# 1.4 Fractions: multiplication  (Page 2/2)

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## Sample set b

Perform the following multiplications.

$\frac{4}{5}\cdot \frac{5}{6}$

$\frac{\stackrel{2}{\overline{)4}}}{\underset{1}{\overline{)5}}}\cdot \frac{\stackrel{1}{\overline{)5}}}{\underset{3}{\overline{)6}}}=\frac{2\cdot 1}{1\cdot 3}=\frac{2}{3}$

Divide 4 and 6 by 2
Divide 5 and 5 by 5

$\frac{8}{\text{12}}\cdot \frac{8}{\text{10}}$

$\frac{\stackrel{4}{\overline{)8}}}{\underset{3}{\overline{)12}}}\cdot \frac{\stackrel{2}{\overline{)8}}}{\underset{5}{\overline{)10}}}=\frac{4\cdot 2}{3\cdot 5}=\frac{8}{\text{15}}$

Divide 8 and 10 by 2.
Divide 8 and 12 by 4.

$8\cdot \frac{5}{\text{12}}=\frac{\stackrel{2}{\overline{)8}}}{1}\cdot \frac{5}{\underset{3}{\overline{)12}}}=\frac{2\cdot 5}{1\cdot 3}=\frac{\text{10}}{3}$

$\frac{\text{35}}{\text{18}}\cdot \frac{\text{63}}{\text{105}}$

$\frac{\stackrel{\stackrel{1}{\overline{)7}}}{\overline{)35}}}{\underset{2}{\overline{)18}}}\frac{\stackrel{7}{\overline{)63}}}{\underset{\underset{3}{\overline{)21}}}{\overline{)105}}}=\frac{1\cdot 7}{2\cdot 3}=\frac{7}{6}$

$\frac{\text{13}}{9}\cdot \frac{6}{\text{39}}\cdot \frac{1}{\text{12}}$

$\frac{\stackrel{1}{\overline{)13}}}{9}\cdot \frac{\stackrel{\stackrel{1}{\overline{)2}}}{\overline{)6}}}{\underset{\underset{1}{\overline{)3}}}{\overline{)39}}}\cdot \frac{1}{\underset{6}{\overline{)\text{12}}}}=\frac{1\cdot 1\cdot 1}{9\cdot 1\cdot 6}=\frac{1}{\text{54}}$

## Practice set b

Perform the following multiplications.

$\frac{2}{3}\cdot \frac{7}{8}$

$\frac{7}{\text{12}}$

$\frac{\text{25}}{\text{12}}\cdot \frac{\text{10}}{\text{45}}$

$\frac{\text{25}}{\text{54}}$

$\frac{\text{40}}{\text{48}}\cdot \frac{\text{72}}{\text{90}}$

$\frac{2}{3}$

$7\cdot \frac{2}{\text{49}}$

$\frac{2}{7}$

$\text{12}\cdot \frac{3}{8}$

$\frac{9}{2}$

$\left(\frac{\text{13}}{7}\right)\left(\frac{\text{14}}{\text{26}}\right)$

1

$\frac{\text{16}}{\text{10}}\cdot \frac{\text{22}}{6}\cdot \frac{\text{21}}{\text{44}}$

$\frac{\text{14}}{5}$

## Multiplying mixed numbers

To perform a multiplication in which there are mixed numbers, it is convenient to first convert each mixed number to an improper fraction, then multiply.

## Sample set c

Perform the following multiplications. Convert improper fractions to mixed numbers.

$1\frac{1}{8}\cdot 4\frac{2}{3}$

Convert each mixed number to an improper fraction.

$1\frac{1}{8}=\frac{8\cdot 1+1}{8}=\frac{9}{8}$

$4\frac{2}{3}=\frac{4\cdot 3+2}{3}=\frac{\text{14}}{3}$

$\frac{\stackrel{3}{\overline{)9}}}{\underset{4}{\overline{)8}}}\cdot \frac{\stackrel{7}{\overline{)14}}}{\underset{1}{\overline{)3}}}=\frac{3\cdot 7}{4\cdot 1}=\frac{\text{21}}{4}=5\frac{1}{4}$

$\text{16}\cdot 8\frac{1}{5}$

Convert $8\frac{1}{5}$ to an improper fraction.

