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How to Divide Fractions

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When we’re dividing one fraction by another fraction, we use the keep, change, flip formula. You keep the first fraction, change division to multiplication and then flip the last fraction.

Formula

Keep, Change, Flip: Fraction over Fraction

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a d}{b c}

Rule

Rewriting a Number into a Fraction

a = \frac{a}{1}

Formula

Keep, Change, Flip: Number over Fraction

\frac{a}{\frac{c}{d}} = \frac{\frac{a}{1}}{\frac{c}{d}} = \frac{a}{1} \cdot \frac{d}{c} = \frac{a d}{c}

Formula

Keep, Change, Flip: Fraction over Number

\frac{\frac{a}{b}}{c} = \frac{\frac{a}{b}}{\frac{c}{1}} = \frac{a}{b} \cdot \frac{1}{c} = \frac{a}{b c}

Division when using complex fraction:

\frac{\frac{2}{6}}{\frac{2}{3}} = \frac{2}{6} \cdot \frac{3}{2} = \frac{2 \cdot 3}{6 \cdot 2} = \frac{3}{6} = \frac{3 \cdot 1}{3 \cdot 2} = \frac{1}{2}

Dividing when using the division symbol:

\frac{2}{5} \div \frac{3}{7} = \frac{2}{5} \cdot \frac{7}{3} = \frac{2 \cdot 7}{5 \cdot 3} = \frac{14}{15}

When you’re dividing a fraction by a number, or a number by a fraction, you rewrite the number as a fraction!

Example 1

Dividing a fraction by a number using complex fraction:

\frac{\frac{4}{3}}{5} = \frac{\frac{4}{3}}{\frac{5}{1}} = \frac{4}{3} \cdot \frac{1}{5} = \frac{4}{15}

Dividing a fraction by a number using division symbol:

\frac{4}{3} \div 5 = \frac{4}{3} \div \frac{5}{1} = \frac{4}{3} \cdot \frac{1}{5} = \frac{4}{15}

Example 2

Dividing a number by a fraction using complex fractions:

\frac{2}{\frac{5}{3}} = \frac{\frac{2}{1}}{\frac{5}{3}} = \frac{2}{1} \cdot \frac{3}{5} = \frac{6}{5}

Dividing a number by a fraction using division symbol:

2 \div \frac{5}{3} = \frac{2}{1} \div \frac{5}{3} = \frac{2}{1} \cdot \frac{3}{5} = \frac{6}{5}

Example 3

Find $\frac{\frac{3}{5}}{\frac{4}{7}}$ .

You can flip the fraction underneath the fraction bar (use the flip formula) and find the answer like this:

\frac{\frac{3}{5}}{\frac{4}{7}} = \frac{3}{5} \div \frac{4}{7} = \frac{3}{5} \cdot \frac{7}{4} = \frac{3 \cdot 7}{5 \cdot 4} = \frac{21}{20}

Example 4

Find $\frac{2}{3} \div \frac{1}{13}$ .

\frac{2}{3} \div \frac{1}{13} = \frac{2}{3} \cdot \frac{13}{1} = \frac{2 \cdot 13}{3 \cdot 1} = \frac{26}{3}

Find $\frac{1}{6} \div 5$ .

\frac{1}{6} \div 5 = \frac{1}{6} \div \frac{5}{1} = \frac{1}{6} \cdot \frac{1}{5} = \frac{1 \cdot 1}{6 \cdot 5} = \frac{1}{30}

Find $7 \div \frac{9}{5}$ .

7 \div \frac{9}{5} = \frac{7}{1} \div \frac{9}{5} = \frac{7}{1} \cdot \frac{5}{9} = \frac{7 \cdot 5}{1 \cdot 9} = \frac{35}{9}

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