# 1.4 Fractions: multiplication

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses multiplication of fractions. By the end of the module students should be able to understand the concept of multiplication of fractions, multiply one fraction by another, multiply mixed numbers and find powers and roots of various fractions.

## Section overview

• Fractions of Fractions
• Multiplication of Fractions
• Multiplication of Fractions by Dividing Out Common Factors
• Multiplication of Mixed Numbers
• Powers and Roots of Fractions

## Fractions of fractions

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 $\frac{1}{3}$ .

## Multiplication of fractions

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:

1. The word "of" translates to the arithmetic operation "times."
2. To multiply two or more fractions, multiply the numerators together and then multiply the denominators together. Reduce if necessary.

$\frac{\text{numerator 1}}{\text{denominator 1}}\cdot \frac{\text{numerator 2}}{\text{denominator 2}}=\frac{\text{numerator 1}}{\text{denominator 1}}\cdot \frac{\text{numerator 2}}{\text{denominator 2}}$

## Sample set a

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.

## Practice set a

Perform the following multiplications.

$\frac{2}{5}\cdot \frac{1}{6}$

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

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

$\frac{2}{9}$

$\frac{4}{9}\cdot \frac{15}{16}$

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

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

$\frac{4}{9}$

$\left(\frac{7}{4}\right)\left(\frac{8}{5}\right)$

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

$\frac{5}{6}\cdot \frac{7}{8}$

$\frac{\text{35}}{\text{48}}$

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

$\frac{\text{10}}{3}$

$\left(\frac{3}{4}\right)\left(\text{10}\right)$

$\frac{\text{15}}{2}$

$\frac{3}{4}\cdot \frac{8}{9}\cdot \frac{5}{\text{12}}$

$\frac{5}{\text{18}}$

## Multiplying fractions by dividing out common factors

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.

## The process of multiplication by dividing out common factors

To multiply fractions by dividing out common factors, divide out factors that are common to both a numerator and a denominator. The factor being divided out can appear in any numerator and any denominator.

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