Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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\[ \frac{2}{5} \cdot \frac{5}{8} \cdot \frac{1}{4} = \frac{2 \cdot 5 \cdot 1}{5 \cdot 8 \cdot 4} = \frac{\frac{1}{)10}}{\frac{160}{16}} = \frac{1}{16} \] | This means that \( \frac{2}{5} \) of \( \frac{5}{8} \) of \( \frac{1}{4} \) of a whole unit is \( \frac{1}{16} \) of the original unit. | No |
\( \frac{4}{5} \cdot \frac{5}{6} \) | \[ \frac{\frac{2}{14}}{\frac{5}{15}} \cdot \frac{\frac{1}{15}}{\frac{6}{3}} = \frac{2 \cdot 1}{1 \cdot 3} = \frac{2}{3} \] | No |
\( \frac{35}{18} \cdot \frac{63}{105} \) | \[\n\frac{\frac{1}{\overset{17}{\overline{){35}}}}}{\frac{\underset{2}{\overline{){18}}}}{2}}\frac{\frac{7}{\overset{7}{\overline{){63}}}}}{\frac{\underset{2}{\overline{){105}}}}{\underset{3}{\overline{){21}}}}} = \frac{1 \cdot 7}{2 \cdot 3} = \frac{7}{6}\n\] | Yes |
[ \frac{13}{9} \cdot \frac{6}{39} \cdot \frac{1}{12} ] | [ \frac{1}{\frac{113}{9}} \cdot \frac{\frac{1}{12}}{\frac{16}{\frac{33}{13}}} \cdot \frac{1}{\frac{112}{6}} = \frac{1 \cdot 1 \cdot 1}{9 \cdot 1 \cdot 6} = \frac{1}{54} ] | No |
\( 1\frac{1}{8} \cdot 4\frac{2}{3} \) | Convert each mixed number to an improper fraction.\n\n\[ 1\frac{1}{8} = \frac{8 \cdot 1 + 1}{8} = \frac{9}{8} \]\n\n\[ 4\frac{2}{3} = \frac{4 \cdot 3 + 2}{3} = \frac{14}{3} \]\n\n\[ \frac{\frac{3}{19}}{\frac{8}{18}} \cdot \frac{\frac{7}{14}}{\frac{13}{13}} = \frac{3 \cdot 7}{4 \cdot 1} = \frac{21}{4} = 5\frac{1}{4} \] | No |
\( {16} \cdot 8\frac{1}{5} \) | Convert \( 8\frac{1}{5} \) to an improper fraction.\n\n\[ 8\frac{1}{5} = \frac{5 \cdot 8 + 1}{5} = \frac{41}{5} \]\n\n\[ \frac{16}{1} \cdot \frac{41}{5} \]\n\nThere are no common factors to divide out.\n\n\[ \frac{16}{1} \cdot \frac{41}{5} = \frac{{16} \cdot {41}}{1 \cdot 5} = \frac{656}{5} = {131}\frac{1}{5} \] | Yes |
\( 9\frac{1}{6} \cdot {12}\frac{3}{5} \) | Convert to improper fractions.\n\n\[ 9\frac{1}{6} = \frac{6 \cdot 9 + 1}{6} = \frac{55}{6} \]\n\n\( {12}\frac{3}{5} = \frac{5 \cdot {12} + 3}{5} = \frac{63}{5} \)\n\n\[ \frac{11}{\frac{55}{0}} \cdot \frac{21}{\frac{63}{5}} = \frac{{11} \cdot {21}}{2 \cdot 1} = \frac{231}{2} = {115}\frac{1}{2} \] | No |
\[ \frac{11}{8} \cdot 4\frac{1}{2} \cdot 3\frac{1}{8} = \frac{11}{8} \cdot \frac{\frac{3}{9}}{\frac{9}{12}} \cdot \frac{\frac{5}{10}}{\frac{13}{13}} \] | \[ = \;\frac{{11} \cdot 3 \cdot 5}{8 \cdot 1 \cdot 1} = \frac{165}{8} = {20}\frac{5}{8} \] | Yes |
\[ {\left( \frac{1}{6}\right) }^{2} = \frac{1}{6} \cdot \frac{1}{6} = \frac{1 \cdot 1}{6 \cdot 6} = \frac{1}{36} \] | \[ {\left( \frac{1}{6}\right) }^{2} = \frac{1}{6} \cdot \frac{1}{6} = \frac{1 \cdot 1}{6 \cdot 6} = \frac{1}{36} \] | Yes |
\( \sqrt{\frac{9}{100}} \) . We’re looking for a number, call it ?, such that when it is squared, \( \frac{9}{100} \) is produced. | \[ \sqrt{\frac{9}{100}} = \frac{3}{10} \] | Yes |
\[ 4\frac{2}{5} \cdot \sqrt{\frac{100}{121}} \] | \[ \frac{\frac{2}{){22}}}{\frac{5}{5}} \cdot \frac{\frac{2}{10}}{\frac{111}{1}} = \frac{2 \cdot 2}{1 \cdot 1} = \frac{4}{1} = 4 \] \[ 4\frac{2}{5} \cdot \sqrt{\frac{100}{121}} = 4 \] | No |
\[ \underbrace{3}\text{and}\frac{4}{3} \] | \[ \frac{3}{4} \cdot \frac{4}{3} = 1 \] | Yes |
\[ \frac{1}{6} \cdot \frac{6}{1} = 1 \] | Notice that we can find the reciprocal of a nonzero number in fractional form by inverting it (exchanging positions of the numerator and denominator). | No |
