Multiplication and Division of Fractions

Multiplying a fraction by a whole or mixed number is similar to multiplying two fractions. When multiplying by a whole number, change the whole number to a fraction with a denominator of 1. Next, multiply the numerators together and the denominators together. Rewrite the final answer as a mixed number.

For example: \(\frac{9}{10}\times 3=\frac{9}{10}\times\frac{3}{1}=\frac{27}{10}=2\frac{7}{10}\).

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KEEP IN MIND

Always change a mixed number to an improper fraction when multiplying by a mixed number.

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When multiplying a fraction by a mixed number or multiplying two mixed numbers, the process is similar.

For example: multiply \(\frac{10}{11}\times3\frac{1}{2}\).

This process can also be used when multiplying a whole number by a mixed number or multiplying two mixed numbers.

Multiplying a Fraction by a Whole or Mixed Number Review

Dividing a Fraction by a Fraction

Some basic steps apply when dividing a fraction by a fraction. The information from the previous two sections is applicable to dividing fractions.

STEP BY STEP

Step 1. Leave the first fraction alone.

Step 2. Find the reciprocal of the second fraction.

Step 3. Multiply the first fraction by the reciprocal of the second fraction.

Step 4. Rewrite the fraction as a mixed number and reduce the fraction completely.

Divide \(\frac{3}{10}\div\frac{1}{2}\).

Find the reciprocal of the second fraction, which is \(\frac{2}{1}\).
Now multiply the fractions, \(\frac{3}{10}\times\frac{2}{1}=\frac{6}{10}\).
Reduce \(\frac{6}{10}\) to \(\frac{3}{5}\).
Divide, \(\frac{4}{5}\div\frac{3}{8}\). Find the reciprocal of the second fraction, which is \(\frac{8}{3}\).
Now multiply the fractions, \(\frac{4}{5}\times\frac{8}{3}=\frac{32}{15}\).
Rewrite the fraction as a mixed number \(\frac{32}{15}=2\frac{2}{15}\).

Dividing a Fraction by a Fraction Review

Dividing a Fraction by a Whole or Mixed Number

Some basic steps apply when dividing a fraction by a whole number or a mixed number.

STEP BY STEP

Step 1. Write any whole number as a fraction with a denominator of 1. Write any mixed numbers as improper fractions.

Step 2. Leave the first fraction (improper fraction) alone.

Step 3. Find the reciprocal of the second fraction.

Step 4. Multiply the first fraction by the reciprocal of the second fraction.

Step 5. Rewrite the fraction as a mixed number and reduce the fraction completely.

Divide \(\frac{3}{10}\div3\).

Rewrite the expression as \(\frac{3}{10}\div\frac{3}{1}\). Find the reciprocal of the second fraction, which is \(\frac{1}{3}\). Multiply the fractions, \(\frac{3}{10}\times\frac{1}{3}=\frac{3}{30}=\frac{1}{3}\). Reduce \(\frac{3}{30}\) to \(\frac{1}{10}\).

Divide, \(2\frac{4}{5}\div1\frac{3}{8}\). Rewrite the expression as \(\frac{14}{5}\div\frac{11}{8}\). Find the reciprocal of the second fraction, which is \(\frac{8}{11}\).

Multiply the fractions, \(\frac{14}{5}\times\frac{8}{11}=\frac{112}{55}=2\frac{2}{55}\). Reduce \(\frac{112}{55}\) to \(2\frac{2}{55}\).

Dividing a Fraction by a Whole or Mixed Number Review

Let’s Review!

• The process to multiply fractions is to multiply the numerators together and multiply the denominators together. When there is a mixed number, change the mixed number to an improper fraction before multiplying.
• The process to divide fractions is to find the reciprocal of the second fraction and multiply the fractions. As with multiplying, change any mixed numbers to improper fractions before dividing.