$8\frac{1}{5}=\frac{5\cdot 8+1}{5}=\frac{\text{41}}{5}$

$\frac{16}{1}\cdot \frac{41}{5}$ .

There are no common factors to divide out.

$\frac{\text{16}}{1}\cdot \frac{\text{41}}{5}=\frac{\text{16}\cdot \text{41}}{1\cdot 5}=\frac{\text{656}}{5}=\text{131}\frac{1}{5}$

$9\frac{1}{6}\cdot \text{12}\frac{3}{5}$

Convert to improper fractions.

$9\frac{1}{6}=\frac{6\cdot 9+1}{6}=\frac{\text{55}}{6}$

$\text{12}\frac{3}{5}=\frac{5\cdot \text{12}+3}{5}=\frac{\text{63}}{5}$

$\frac{\stackrel{\text{11}}{\overline{)55}}}{\underset{2}{\overline{)6}}}\cdot \frac{\stackrel{\text{21}}{\overline{)63}}}{\underset{1}{\overline{)5}}}=\frac{\text{11}\cdot \text{21}}{2\cdot 1}=\frac{\text{231}}{2}=\text{115}\frac{1}{2}$

$\begin{array}{ccc}\hfill \frac{11}{8}\cdot 4\frac{1}{2}\cdot 3\frac{1}{8}& =& \frac{11}{8}\cdot \frac{\stackrel{3}{\overline{)9}}}{\underset{1}{\overline{)2}}}\cdot \frac{\stackrel{5}{\overline{)10}}}{\underset{1}{\overline{)3}}}\hfill \\ & =& \frac{11\cdot 3\cdot 5}{8\cdot 1\cdot 1}=\frac{165}{8}=20\frac{5}{8}\hfill \end{array}$

## Practice set c

Perform the following multiplications. Convert improper fractions to mixed numbers.

$2\frac{2}{3}\cdot 2\frac{1}{4}$

6

$6\frac{2}{3}\cdot 3\frac{3}{\text{10}}$

22

$7\frac{1}{8}\cdot \text{12}$

$\text{85}\frac{1}{2}$

$2\frac{2}{5}\cdot 3\frac{3}{4}\cdot 3\frac{1}{3}$

30

## Sample set d

Find the value of each of the following.

${\left(\frac{1}{6}\right)}^{2}=\frac{1}{6}\cdot \frac{1}{6}=\frac{1\cdot 1}{6\cdot 6}=\frac{1}{\text{36}}$

$\sqrt{\frac{9}{\text{100}}}$ . We’re looking for a number, call it ?, such that when it is squared, $\frac{9}{\text{100}}$ is produced.

${\left(?\right)}^{2}=\frac{9}{\text{100}}$

We know that

${3}^{2}=9$ and ${\text{10}}^{2}=\text{100}$

We’ll try $\frac{3}{\text{10}}$ . Since

${\left(\frac{3}{\text{10}}\right)}^{2}=\frac{3}{\text{10}}\cdot \frac{3}{\text{10}}=\frac{3\cdot 3}{\text{10}\cdot \text{10}}=\frac{9}{\text{100}}$

$\sqrt{\frac{9}{\text{100}}}=\frac{3}{\text{10}}$

$4\frac{2}{5}\cdot \sqrt{\frac{\text{100}}{\text{121}}}$

$\frac{\stackrel{2}{\overline{)22}}}{\underset{1}{\overline{)5}}}\cdot \frac{\stackrel{2}{\overline{)10}}}{\underset{1}{\overline{)11}}}=\frac{2\cdot 2}{1\cdot 1}=\frac{4}{1}=4$

$4\frac{2}{5}\cdot \sqrt{\frac{\text{100}}{\text{121}}}=4$

## Practice set d

Find the value of each of the following.