\( \frac{1}{3} \div \frac{3}{4} \) | The divisor is \( \frac{3}{4} \). Its reciprocal is \( \frac{4}{3} \). Multiply \( \frac{1}{3} \) by \( \frac{4}{3} \). \( \frac{1}{3} \cdot \frac{4}{3} = \frac{1 \cdot 4}{3 \cdot 3} = \frac{4}{9} \). \[ \frac{1}{3} \div \frac{3}{4} = \frac{4}{9} \] | Yes |
\( \frac{3}{8} \div \frac{5}{4} \) | The divisor is \( \frac{5}{4} \). Its reciprocal is \( \frac{4}{5} \). Multiply \( \frac{3}{8} \) by \( \frac{4}{5} \). \[ \frac{3}{8} \cdot \frac{4}{5} = \frac{3 \cdot 4}{8 \cdot 5} = \frac{12}{40} = \frac{3}{10} \] \[ \frac{3}{8} \div \frac{5}{4} = \frac{3}{10} \] | Yes |
\( \frac{5}{6} \div \frac{5}{12} \) | The divisor is \( \frac{5}{12} \). Its reciprocal is \( \frac{12}{5} \). Multiply \( \frac{5}{6} \) by \( \frac{12}{5} \). \[ \frac{5}{6} \div \frac{5}{12} = 2 \] | Yes |
Example 4.52\n\n\( 2\frac{2}{9} \div 3\frac{1}{3} \) . Convert each mixed number to an improper fraction. | \n\n\[ 2\frac{2}{9} = \frac{9 \cdot 2 + 2}{9} = \frac{20}{9}. \]\n\n\[ 3\frac{1}{3} = \frac{3 \cdot 3 + 1}{3} = \frac{10}{3}\text{.} \]\n\n\( \frac{20}{9} \div \frac{10}{3} \) The divisor is \( \frac{10}{3} \) . Its reciprocal is \( \frac{3}{10} \) . Multiply \( \frac{20}{9} \) by \( \frac{3}{10} \) .\n\n\[ \frac{\frac... | No |
\( \frac{12}{11} \div 8 \) | First conveniently write 8 as \( \frac{8}{1} \). \( \frac{12}{11} \div \frac{8}{1} \). The divisor is \( \frac{8}{1} \). Its reciprocal is \( \frac{1}{8} \). Multiply \( \frac{12}{11} \) by \( \frac{1}{8} \). \[ \frac{12}{11} \cdot \frac{1}{8} = \frac{12 \cdot 1}{11 \cdot 8} = \frac{12}{88} = \frac{3}{22} \] \[ \frac{1... | Yes |
\( \frac{7}{8} \div \frac{21}{20} \cdot \frac{3}{35} \) | \[ \frac{7}{8} \div \frac{21}{20} \cdot \frac{3}{25} = \frac{1}{14} \] | No |
How many \( 2\frac{3}{8} \) -inch-wide packages can be placed in a box 19 inches wide? | The problem is to determine how many two and three eighths are contained in 19, that is, what is \( {19} \div 2\frac{3}{8} \) ?\n\n\( 2\frac{3}{8} = \frac{19}{8} \) Convert the divisor \( 2\frac{3}{8} \) to an improper fraction.\n\n\( {19} = \frac{19}{1} \) Write the dividend 19 as \( \frac{19}{1} \) .\n\n\( \frac{19}{... | Yes |
\( \frac{3}{7} + \frac{2}{7} \) . The denominators are the same. Add the numerators and place that sum over 7 . | \[\n\frac{3}{7} + \frac{2}{7} = \frac{3 + 2}{7} = \frac{5}{7}\n\] | Yes |
\( \frac{1}{8} + \frac{3}{8} \) | The denominators are the same. Add the numerators and place the sum over 8 . Reduce.\n\n\[ \frac{1}{8} + \frac{3}{8} = \frac{1 + 3}{8} = \frac{4}{8} = \frac{1}{2} \] | Yes |
\( \frac{4}{9} + \frac{5}{9} \) | The denominators are the same. Add the numerators and place the sum over 9 . \[ \frac{4}{9} + \frac{5}{9} = \frac{4 + 5}{9} = \frac{9}{9} = 1 \] | Yes |
\( \frac{7}{8} + \frac{5}{8} \) | The denominators are the same. Add the numerators and place the sum over 8 . \[ \frac{7}{8} + \frac{5}{8} = \frac{7 + 5}{8} = \frac{12}{8} = \frac{3}{2} \] | Yes |
To see what happens if we mistakenly add the denominators as well as the numerators, let's add\n\n\\[ \n\\frac{1}{2} + \\frac{1}{2} \n\\] | Adding the numerators and mistakenly adding the denominators produces\n\n\\[ \n\\frac{1}{2} + \\frac{1}{2} = \\frac{1 + 1}{2 + 2} = \\frac{2}{4} = \\frac{1}{2} \n\\]\n\nThis means that two \\( \\frac{1}{2} \\) ’s is the same as one \\( \\frac{1}{2} \\) . Preposterous! We do not add denominators. | Yes |
\( \frac{3}{5} - \frac{1}{5} \) | The denominators are the same. Subtract the numerators. Place the difference over 5 . \[ \frac{3}{5} - \frac{1}{5} = \frac{3 - 1}{5} = \frac{2}{5} \] | Yes |
