1.4 Fractions: multiplication

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" Indicates Multiplication The word "of" translates to the arithmetic operation "times."
2. The Method of Multiplying Fractions 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.

Practice set a

Perform the following multiplications.

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.

$\frac{9}{\text{16}}\cdot \frac{\text{10}}{\text{21}}=\frac{\stackrel{3}{\cancel{9}}}{\underset{8}{\cancel{16}}}\cdot \frac{\stackrel{5}{\cancel{10}}}{\underset{7}{\cancel{21}}}=\frac{3\cdot 5}{8\cdot 7}=\frac{\text{15}}{\text{56}}$

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