Multiplying and Dividing Fractions
Theory
To multiply fractions, cancel any top with any bottom, then multiply the tops and multiply the bottoms. To divide, multiply by the reciprocal of the second fraction: \(\dfrac{4}{9} \div \dfrac{2}{3} = \dfrac{4}{9} \times \dfrac{3}{2} = \dfrac{2}{3}\). This Year 8 maths page shows how.
Multiplying. Cancel any numerator with any denominator, then multiply numerators together and denominators together: \(\dfrac{4}{10} \times \dfrac{6}{8} = \dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{3}{10}\). ‘Of’ means times.
Mixed numerals first. Change every mixed numeral to an improper fraction, and write a whole number over \(1\): \(2\dfrac{1}{5} \times 1\dfrac{1}{4} = \dfrac{11}{5} \times \dfrac{5}{4} = \dfrac{11}{4} = 2\dfrac{3}{4}\).
The reciprocal swaps the numerator and the denominator: the reciprocal of \(\dfrac{4}{7}\) is \(\dfrac{7}{4}\), of \(2\dfrac{1}{4} = \dfrac{9}{4}\) is \(\dfrac{4}{9}\), and of \(-6\) is \(-\dfrac{1}{6}\). A number times its reciprocal is \(1\).
Dividing: keep, change, flip. \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}\). Flip the second fraction first, then cancel.
Signs. Count the negatives: an odd number gives a negative answer, an even number a positive one. Find the sign, work with the sizes, then put the sign back.
Order of operations. Brackets, then powers, then \(\times\) and \(\div\) from left to right, then \(+\) and \(-\). A square is never negative: \(\left(-\dfrac{2}{5}\right)^2 = \dfrac{4}{25}\).
Size. Multiplying by a number between \(0\) and \(1\) makes a number smaller; dividing by one makes it bigger.
In problems. A fraction of what is left is a product. Speed \(=\) distance \(\div\) time, a missing side \(=\) area \(\div\) the known side, and a steady change \(=\) rate \(\times\) time.
Pronumerals on top. The pronumeral stays on top and the numbers cancel as usual: \(\dfrac{3x}{4} \times \dfrac{2}{9} = \dfrac{x}{6}\).
Reciprocals:
| Number | As one fraction | Reciprocal |
|---|---|---|
| \(\dfrac{4}{7}\) | \(\dfrac{4}{7}\) | \(\dfrac{7}{4}\) |
| \(2\dfrac{1}{4}\) | \(\dfrac{9}{4}\) | \(\dfrac{4}{9}\) |
| \(-6\) | \(-\dfrac{6}{1}\) | \(-\dfrac{1}{6}\) |
| \(-1\dfrac{3}{5}\) | \(-\dfrac{8}{5}\) | \(-\dfrac{5}{8}\) |
Multiplying fractions
- Change mixed numerals to improper fractions; put whole numbers over \(1\).
- Decide the sign by counting the negatives.
- Cancel any top with any bottom.
- Multiply across and write the answer as a mixed numeral if needed.
Dividing fractions
- Change to improper fractions.
- Keep the first fraction, change \(\div\) to \(\times\), flip the second.
- Cancel and multiply as above.
(i) \(\dfrac{4}{10} \times \dfrac{6}{8}\)
(ii) \(2\dfrac{1}{2} \times 1\dfrac{1}{4}\)
| \(\dfrac{4}{10} \times \dfrac{6}{8}\) | \(=\) | \(\dfrac{2}{5} \times \dfrac{3}{4}\) |
| \(\) | \(=\) | \(\dfrac{2 \times 3}{5 \times 4}\) |
| \(\) | \(=\) | \(\dfrac{1}{5} \times \dfrac{3}{2} = \dfrac{3}{10}\) |
| \(2\dfrac{1}{2} \times 1\dfrac{1}{4}\) | \(=\) | \(\dfrac{5}{2} \times \dfrac{5}{4}\) |
| \(\) | \(=\) | \(\dfrac{25}{8} = 3\dfrac{1}{8}\) |
(i) \(\dfrac{3}{10}\)
(ii) \(3\dfrac{1}{8}\)
(i) \(\dfrac{4}{9} \div \dfrac{2}{3}\)
(ii) \(3\dfrac{1}{3} \div \dfrac{5}{9}\)
| \(\dfrac{4}{9} \div \dfrac{2}{3}\) | \(=\) | \(\dfrac{4}{9} \times \dfrac{3}{2}\) |
| \(\) | \(=\) | \(\dfrac{2}{3} \times \dfrac{1}{1} = \dfrac{2}{3}\) |
| \(3\dfrac{1}{3} \div \dfrac{5}{9}\) | \(=\) | \(\dfrac{10}{3} \times \dfrac{9}{5}\) |
| \(\) | \(=\) | \(\dfrac{2}{1} \times \dfrac{3}{1} = 6\) |
(i) \(\dfrac{2}{3}\)
(ii) \(6\)
(ii) Evaluate \(\dfrac{7}{12} \div \left(-\dfrac{7}{8}\right)\).
