Dividing Fractions
Dividing Fractions asks how many of one fraction fit into another, in NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01). To divide, multiply by the reciprocal of the second fraction (turn it upside down), so \(\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2\).
You will find reciprocals, divide a fraction by a fraction, a whole number by a fraction and a fraction by a whole number, cancel before multiplying, write answers as mixed numerals, and check by multiplying back. Cutting ribbon into equal pieces and sharing leftover pizza are everyday examples. It is part of Year 7 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. \(\dfrac{12}{5} \times \dfrac{3}{8} = \dfrac{9}{10}\).