${\left(\frac{1}{8}\right)}^{2}$

$\frac{1}{64}$

${\left(\frac{3}{\text{10}}\right)}^{2}$

$\frac{9}{100}$

$\sqrt{\frac{4}{9}}$

$\frac{2}{3}$

$\sqrt{\frac{1}{4}}$

$\frac{1}{2}$

$\frac{3}{8}\cdot \sqrt{\frac{1}{9}}$

$\frac{1}{8}$

$9\frac{1}{3}\cdot \sqrt{\frac{\text{81}}{\text{100}}}$

$8\frac{2}{5}$

$2\frac{8}{\text{13}}\cdot \sqrt{\frac{\text{169}}{\text{16}}}$

$8\frac{1}{2}$

## Exercises

For the following six problems, use the diagrams to find each of the following parts. Use multiplication to verify your re­sult.

$\frac{3}{4}$ of $\frac{1}{3}$

$\frac{1}{4}$

$\frac{2}{3}$ of $\frac{3}{5}$

$\frac{2}{7}$ of $\frac{7}{8}$

$\frac{1}{4}$

$\frac{5}{6}$ of $\frac{3}{4}$

$\frac{1}{8}$ of $\frac{1}{8}$

$\frac{1}{\text{64}}$

$\frac{7}{\text{12}}$ of $\frac{6}{7}$

For the following problems, find each part without using a diagram.

$\frac{1}{2}$ of $\frac{4}{5}$

$\frac{2}{5}$

$\frac{3}{5}$ of $\frac{5}{\text{12}}$

$\frac{1}{4}$ of $\frac{8}{9}$

$\frac{2}{9}$

$\frac{3}{\text{16}}$ of $\frac{\text{12}}{\text{15}}$

$\frac{4}{\text{15}}$

$\frac{3}{5}$

$2$

52

$\frac{4}{9}$

$\frac{1}{6}\text{of}\frac{\text{12}}{\text{13}}\text{of}\frac{\text{26}}{\text{36}}$

$\frac{1}{2}\text{of}\frac{1}{3}\text{of}\frac{1}{4}$

$\frac{1}{\text{24}}$

$1\frac{3}{7}\text{of 5}\frac{1}{5}\text{of 8}\frac{1}{3}$

$2\frac{4}{5}\text{of 5}\frac{5}{6}\text{of 7}\frac{5}{7}$

126

For the following problems, find the products. Be sure to reduce.

$\frac{1}{3}\cdot \frac{2}{3}$

$\frac{1}{2}\cdot \frac{1}{2}$

$\frac{1}{4}$

$\frac{3}{4}\cdot \frac{3}{8}$

$\frac{2}{5}\cdot \frac{5}{6}$

$\frac{1}{3}$

$\frac{3}{8}\cdot \frac{8}{9}$

$\frac{5}{6}\cdot \frac{\text{14}}{\text{15}}$

$\frac{7}{9}$

$\frac{4}{7}\cdot \frac{7}{4}$

$\frac{3}{\text{11}}\cdot \frac{\text{11}}{3}$

1

$\frac{9}{\text{16}}\cdot \frac{\text{20}}{\text{27}}$

$\frac{\text{35}}{\text{36}}\cdot \frac{\text{48}}{\text{55}}$

$\frac{\text{28}}{\text{33}}$

$\frac{\text{21}}{\text{25}}\cdot \frac{\text{15}}{\text{14}}$

$\frac{\text{76}}{\text{99}}\cdot \frac{\text{66}}{\text{38}}$

$\frac{4}{3}$

$\frac{3}{7}\cdot \frac{\text{14}}{\text{18}}\cdot \frac{6}{2}$

$\frac{4}{\text{15}}\cdot \frac{\text{10}}{3}\cdot \frac{\text{27}}{2}$

12

$\frac{\text{14}}{\text{15}}\cdot \frac{\text{21}}{\text{28}}\cdot \frac{\text{45}}{7}$

$\frac{8}{3}\cdot \frac{\text{15}}{4}\cdot \frac{\text{16}}{\text{21}}$

$7\frac{\text{13}}{\text{21}}\text{or}\frac{\text{160}}{\text{21}}$

$\frac{\text{18}}{\text{14}}\cdot \frac{\text{21}}{\text{35}}\cdot \frac{\text{36}}{7}$