\( \frac{8}{6} - \frac{2}{6} \) . The denominators are the same. Subtract the numerators. Place the difference over 6. | \[\n\frac{8}{6} - \frac{2}{6} = \frac{8 - 2}{6} = \frac{6}{6} = 1\n\] | Yes |
\( \frac{16}{9} - \frac{2}{9} \) . The denominators are the same. Subtract numerators and place the difference over 9 . | \[\n\frac{16}{9} - \frac{2}{9} = \frac{{16} - 2}{9} = \frac{14}{9}\n\] | Yes |
To see what happens if we mistakenly subtract the denominators, let's consider\n\n\[ \frac{7}{15} - \frac{4}{15} = \frac{7 - 4}{{15} - {15}} = \frac{3}{0} \] | We get division by zero, which is undefined. We do not subtract denominators. | Yes |
\( \frac{1}{6} + \frac{3}{4} \) . The denominators are not the same. Find the LCD of 6 and 4. | \[ \left. \begin{array}{l} 6 = 2 \cdot 3 \\ 4 = {2}^{2} \end{array}\right\} \text{The}\mathrm{{LCD}} = {2}^{2} \cdot 3 = 4 \cdot 3 = {12} \] Write each of the original fractions as a new, equivalent fraction having the common denominator 12. \[ \frac{1}{6} + \frac{3}{4} = \frac{}{12} + \frac{}{12} \] To find a new nume... | Yes |
\( \frac{1}{2} + \frac{2}{3} \) . The denominators are not the same. Find the LCD of 2 and 3 . | \[ \mathrm{{LCD}} = 2 \cdot 3 = 6 \] Write each of the original fractions as a new, equivalent fraction having the common denominator 6. \[ \frac{1}{2} + \frac{2}{3} = \frac{}{6} + \frac{}{6} \] To find a new numerator, we divide the original denominator into the LCD. Since the original denominator is being multiplied ... | Yes |
\( \frac{5}{9} - \frac{5}{12} \) . The denominators are not the same. Find the LCD of 9 and 12. | \[ \left. \begin{matrix} 9 = 3 \cdot 3 = {3}^{2} \\ {12} = 2 \cdot 6 = 2 \cdot 2 \cdot 3 = {2}^{2} \cdot 3 \end{matrix}\right\} \;\text{ LCD } = {2}^{2} \cdot {3}^{2} = 4 \cdot 9 = {36} \] \[ \frac{5}{9} - \frac{5}{12} = \frac{}{36} - \frac{}{36} \] \( {36} \div 9 = 4 \) Multiply the numerator 5 by 4 . \( {36} \div {12... | Yes |
\( \frac{5}{6} - \frac{1}{8} + \frac{7}{16} \) The denominators are not the same. Find the LCD of 6,8, and 16 | \[ 6 = \;2 \cdot 3 \] \[ 8 = \;2 \cdot 4 = 2 \cdot 2 \cdot 2 = {2}^{3}\;\} \text{The LCD is}{2}^{4} \cdot 3 = {48} \] \[ {16} = 2 \cdot 8 = 2 \cdot 2 \cdot 4 = 2 \cdot 2 \cdot 2 \cdot 2 = {2}^{4} \] \[ \frac{5}{6} - \frac{1}{8} + \frac{7}{16} = \frac{}{48} - \frac{}{48} + \frac{}{48} \] \[ {48} \div 6 = 8 \] Multiply t... | Yes |
\( 8\frac{3}{5} + 5\frac{1}{4} \) . Convert each mixed number to an improper fraction. | \[ 8\frac{3}{5} = \frac{5 \cdot 8 + 3}{5} = \frac{{40} + 3}{5} = \frac{43}{5} \] \[ 5\frac{1}{4} = \frac{4 \cdot 5 + 1}{4} = \frac{{20} + 1}{4} = \frac{21}{4} \] Now add the improper fractions \( \frac{43}{5} \) and \( \frac{21}{4} \) . \[ \frac{43}{5} + \frac{21}{4} \] The LCD \( = {20} \) . \[ \frac{43}{5} + \frac{21... | Yes |
\( 3\frac{1}{8} - \frac{5}{6} \) . Convert the mixed number to an improper fraction. | \[ 3\frac{1}{8} = \frac{3 \cdot 8 + 1}{8} = \frac{{24} + 1}{8} = \frac{25}{8} \] \[ \frac{25}{8} - \frac{5}{6}\text{The LCD} = {24}\text{.} \] \[ \frac{25}{8} - \frac{5}{6} = \frac{{25} \cdot 3}{24} - \frac{5 \cdot 4}{24} \] \[ = \;\frac{75}{24} - \frac{20}{24} \] \[ = \;\frac{{75} - {20}}{24} \] \( = \;\frac{55}{24}\;... | Yes |
Compare \( \frac{8}{9} \) and \( \frac{14}{15} \) . | Convert each fraction to an equivalent fraction with the LCD as the denominator. Find the LCD.\n\n\[ \left. \begin{matrix} 9 & = & {3}^{2} \\ {15} & = & 3 \cdot 5 \end{matrix}\right\} \text{ The }\mathrm{{LCD}} = {3}^{2} \cdot 5 = 9 \cdot 5 = {45} \]\n\n\[ \frac{8}{9} = \frac{8 \cdot 5}{45} = \frac{40}{45} \]\n\n\[ \fr... | Yes |