(iii) Write the reciprocal of \(-1\dfrac{3}{5}\).
(i) One negative, so the answer is negative.
(ii) One negative again: flip the divisor, then cancel.
| \(\dfrac{3}{8} \times \dfrac{4}{9}\) | \(=\) | \(\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}\) |
| \(\dfrac{7}{12} \times \dfrac{8}{7}\) | \(=\) | \(\dfrac{1}{3} \times \dfrac{2}{1} = \dfrac{2}{3}\) |
| \(-1\dfrac{3}{5}\) | \(=\) | \(-\dfrac{8}{5}\) |
(i) \(-\dfrac{1}{6}\)
(ii) \(-\dfrac{2}{3}\)
(iii) \(-\dfrac{5}{8}\) (the sign stays).
(i) \(2\dfrac{5}{8} \times 1\dfrac{3}{7}\)
(ii) \(-2\dfrac{1}{2} \div 1\dfrac{7}{8}\)
| \(\dfrac{21}{8} \times \dfrac{10}{7}\) | \(=\) | \(\dfrac{3}{4} \times \dfrac{5}{1}\) |
| \(\) | \(=\) | \(\dfrac{15}{4} = 3\dfrac{3}{4}\) |
| \(\dfrac{5}{2} \times \dfrac{8}{15}\) | \(=\) | \(\dfrac{1}{1} \times \dfrac{4}{3}\) |
| \(\) | \(=\) | \(1\dfrac{1}{3}\) |
(i) \(3\dfrac{3}{4}\)
(ii) One negative, so \(-1\dfrac{1}{3}\).
| \(\left(-\dfrac{3}{4}\right)^2\) | \(=\) | \(\dfrac{9}{16}\) |
| \(\dfrac{9}{16} \div \dfrac{9}{8}\) | \(=\) | \(\dfrac{9}{16} \times \dfrac{8}{9} = \dfrac{1}{2}\) |
| \(\dfrac{1}{2} + \dfrac{1}{4}\) | \(=\) | \(\dfrac{3}{4}\) |
\(\dfrac{3}{4}\)
(ii) A cyclist rides \(10\dfrac{1}{2}\) km in \(\dfrac{3}{4}\) of an hour. Find the average speed.
(iii) A tank leaks \(\dfrac{3}{8}\) L every hour. Write the change over \(5\dfrac{1}{3}\) hours as a signed number.
(i) Scale by \(\dfrac{6}{4}\).
(ii) Speed \(=\) distance \(\div\) time.
(iii) A leak is negative: \(-\dfrac{3}{8} \times 5\dfrac{1}{3}\).
| \(\dfrac{4}{3} \times \dfrac{6}{4}\) | \(=\) | \(2\) |
| \(\dfrac{21}{2} \div \dfrac{3}{4}\) | \(=\) | \(\dfrac{21}{2} \times \dfrac{4}{3} = 14\) |
| \(-\dfrac{3}{8} \times \dfrac{16}{3}\) | \(=\) | \(-2\) |
(i) \(2\) cups
(ii) \(14\) km/h
(iii) \(-2\) L
(ii) Simplify \(\dfrac{3x}{4} \times \dfrac{2}{9}\) and \(\dfrac{x}{6} \div \dfrac{2}{3}\).
| \(\dfrac{2}{3} \times \dfrac{3}{5}\) | \(=\) | \(\dfrac{2}{5}\) |
| \(\dfrac{3}{5} - \dfrac{2}{5}\) | \(=\) | \(\dfrac{1}{5}\) |
| \(\dfrac{3x}{4} \times \dfrac{2}{9}\) | \(=\) | \(\dfrac{x}{2} \times \dfrac{1}{3} = \dfrac{x}{6}\) |
| \(\dfrac{x}{6} \times \dfrac{3}{2}\) | \(=\) | \(\dfrac{x}{4}\) |
(i) \(\dfrac{2}{5}\) of the whole cake is eaten and \(\dfrac{1}{5}\) remains.
(ii) \(\dfrac{x}{6}\) and \(\dfrac{x}{4}\)
Common pitfalls
Frequently asked questions
How do you multiply fractions?
Cancel any numerator with any denominator, then multiply the numerators together and the denominators together.
How do you divide fractions?
Keep the first fraction, change divide to multiply and flip the second fraction: \(\dfrac{1}{2} \div \dfrac{3}{5} = \dfrac{1}{2} \times \dfrac{5}{3} = \dfrac{5}{6}\).
What is a reciprocal?
The fraction turned upside down. A number multiplied by its reciprocal gives 1.
How do I multiply mixed numerals?
Change them to improper fractions first, then cancel and multiply.
Why does dividing by a fraction make the answer bigger?
Dividing by \(\dfrac{1}{2}\) asks how many halves fit, and there are two halves in every whole, so the answer doubles.