Dividing Fractions
Every question with a fully worked solution.
- Dividing Fractions - Video - Dividing Fractions by Fractions | How to Divide a Fraction by a Fraction Watch
Theory
Dividing by a fraction asks how many of that fraction fit. To divide, multiply by the reciprocal of the second number: keep, change, flip. This Year 7 maths page explains reciprocals, shows why \(2 \div \dfrac{1}{3} = 6\), and works through divisions, sharing, cutting and speed problems.
\(2 \div \dfrac{1}{3}\) asks how many thirds fit into 2. Each whole holds \(3\) thirds, so \(2 \div \dfrac{1}{3} = 2 \times 3 = 6\).
The reciprocal of \(\dfrac{a}{b}\) is \(\dfrac{b}{a}\): swap the numerator and the denominator. The reciprocal of \(7\) is \(\dfrac{1}{7}\). A number times its reciprocal is \(1\): \(\dfrac{2}{5} \times \dfrac{5}{2} = 1\).
Keep, change, flip: keep the first fraction, change \(\div\) to \(\times\), flip the second: \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}\). For example \(\dfrac{1}{2} \div \dfrac{5}{6} = \dfrac{1}{2} \times \dfrac{6}{5} = \dfrac{3}{5}\).
Dividing by a fraction between \(0\) and \(1\) gives a bigger answer: \(2 \div \dfrac{1}{3} = 6\).
Keep, change, flip:
How to divide fractions
- Flip the second number (the divisor); write a whole number over \(1\) first.
- Change \(\div\) to \(\times\).
- Cancel a top with a bottom, then multiply.
- Write an improper answer as a mixed numeral, and check.
| \(\dfrac{2}{3} \div \dfrac{7}{5}\) | \(=\) | \(\dfrac{2}{3} \times \dfrac{5}{7}\) |
| \(\) | \(=\) | \(\dfrac{10}{21}\) |
\(\dfrac{10}{21}\)
| \(\dfrac{7}{8} \times \dfrac{4}{3}\) | \(=\) | \(\dfrac{7}{2} \times \dfrac{1}{3}\) |
| \(\) | \(=\) | \(\dfrac{7}{6} = 1\dfrac{1}{6}\) |
\(1\dfrac{1}{6}\)
| \(\dfrac{3}{5} \div \dfrac{1}{10}\) | \(=\) | \(\dfrac{3}{5} \times \dfrac{10}{1}\) |
| \(\) | \(=\) | \(6\) |
\(\dfrac{3}{5} = \dfrac{6}{10}\), so \(6\) tenths fit.
| \(\dfrac{9}{10} \times \dfrac{8}{3}\) | \(=\) | \(\dfrac{3}{5} \times \dfrac{4}{1}\) |
| \(\) | \(=\) | \(\dfrac{12}{5} = 2\dfrac{2}{5}\) |
Check: \(\dfrac{12}{5} \times \dfrac{3}{8} = \dfrac{9}{10}\).
(ii) How many \(\dfrac{2}{5}\) m pieces can be cut from \(8\) m of rope?
| \(\dfrac{4}{5} \times \dfrac{1}{6}\) | \(=\) | \(\dfrac{2}{15}\) |
| \(\dfrac{8}{1} \times \dfrac{5}{2}\) | \(=\) | \(20\) |
(i) \(\dfrac{2}{15}\) L
(ii) \(20\) pieces
| \(\dfrac{3}{4} \div \dfrac{3}{10}\) | \(=\) | \(\dfrac{3}{4} \times \dfrac{10}{3}\) |
| \(\) | \(=\) | \(\dfrac{5}{2} = 2\dfrac{1}{2}\) |
Speed \(=\) distance \(\div\) time: \(2\dfrac{1}{2}\) km per hour.
Common pitfalls
Frequently asked questions
What is a reciprocal?
The fraction turned upside down. The reciprocal of \(\dfrac{2}{5}\) is \(\dfrac{5}{2}\).
How do you divide fractions?
Keep the first fraction, change divide to multiply, and flip the second fraction.
Why does dividing by a half double a number?
Dividing by \(\dfrac{1}{2}\) asks how many halves fit. Each whole holds \(2\) halves.
How do you divide a fraction by a whole number?
Multiply by its reciprocal. \(\dfrac{4}{5} \div 6 = \dfrac{4}{5} \times \dfrac{1}{6} = \dfrac{2}{15}\).
How do you check a division?
Multiply the answer by the divisor. \(\dfrac{12}{5} \times \dfrac{3}{8} = \dfrac{9}{10}\).