Write \( \frac{5}{6},\frac{7}{10} \), and \( \frac{13}{15} \) in order from smallest to largest. | Convert each fraction to an equivalent fraction with the LCD as the denominator.\n\nFind the LCD.\n\n\[ 6 = 2 \cdot 3 \]\n\n\[ {10} = 2 \cdot 5\} \text{The}\mathrm{{LCD}} = 2 \cdot 3 \cdot 5 = {30} \]\n\n\[ {15} = 3 \cdot 5 \]\n\n\[ \frac{5}{6} = \frac{5 \cdot 5}{30} = \frac{25}{30} \]\n\n\[ \frac{7}{10} = \frac{7 \cdo... | Yes |
Compare \( 8\frac{6}{7} \) and \( 6\frac{3}{4} \) . | To compare mixed numbers that have different whole number parts, we need only compare whole number parts. Since \( 6 < 8 \) ,\n\n\[ 6\frac{3}{4} < 8\frac{6}{7} \] | No |
Compare \( 4\frac{5}{8} \) and \( 4\frac{7}{12} \) | To compare mixed numbers that have the same whole number parts, we need only compare fractional parts.\n\n\[ \left. \begin{matrix} 8 & = & {2}^{3} \\ {12} & = & {2}^{2} \cdot 3 \end{matrix}\right\} \text{The LCD} = {2}^{3} \cdot 3 = 8 \cdot 3 = {24} \]\n\n\[ \frac{5}{8} = \frac{5 \cdot 3}{24} = \frac{15}{24} \]\n\n\[ \... | Yes |
\( \frac{\frac{3}{8}}{\frac{15}{16}} \) | Convert this complex fraction to a simple fraction by performing the indicated division.\n\n\( \frac{3}{\frac{15}{16}} = \frac{3}{8} \div \frac{15}{16}\; \) The divisor is \( \frac{15}{16} \). Invert \( \frac{15}{16} \) and multiply.\n\n\[ = \frac{\frac{1}{)3}}{\frac{)8}{1}} \cdot \frac{\frac{2}{){16}}}{\frac{){15}}{5}... | No |
\( \frac{\frac{4}{9}}{6} \) Write 6 as \( \frac{6}{1} \) and divide. | \[\n\frac{\frac{4}{9}}{\frac{6}{1}} = \;\frac{4}{9} \div \frac{6}{1}\n\]\n\n\[\n= \frac{\frac{2}{)4}}{\frac{)4}{9}} \cdot \frac{1}{\frac{)6}{)6}} = \frac{2 \cdot 1}{9 \cdot 3} = \frac{2}{27}\n\] | Yes |
\( \frac{5 + \frac{3}{4}}{46} \) Simplify the numerator. | \[ \frac{\frac{4 \cdot 5 + 3}{4}}{46} = \frac{\frac{{20} + 3}{4}}{46} = \frac{\frac{23}{4}}{46}\;\text{Write 46 as}\frac{46}{1}\text{.} \]\n\[ \frac{\frac{23}{4}}{\frac{46}{1}} = \frac{23}{4} \div \frac{46}{1} \]\n\[ = \;\frac{\frac{1}{\underset{4}{\overline{){23}}}}}{4} \cdot \frac{1}{\underset{2}{\overline{){46}}}} =... | Yes |
\[ \frac{\frac{1}{4} + \frac{3}{8}}{\frac{1}{2} + \frac{13}{24}} = \frac{\frac{2}{8} + \frac{3}{8}}{\frac{12}{24} + \frac{13}{24}} = \frac{\frac{2 + 3}{8}}{\frac{{12} + {13}}{24}} = \frac{\frac{5}{8}}{\frac{25}{24}} = \frac{5}{8} \div \frac{25}{24}} \] | \[ \frac{5}{8} \div \frac{25}{24} = \frac{\frac{1}{5}}{\frac{5}{8}} \cdot \frac{\frac{3}{24}}{\frac{25}{25}} = \frac{1 \cdot 3}{1 \cdot 5} = \frac{3}{5} \] | Yes |
Example 5.25\n\n\\[ \n\\frac{11 + \\frac{3}{10}}{4\\frac{4}{5}} \n\\] | \\[ \n\\frac{11 + \\frac{3}{10}}{4\\frac{4}{5}} = \\frac{\\frac{11 \\cdot 10 + 3}{10}}{\\frac{4 \\cdot 5 + 4}{5}} = \\frac{\\frac{110 + 3}{10}}{\\frac{20 + 4}{5}} = \\frac{\\frac{113}{10}}{\\frac{24}{5}} = \\frac{113}{10} \\div \\frac{24}{5} \n\\]\n\n\\[ \n\\frac{113}{10} \\div \\frac{24}{5} = \\frac{113}{\\frac{110}{2... | Yes |
\( \frac{1}{4} + \frac{5}{8} \cdot \frac{2}{15} \) | (a) Multiply first.\n\n\[ \frac{1}{4} + \frac{\frac{1}{5}}{\frac{8}{9}} \cdot \frac{\frac{1}{2}}{\frac{15}{3}} = \frac{1}{4} + \frac{1 \cdot 1}{4 \cdot 3} = \frac{1}{4} + \frac{1}{12} \]\n\n(b) Now perform this addition. Find the LCD.\n\n\[ \left. \begin{array}{l} 4 = {2}^{2} \\ {12} = {2}^{2} \cdot 3 \end{array}\right... | No |
