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We know that a fraction represents a part of a whole quantity. For example, two fifths of one unit can be represented by
$\frac{2}{5}$ of the whole is shaded.
A natural question is, what is a fractional part of a fractional quantity, or, what is a fraction of a fraction? For example, what $\frac{2}{3}$ of $\frac{1}{2}$ ?
We can suggest an answer to this question by using a picture to examine $\frac{2}{3}$ of $\frac{1}{2}$ .
First, let’s represent $\frac{1}{2}$ .
$\frac{1}{2}$ of the whole is shaded.
Then divide each of the $\frac{1}{2}$ parts into 3 equal parts.
Each part is $\frac{1}{6}$ of the whole.
Now we’ll take $\frac{2}{3}$ of the $\frac{1}{2}$ unit.
$\frac{2}{3}$ of $\frac{1}{2}$ is $\frac{2}{6}$ , which reduces to $\frac{1}{3}$ .
Now we ask, what arithmetic operation (+, –, ×, ÷) will produce $\frac{2}{6}$ from $\frac{2}{3}$ of $\frac{1}{2}$ ?
Notice that, if in the fractions $\frac{2}{3}$ and $\frac{1}{2}$ , we multiply the numerators together and the denominators together, we get precisely $\frac{2}{6}$ .
$\frac{2\cdot 1}{3\cdot 2}=\frac{2}{6}$
This reduces to $\frac{1}{3}$ as before.
Using this observation, we can suggest the following:
$\frac{\text{numerator 1}}{\text{denominator 1}}\cdot \frac{\text{numerator 2}}{\text{denominator 2}}=\frac{\text{numerator 1}}{\text{denominator 1}}\cdot \frac{\text{numerator 2}}{\text{denominator 2}}$
Perform the following multiplications.
$\begin{array}{ccc}\frac{3}{4}\cdot \frac{1}{6}=\frac{3\cdot 1}{4\cdot 6}\hfill & =\frac{3}{\text{24}}& \text{Now, reduce.}\end{array}$
$=\frac{\stackrel{1}{\overline{)3}}}{\underset{8}{\overline{)24}}}=\frac{1}{8}$
Thus
$\frac{3}{4}\cdot \frac{1}{6}=\frac{1}{8}$
This means that $\frac{3}{4}$ of $\frac{1}{6}$ is $\frac{1}{8}$ , that is, $\frac{3}{4}$ of $\frac{1}{6}$ of a unit is $\frac{1}{8}$ of the original unit.
$\frac{3}{8}\cdot 4$ . Write 4 as a fraction by writing $\frac{4}{1}$
$\frac{3}{8}\cdot \frac{4}{1}=\frac{3\cdot 4}{8\cdot 1}=\frac{\text{12}}{8}=\frac{\stackrel{3}{\overline{)12}}}{\underset{2}{\overline{)8}}}=\frac{3}{2}$
$\frac{3}{8}\cdot 4=\frac{3}{2}$
This means that $\frac{3}{8}$ of 4 whole units is $\frac{3}{2}$ of one whole unit.
$\frac{2}{5}\cdot \frac{5}{8}\cdot \frac{1}{4}=\frac{2\cdot 5\cdot 1}{5\cdot 8\cdot 4}=\frac{\stackrel{1}{\overline{)10}}}{\underset{\text{16}}{\overline{)160}}}=\frac{\stackrel{}{1}}{\underset{}{\text{16}}}$
This means that $\frac{2}{5}$ of $\frac{5}{8}$ of $\frac{1}{4}$ of a whole unit is $\frac{1}{\text{16}}$ of the original unit.
Perform the following multiplications.
$\frac{2}{5}\cdot \frac{1}{6}$
$\frac{1}{\text{15}}$
$\frac{1}{4}\cdot \frac{8}{9}$
$\frac{2}{9}$
$\frac{4}{9}\cdot \frac{15}{16}$
$\frac{5}{\text{12}}$
$\left(\frac{2}{3}\right)\left(\frac{2}{3}\right)$
$\frac{4}{9}$
$\left(\frac{7}{4}\right)\left(\frac{8}{5}\right)$
$\frac{\text{14}}{5}$
$\frac{5}{6}\cdot \frac{7}{8}$
$\frac{\text{35}}{\text{48}}$
$\frac{2}{3}\cdot 5$
$\frac{\text{10}}{3}$
$\left(\frac{3}{4}\right)\left(\text{10}\right)$
$\frac{\text{15}}{2}$
$\frac{3}{4}\cdot \frac{8}{9}\cdot \frac{5}{\text{12}}$
$\frac{5}{\text{18}}$
We have seen that to multiply two fractions together, we multiply numerators together, then denominators together, then reduce to lowest terms, if necessary. The reduction can be tedious if the numbers in the fractions are large. For example,
$\frac{9}{\text{16}}\cdot \frac{\text{10}}{\text{21}}=\frac{9\cdot \text{10}}{\text{16}\cdot \text{21}}=\frac{\text{90}}{\text{336}}=\frac{\text{45}}{\text{168}}=\frac{\text{15}}{\text{28}}$
We avoid the process of reducing if we divide out common factors before we multiply.
$\frac{9}{\text{16}}\cdot \frac{\text{10}}{\text{21}}=\frac{\stackrel{3}{\overline{)9}}}{\underset{8}{\overline{)16}}}\cdot \frac{\stackrel{5}{\overline{)10}}}{\underset{7}{\overline{)21}}}=\frac{3\cdot 5}{8\cdot 7}=\frac{\text{15}}{\text{56}}$
Divide 3 into 9 and 21, and divide 2 into 10 and 16. The product is a fraction that is reduced to lowest terms.
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