$\frac{3}{5}\cdot \text{20}$

12

$\frac{8}{9}\cdot \text{18}$

$\frac{6}{\text{11}}\cdot \text{33}$

18

$\frac{\text{18}}{\text{19}}\cdot \text{38}$

$\frac{5}{6}\cdot \text{10}$

$\frac{\text{25}}{3}\text{or 8}\frac{1}{3}$

$\frac{1}{9}\cdot 3$

$5\cdot \frac{3}{8}$

$\frac{\text{15}}{8}\text{=1}\frac{7}{8}$

$\text{16}\cdot \frac{1}{4}$

$\frac{2}{3}\cdot \text{12}\cdot \frac{3}{4}$

6

$\frac{3}{8}\cdot \text{24}\cdot \frac{2}{3}$

$\frac{5}{\text{18}}\cdot \text{10}\cdot \frac{2}{5}$

$\frac{\text{10}}{9}\text{=1}\frac{1}{9}$

$\frac{\text{16}}{\text{15}}\cdot \text{50}\cdot \frac{3}{\text{10}}$

$5\frac{1}{3}\cdot \frac{\text{27}}{\text{32}}$

$\frac{9}{2}\text{=4}\frac{1}{2}$

$2\frac{6}{7}\cdot 5\frac{3}{5}$

$6\frac{1}{4}\cdot 2\frac{4}{\text{15}}$

$\frac{\text{85}}{6}\text{=14}\frac{1}{6}$

$9\frac{1}{3}\cdot \frac{9}{\text{16}}\cdot 1\frac{1}{3}$

$3\frac{5}{9}\cdot 1\frac{\text{13}}{\text{14}}\cdot \text{10}\frac{1}{2}$

72

$\text{20}\frac{1}{4}\cdot 8\frac{2}{3}\cdot \text{16}\frac{4}{5}$

${\left(\frac{2}{3}\right)}^{2}$

$\frac{4}{9}$

${\left(\frac{3}{8}\right)}^{2}$

${\left(\frac{2}{\text{11}}\right)}^{2}$

$\frac{4}{\text{121}}$

${\left(\frac{8}{9}\right)}^{2}$

${\left(\frac{1}{2}\right)}^{2}$

$\frac{1}{4}$

${\left(\frac{3}{5}\right)}^{2}\cdot \frac{\text{20}}{3}$

${\left(\frac{1}{4}\right)}^{2}\cdot \frac{\text{16}}{\text{15}}$

$\frac{1}{\text{15}}$

${\left(\frac{1}{2}\right)}^{2}\cdot \frac{8}{9}$

${\left(\frac{1}{2}\right)}^{2}\cdot {\left(\frac{2}{5}\right)}^{2}$

$\frac{1}{\text{25}}$

${\left(\frac{3}{7}\right)}^{2}\cdot {\left(\frac{1}{9}\right)}^{2}$

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers.

$\sqrt{\frac{4}{9}}$

$\frac{2}{3}$

$\sqrt{\frac{\text{16}}{\text{25}}}$

$\sqrt{\frac{\text{81}}{\text{121}}}$

$\frac{9}{\text{11}}$

$\sqrt{\frac{\text{36}}{\text{49}}}$

$\sqrt{\frac{\text{144}}{\text{25}}}$

$\frac{\text{12}}{5}=2\frac{2}{5}$

$\frac{2}{3}\cdot \sqrt{\frac{9}{\text{16}}}$

$\frac{3}{5}\cdot \sqrt{\frac{\text{25}}{\text{81}}}$

$\frac{1}{3}$

${\left(\frac{8}{5}\right)}^{2}\cdot \sqrt{\frac{\text{25}}{\text{64}}}$

${\left(1\frac{3}{4}\right)}^{2}\cdot \sqrt{\frac{4}{\text{49}}}$

$\frac{7}{8}$

${\left(2\frac{2}{3}\right)}^{2}\cdot \sqrt{\frac{\text{36}}{\text{49}}}\cdot \sqrt{\frac{\text{64}}{\text{81}}}$

## Exercises for review

( [link] ) How many thousands in 342,810?

2

( [link] ) Find the sum of 22, 42, and 101.

( [link] ) Is 634,281 divisible by 3?

yes

( [link] ) Is the whole number 51 prime or composite?

( [link] ) Reduce $\frac{\text{36}}{\text{150}}$ to lowest terms.

$\frac{6}{\text{25}}$

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