\\( 2\\frac{7}{8} + \\sqrt{\\frac{25}{36}} \\div \\left( {2\\frac{1}{2} - 1\\frac{1}{3}}\\right) \\) | (a) Begin by operating inside the parentheses.\n\n\\[ \n2\\frac{1}{2} - 1\\frac{1}{3} = \\frac{2 \\cdot 2 + 1}{2} - \\frac{1 \\cdot 3 + 1}{3} = \\frac{5}{2} - \\frac{4}{3} \n\\]\n\n\\[ \n= \\;\\frac{15}{6} - \\frac{8}{6} = \\frac{{15} - 8}{6} = \\frac{7}{6} \n\\]\n\n(b) Now simplify the square root.\n\n\\[ \n\\sqrt{\\f... | Yes |
Example 6.1\n\n6.8\n\nsix and eight tenths | NOTE: Some people read this as \ | No |
Example 6.3 | 0.0019  | No |
In this problem, the indication is that any whole number is a decimal fraction. Whole numbers are often called decimal numbers. | \[ {81} = {81.0} \] | Yes |
Example 6.5\n\nThirty-one and twelve hundredths.\n\nThe decimal position indicated is the hundredths position. | 31.12 | Yes |
Two and three hundred-thousandths. | The decimal position indicated is the hundred thousandths. We'll need to insert enough zeros to the immediate right of the decimal point in order to locate the 3 in the correct position.\n\n## 2.00003 | Yes |
Six thousand twenty-seven and one hundred four millionths. | The decimal position indicated is the millionths position. We'll need to insert enough zeros to the immediate right of the decimal point in order to locate the 4 in the correct position.\n\n## 6,027.000104 | Yes |
\\( {0.6}_{t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t - t... | Reading: six tenths \\( \\rightarrow \\frac{6}{10} \\) . \n\nReduce: \\( \\frac{3}{5} \\) . | Yes |
Example 6.14\n\n\( {4.006}\frac{1}{4} \) | Note that \( {4.006}\frac{1}{4} = 4 + {.006}\frac{1}{4} \)\n\n\[ 4 + {.006}\frac{1}{4} = \;4 + \frac{6\frac{1}{4}}{1000} \]\n\n\[ = \;4 + \frac{\frac{25}{4}}{\frac{1000}{1}} \]\n\n\[ = 4 + \frac{\frac{1}{25}}{4} \cdot \frac{1}{\frac{1000}{40}} \]\n\n\[ = \;4 + \frac{1 \cdot 1}{4 \cdot {40}} \]\n\n\[ = \;4 + \frac{1}{16... | Yes |
Round 32.116 to the nearest hundredth. | (2b) The digit immediately to the right is 6, and \( 6 > 5 \), so we add 1 to the round-off digit: \( 1 + 1 = 2 \)\n\n(3a) The round-off digit is to the right of the decimal point, so we eliminate all digits to its right. 32.12\n\nThe number 32.116 rounded to the nearest hundredth is 32.12. | Yes |
Round 633.14216 to the nearest hundred. | (1) — hundreds position\n\n(2a) The digit immediately to the right is 3, and \( 3 < 5 \) so we leave the round-off digit unchanged.\n\n(3b) The round-off digit is to the left of 0 , so we replace all the digits between it and the decimal point with zeros and eliminate the decimal point and all the decimal digits. 600\n... | Yes |
60.98 rounded to the nearest one is 61. | Sometimes we hear a phrase such as \ | No |
Example 6.22\n\n\( {9.813} + {2.140} \) | 9.813 The decimal points are aligned in the same column.\n\n\( + {2.140} \)\n\n11.953 | Yes |
Find the sum of 6.88106 and 3.5219 and round it to three decimal places. | 6.88106\n+3.5219 Write a 0 in the ten thousandths position.\n11\n6.88106\n+ {3.52190}\n10.40296\nWe need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9 > 5, we get\n10.403 | Yes |
Example 6.28\n\n\( {42.0638} + {126.551} \) | <table><thead><tr><th></th><th></th><th>Display Reads</th></tr></thead><tr><td>Type</td><td>42.0638</td><td>42.0638</td></tr><tr><td>Press</td><td>\( + \)</td><td>42.0638</td></tr><tr><td>Type</td><td>126.551</td><td>126.551</td></tr><tr><td>Press</td><td>\( = \)</td><td>168.6148</td></tr></table>\n\nTable 6.11\n\n\n\n... | Yes |
