downstream at the rate of 16 km / hr and upstream at 8 km / hr . find man' s rate in still water and the rate of current ? a ) 12,4 , b ) 6,3 , c ) 9,3 , d ) 6,6 , e ) none of these

"explanation : rate of still water = 1 / 2 ( 16 + 8 ) = 12 km / hr rate of current = 1 / 2 ( 16 - 8 ) = 4 km / hr answer : option a"

Work through the equation methodically, ensuring clarity at each step.

find the ones digit of 73 ^ 352 a ) 1 , b ) 5 , c ) 6 , d ) 7 , e ) 9

"the units digit of 73 ^ 352 will be the same as the units digit of 3 ^ 352 . 3 ^ 1 = 3 - - > the units digit is 3 ; 3 ^ 2 = 9 - - > the units digit is 9 ; 3 ^ 3 = 27 - - > the units digit is 7 ; 3 ^ 4 = 81 - - > the units digit is 1 ; 3 ^ 5 = 243 - - > the units digit is 3 again ; . . . so , as you can see the units digit repeats in blocks of 4 : { 3 , 9 , 7 , 1 } , { 3 , 9 , 7 , 1 } , . . . now , since 352 = ( multiple of 4 ) + 0 , since 3 is an odd number , the units digit of 3 ^ 352 will be 1 . answer : a ."

Explain the solution process for the given equation.

- 64 x 29 + 165 = ? a ) 2436 , b ) 2801 , c ) - 2801 , d ) - 1601 , e ) none of them

"given exp . = - 64 x ( 30 - 1 ) + 165 = - ( 64 x 30 ) + 64 + 165 = - 1830 + 229 = - 1601 answer is d"

Explain the solution process for the given equation.

a watch was sold at a loss of 10 % . if it was sold for rs . 168 more , there would have been a gain of 4 % . what is the cost price ? a ) s : 1000 , b ) s : 1067 , c ) s : 1200 , d ) s : 1028 , e ) s : 1027

"90 % 104 % - - - - - - - - 14 % - - - - 168 100 % - - - - ? = > rs : 1200 answer : c"

Explain the solution process for the given equation.

how many integers are divisible by 5 between 10 ! and 10 ! + 20 inclusive ? a ) 5 , b ) 7 , c ) 8 , d ) 9 , e ) 10

a - 7 10 ! is divisible by 5 there are 4 numbers between 10 ! and 10 ! + 20 that are divisible by 5 . hence 5

Explain the solution process for the given equation.

in a flight of 600 km , an aircraft was slowed down due to bad weather . its average speed for the trip was reduced by 200 km / hr and the time of flight increased by 30 minutes . the duration of the flight is a ) 3 hr . , b ) 5 hr . , c ) 2 hr . , d ) 7 hr . , e ) 1 hr .

"e 1 hr . let the duration of the flight be x hours . then , 600 / x - 600 / 0.5 x = 200 - - > 600 / x - 1200 / ( 2 x + 1 ) = 200 - - > x ( 2 x + 1 ) = 3 - - > 2 x ^ 2 + x - 3 = 0 - - > ( 2 x + 3 ) ( x - 1 ) = 0 - - > x = 1 hr ."

Provide a detailed solution to the given mathematical problem.

martin bought 10 concert tickets , some at the full price of $ 2.00 per ticket , and some at a discounted price of $ 1.60 per ticket . if he spent a total of $ 17.60 , how many discounted tickets did he buy ? a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7

"let x be the number of tickets he bought at $ 2 per ticket . then 2 x + ( 10 - x ) 1.6 = 17.6 0.4 x = 1.6 = > x = 4 discounted tickets = 10 - x = 6 ans : d"

Solve the problem systematically, showing each step clearly.

a man can do a piece of work in 5 days , but with the help of his son , he can do it in 3 days . in what time can the son do it alone ? a ) 13 / 2 , b ) 7 , c ) 15 / 2 , d ) 8 , e ) 9

"son' s 1 day' s work = ( 1 / 3 ) - ( 1 / 5 ) = 2 / 5 the son alone can do the work in 15 / 2 days answer is c"

Break down and solve the equation step by step.

a cycle is bought for rs . 670 and sold for rs . 780 , find the gain percent ? a ) 70 % , b ) 20 % , c ) 16 % , d ) 19 % , e ) 80 %

"explanation : 670 - - - - 110 100 - - - - ? = > 16 % answer : c"

Solve the problem systematically, showing each step clearly.

in a certain corporation , there are 300 male employees and 150 female employees . it is known that 10 % of the male employees have advanced degrees and 40 % of the females have advanced degrees . if one of the 450 employees is chosen at random , what is the probability this employee has an advanced degree or is female ? a ) 1 / 5 , b ) 5 / 15 , c ) 6 / 15 , d ) 3 / 10 , e ) 4 / 15

p ( female ) = 150 / 450 = 1 / 3 p ( male with advanced degree ) = 0.1 * 300 / 450 = 30 / 450 = 1 / 15 the sum of the probabilities is 6 / 15 = 2 / 5 the answer is c .

Explain the solution process for the given equation.

if ( 2 to the x ) - ( 2 to the ( x - 2 ) ) = 3 ( 2 to the 10 ) , what is the value of x ? a ) 9 , b ) 11 , c ) 13 , d ) 12 , e ) 17

"( 2 to the power x ) - ( 2 to the power ( x - 2 ) ) = 3 ( 2 to the power 10 ) 2 ^ x - 2 ^ ( x - 2 ) = 3 . 2 ^ 10 hence x = 12 . answer is d"

Work through the equation methodically, ensuring clarity at each step.

a worker' s daily wage is increased by 50 % and the new wage is $ 51 per day . what was the worker' s daily wage before the increase ? a ) $ 28 , b ) $ 32 , c ) $ 34 , d ) $ 35 , e ) $ 40

"let x be the daily wage before the increase . 1.5 x = $ 51 x = $ 34 the answer is c ."

Break down and solve the equation step by step.

the manager of a produce market purchased a quantity of tomatoes for $ 0.80 per pound . due to improper handling , 20 percent of the tomatoes , by weight , were ruined and discarded . at what price per pound should the manager sell the remaining tomatoes if she wishes to make a profit on the sale of the tomatoes equal to 8 percent of the cost of the tomatoes . a ) $ 0.94 , b ) $ 0.96 , c ) $ 0.98 , d ) $ 1.00 , e ) $ 1.09

"assume the manager bought 100 tomatoes . cost price = 80 given : 20 % are damaged - - > available tomatoes to sell = 80 80 * x - 80 = 0.08 * 80 80 x - 80 = 6.4 80 x = 86.64 x = 86.64 / 80 = 87 / 80 ( approx ) = 1.0875 x is slightly under 1.0875 = 1.09 answer : e"

Explain the solution process for the given equation.

consider the word rotor . whichever way you read it , from left to right or from right to left , you get the same word . such a word is known as palindrome . find the maximum possible number of 5 - letter palindromes ? a ) 56678 , b ) 16789 , c ) 17576 , d ) 35142 , e ) none of these

"explanation : the first letter from the right can be chosen in 26 ways because there are 26 alphabets . having chosen this , the second letter can be chosen in 26 ways . = > the first two letters can be chosen in 26 * 26 = 676 wayshaving chosen the first two letters , the third letter can be chosen in 26 ways . = > all the three letters can be chosen in 676 * 26 = 17576 ways . it implies that the maximum possible number of five letter palindromes is 17576 because the fourth letter is the same as the second letter and the fifth letter is the same as the first letter . answer : c"