Find the product of 0.251 and 0.00113 and round to three decimal places. | 0.00028363\n\nNow, rounding to three decimal places, we get\n\n\\( {0.251} \\cdot {0.00113} = {0.000} \\) ,\n\n- to three decimal places. | Yes |
\( {2.58} \cdot {8.61} \) | <table><thead><tr><th></th><th></th><th>Display Reads</th></tr></thead><tr><td>Type</td><td>2.58</td><td>2.58</td></tr><tr><td>Press</td><td>×</td><td>2.58</td></tr><tr><td>Type</td><td>8.61</td><td>8.61</td></tr><tr><td>Press</td><td>\\( = \\)</td><td>22.2138</td></tr></table>\n\nTable 6.13\n\nThe product is 22.2138. | Yes |
\( {0.006} \cdot {0.0042} \) | <table><thead><tr><th></th><th></th><th>Display Reads</th></tr></thead><tr><td>Type</td><td>.006</td><td>.006</td></tr><tr><td>Press</td><td>\( \times \)</td><td>.006</td></tr><tr><td>Type</td><td>.0042</td><td>0.0042</td></tr><tr><td>Press</td><td>\( = \)</td><td>0.0000252</td></tr></table>\n\nTable 6.14\n\nWe know th... | Yes |
\( {100} \cdot {34.876} \) . Since there are 2 zeros in 100, Move the decimal point in 34.876 two places to the right. | \[\n{100} \cdot {34.876} = {3487.6}\n\] | Yes |
\( {10},{000} \cdot {56.82} \) | Since there are 4 zeros in 10,000, move the decimal point in 56.82 four places to the right. We will have to add two zeros in order to obtain the four places.\n\n\[ \n{10},{000} \cdot {56.82} = {56}\underbrace{8200}. \n\]\n\n\[ \n= {568},{200} \n\]\n\nSince there is no fractional part, we can drop the decimal point. | Yes |
Find 4.1 of 3.8. | Translating \ | No |
\( {0.02068} \div 4 \) | 0. \( \widehat{00517} \) \n\n20 \n\n\( \frac{4}{28} \) \n\n\[ \n\frac{28}{0} \n\] \n\nPlace zeros in the tenths and hundredths positions. (See Step 3.) \n\nThus, \( {0.02068} \div 4 = {0.00517} \). | No |
\( {12} \div {0.00032} \) | ## \( {0.00032}\overset{―}{){12.00000}} \)\n\nThe divisor has 5 decimal places.\n\nMove the decimal point of both the divisor and the dividend 5 places to the right. We will need to add 5 zeros to 12 .\n\nSet the decimal place and divide.\n\n## \( 0.\underbrace{00032},\underbrace{12.00000} \)\n\nThis is now the same as... | Yes |
\( {0.5696376} \div {0.00123} \) | <table><thead><tr><th></th><th></th><th>Display Reads</th></tr></thead><tr><td>Type</td><td>.5696376</td><td>0.5696376</td></tr><tr><td>Press</td><td>\( \div \)</td><td>0.5696376</td></tr><tr><td>Type</td><td>.00123</td><td>0.00123</td></tr><tr><td>Press</td><td>\( = \)</td><td>463.12</td></tr></table>\n\nTable 6.23\n\... | Yes |
Example 6.53 \( {3.28} \div {10},{000} \) | Since there are 4 zeros in this power of 10, we move the decimal point 4 places to the left. To do so, we need to add three zeros.\n\n\[ \underbrace{0003.28} \div {10},{000} = {0.000328} \] | Yes |
Divide 2 by 11 and round to 3 decimal places. | Since we wish to round the quotient to three decimal places, we'll carry out the division so that the quotient has four decimal places. The number .1818 rounded to three decimal places is .182 . Thus, correct to three decimal places, \[ 2 \div {11} = {0.182} \] | Yes |
Divide 1 by 6. | \[ \text{6)}\frac{.166}{1.000} \] \[ \left. \begin{array}{r} 6 \\ {40} \\ \frac{36}{40} \\ \frac{36}{4} \end{array}\right\} \] We see that this \ | No |
\( \frac{1}{5} \) Divide 1 by 5. | .2\n\n5) \( {1.0} \)\n\n\( \underline{1.0} \)\n\n0\n\nThus, \( \frac{1}{5} = {0.2} \) | Yes |
\( \frac{5}{6} \) . Divide 5 by 6 . | _ This recurring remainder indicates that the division is nontermin-\n\n\( \frac{5}{6} = {0.833}\cdots \) We are to round to two decimal places.\n\nThus, \( \frac{5}{6} = {0.83} \) to two decimal places. | No |
\( {0.38} \cdot \frac{1}{4} \) . Convert both numbers to decimals or both numbers to fractions. We’ll convert to decimals. | To convert \( \frac{1}{4} \) to a decimal, divide 1 by 4 . Now multiply 0.38 and .25 . .38 \( \times {.25} \) 190 76 .0950 Thus, \( {0.38} \cdot \frac{1}{4} = {0.095} \). | Yes |
\( \frac{5}{13}\left( {\frac{4}{5} - {0.28}}\right) \) Convert 0.28 to a fraction. | \[ \frac{5}{13}\left( {\frac{4}{5} - \frac{28}{100}}\right) = \frac{5}{13}\left( {\frac{4}{5} - \frac{7}{25}}\right) \] \[ = \frac{5}{13}\left( {\frac{20}{25} - \frac{7}{25}}\right) \] \[ = \;\frac{1}{5} \] | Yes |
\[ \frac{0.125}{1\frac{1}{3}} + \frac{1}{16} - {0.1211} = \;\frac{\frac{125}{1000}}{\frac{4}{3}} + \frac{1}{16} - {0.1211} \] | \[ = \;\frac{\frac{1}{8}}{\frac{4}{3}} + \frac{1}{16} - {0.1211} \] \[ = \;\frac{1}{8} \cdot \frac{3}{4} + \frac{1}{16} - {0.1211} \] \[ = \;\frac{3}{32} + \frac{1}{16} - {0.1211} \] \[ = \;\frac{3}{32} + \frac{2}{32} - {0.1211} = \frac{5}{32} - {0.1211} \] \[ = \;{0.15625} - {0.1211} \] \[ = \;{0.03515} \] Convert thi... | Yes |
Compare 8 miles and 3 miles by subtraction. | \\( 8 \\) mile \\( - 3 \\) miles \\( = 5 \\) miles\n\nThis means that 8 miles is 5 miles more than 3 miles. | Yes |
Compare 36 and 4 by division. | \[ {36} \div 4 = 9 \] This means that 36 is 9 times as large as 4 . Recall that \( {36} \div 4 = 9 \) can be expressed as \( \frac{36}{4} = 9 \) . | Yes |
Compare 30 miles and 2 gallons by division. | \[ \frac{{30}\text{ miles }}{2\text{ gallons }} = \frac{{15}\text{ miles }}{1\text{ gallon }} \] | Yes |
The ratio 30 to 2 can be expressed as \( \frac{30}{2} \) . Reducing, we get \( \frac{15}{1} \) . | The ratio 30 to 2 is equivalent to the ratio 15 to 1 . | Yes |
\( \frac{{10}\text{ items }}{5\text{ dollars }} = \frac{2\text{ items }}{1\text{ dollar }} \) | 10 items is to 5 dollars as 2 items is to 1 dollar | Yes |
50 milligrams of vitamin \( \mathrm{C} \) is to 1 tablet as 300 milligrams of vitamin \( \mathrm{C} \) is to 6 tablets. | \[ \frac{50}{1} = \frac{300}{6} \] | Yes |
\( \frac{x}{4} = \frac{20}{16} \) . Find the cross product. | \[ \n{16} \cdot x = {20} \cdot 4 \n\] \n\n\( {16} \cdot x = {80}\; \) Divide the product 80 by the known factor 16 . \n\n\[ \nx\; = \;\frac{80}{16} \n\] \n\n\( x = 5\; \) The unknown number is 5 . \n\nThis mean that \( \frac{5}{4} = \frac{20}{16} \), or 5 is to 4 as 20 is to 16 . | Yes |
\( \frac{5}{x} = \frac{20}{16} \) . Find the cross product. | \( 5 \cdot {16} = {20} \cdot x \)\n\n\( {80} = {20} \cdot x \) Divide the product 80 by the known factor 20 .\n\n\[ \frac{80}{20} = x \]\n\n\[ 4 = x\;\text{The unknown number is 4.} \] | Yes |
\( \frac{16}{3} = \frac{64}{x} \) Find the cross product. | \[ \n{16} \cdot x = {64} \cdot 3 \n\] \n\[ \n{16} \cdot x = {192}\;\text{Divide 192 by 16.} \n\] \n\[ \nx = \frac{192}{16} \n\] \n\[ \nx = {12} \n\] \nThe unknown number is 12 . \n\nThe means that \( \frac{16}{3} = \frac{64}{12} \), or, \( {16} \) is to 3 as 64 is to 12 . | Yes |
\( \frac{9}{8} = \frac{x}{40} \) Find the cross products. | \[ 9 \cdot {40} = 8 \cdot x \] \[ {360} = 8 \cdot x\; \] Divide \( {360} \) by 8 . \[ \frac{360}{8} = x \] \[ {45} = x\; \] The unknown number is 45 . | Yes |
On a map, 2 inches represents 25 miles. How many miles are represented by 8 inches? | Step 1: The unknown quantity is miles.\n\nLet \( x = \) number of miles represented by 8 inches\n\nStep 2: The three specified numbers are\n\n2 inches\n\n\( {25} \) miles\n\n8 inches\n\nStep 3: The comparisons are\n\n2 inches to \( {25} \) miles \( \rightarrow \frac{2\text{ inches }}{{25}\text{ miles }} \)\n\n8 inches ... | Yes |
The ratio \( \frac{165}{100} \) can be written as \( {165}\% \) . | We read \( {165}\% \) as \ | No |
Convert \( {12}\% \) to a decimal. | \( {12}\% = \frac{12}{100} = {0.12} \) | Yes |
Convert 0.75 to a percent. | \[ {0.75} = \frac{75}{100} = {75}\% \] | Yes |
Convert \( \frac{3}{5} \) to a percent. | We see in Example 7.28 that we can convert a decimal to a percent. We also know that we can convert a fraction to a decimal. Thus, we can see that if we first convert the fraction to a decimal, we can then convert the decimal to a percent.\n\n\[ \frac{3}{5} \rightarrow \begin{matrix} 5\overset{6}{\overline{){3.0}}} \\ ... | Yes |
Convert \( {42}\% \) to a fraction. | \[ {42}\% = \frac{42}{100} = \frac{21}{50} \] | Yes |
Convert \( \frac{2}{3}\% \) to a fraction. | \[ \frac{2}{3}\% = \frac{2}{3}\text{ of }1\% = \frac{\frac{1}{)2}}{\frac{2}{3}} \cdot \frac{1}{\frac{100}{50}} \] \[ = \;\frac{1 \cdot 1}{3 \cdot {50}} \] \[ = \;\frac{1}{150} \] | No |
Convert \( \frac{5}{8}\% \) to a decimal. | \[ \frac{5}{8}\% = \frac{5}{8}\text{of}1\% = \frac{5}{8} \cdot \frac{1}{100} \]\n\[ = {0.625} \cdot {0.01} \]\n\[ = \;{0.00625} \] | Yes |
Convert \( \frac{2}{3}\% \) to a three-place decimal. | 1. Convert \( \frac{2}{3} \) to a decimal.\n\nSince we wish the resulting decimal to have three decimal digits, and removing the percent sign will account for two of them, we need to round \( \frac{2}{3} \) to one place \( \left( {2 + 1 = 3}\right) \) .\n\n\( \frac{2}{3}\% = {0.7}\% \) to one decimal place. \( \left( {... | Yes |
What number is 30% of 50? | \[ \begin{matrix} \text{ (percentage) } & = & \text{ (percent) } & . & \text{ (base) } \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \end{matrix} \] \[ P = 30% \cdot 50 \text{Convert } 30% \text{ to a decimal.} \] \[ P = .30 \cdot 50 \text{Multiply.} \] \[ P = 15 \] Thus, 15 is \( 30% \) of 50. | Yes |
\[ \text{What number is}\;{36}\% \;\text{of}\;{95}?\;\text{Missing product statement.} \] | \[ \begin{matrix} \text{(percentage)} & = & \text{(percent)} & \cdot & \text{(base)} \\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \end{matrix} \] \[ P\; = \;{36}\% \; \cdot \;{95}\;\text{Convert}\;{36}\% \text{to a decimal.} \] \[ P\; = \;{.36}\; \cdot \;{95}\;\text{Multiply} \] \[ P\; = \;{34.2} \... | No |
A salesperson, who gets a commission of \( {12}\% \) of each sale she makes, makes a sale of \( \$ 8,{400.00} \) . How much is her commission? | We need to determine what part of \( \$ 8,{400.00} \) is to be taken. What part indicates percentage.\n\n\[ \text{What number is}\;{12}\% \;\text{of}\;8,{400.00}?\;\text{Missing product statement.} \]\n\n\[ \begin{matrix} \text{ (percentage) } & = & \text{ (percent) } \\ \downarrow & \downarrow & \downarrow \end{matrix... | Yes |
A student who makes \$125 a month working part-time receives a 4\% salary raise. What is the student's new monthly salary? | With a 4\% raise, this student will make 100\% of the original salary + 4\% of the original salary. This means the new salary will be 104\% of the original salary. We need to determine what part of \$125 is to be taken. What part indicates percentage.\n\n\\[ \n\\text{What number is}\\,{104}\\% \\;\\text{of}\\,{125}\\;\... | Yes |
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