Multiplying Fractions
Multiplying Fractions shows how to find a fraction of a fraction in NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01). Multiply the numerators together and the denominators together, cancelling common factors first to keep the numbers small, so \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\).
You will model products with an area model, multiply fractions by whole numbers, multiply three or four fractions at once, and see why multiplying by a fraction less than 1 makes a number smaller. How much of a pizza was eaten, the area of a tile and the share of a class that plays netball are typical problems. It is part of Year 7 Fractions.
Every question with a fully worked solution.
- Multiplying Fractions - Video - How to Multiply Two Fractions Watch
Theory
To multiply fractions, multiply the numerators and multiply the denominators. Cancel common factors first (a top with a bottom) to keep the numbers small. This Year 7 maths page shows why \(\dfrac{1}{2} \times \dfrac{3}{5} = \dfrac{3}{10}\), how to cancel, how to multiply a whole number by a fraction and how to multiply three or more fractions.
Multiply the numerators and multiply the denominators separately: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}\). For example \(\dfrac{3}{5} \times \dfrac{4}{7} = \dfrac{12}{35}\).
Of means multiply: \(\dfrac{3}{4}\) of \(\dfrac{1}{2}\) is \(\dfrac{3 \times 1}{4 \times 2} = \dfrac{3}{8}\).
Cancel first with larger fractions: divide a top and a bottom by a common factor before multiplying. In \(\dfrac{3}{4} \times \dfrac{5}{12}\), the \(3\) and \(12\) share \(3\), giving \(\dfrac{1}{4} \times \dfrac{5}{4} = \dfrac{5}{16}\).
Write a whole number over \(1\): \(3 = \dfrac{3}{1}\). Multiplying by a fraction between \(0\) and \(1\) makes a number smaller: \(12 \times \dfrac{2}{3} = 8\).
Tops times tops, bottoms times bottoms:
How to multiply fractions
- Write any whole number over \(1\).
- Cancel: divide a top and a bottom by a common factor.
- Multiply the tops and multiply the bottoms.
- Write an improper answer as a mixed numeral.
| \(\dfrac{3}{4} \times \dfrac{5}{6}\) | \(=\) | \(\dfrac{1}{4} \times \dfrac{5}{2}\) |
| \(\) | \(=\) | \(\dfrac{5}{8}\) |
The \(3\) and \(6\) share \(3\). The answer is \(\dfrac{5}{8}\).
Each pair \(\dfrac{2}{3} \times \dfrac{3}{2}\) and \(\dfrac{4}{5} \times \dfrac{5}{4}\) cancels to \(1\).
| \(1 \times 1 \times \dfrac{1}{7}\) | \(=\) | \(\dfrac{1}{7}\) |
\(\dfrac{1}{7}\)
(i) \(\dfrac{3}{7} \times \dfrac{2}{5}\)
(ii) \(5 \times \dfrac{2}{9}\)
| \(\dfrac{3}{7} \times \dfrac{2}{5}\) | \(=\) | \(\dfrac{6}{35}\) |
| \(\dfrac{5}{1} \times \dfrac{2}{9}\) | \(=\) | \(\dfrac{10}{9} = 1\dfrac{1}{9}\) |
(i) \(\dfrac{6}{35}\)
(ii) \(1\dfrac{1}{9}\)
| \(\dfrac{9}{10} \times \dfrac{4}{15}\) | \(=\) | \(\dfrac{3}{5} \times \dfrac{2}{5}\) |
| \(\) | \(=\) | \(\dfrac{6}{25}\) |
\(9\) and \(15\) share \(3\); \(4\) and \(10\) share \(2\).
| \(1 - \dfrac{3}{5}\) | \(=\) | \(\dfrac{2}{5}\) |
| \(\dfrac{2}{5} \times \dfrac{5}{6}\) | \(=\) | \(\dfrac{1}{3}\) |
\(\dfrac{1}{3}\) of Year 7 walked.
| \(\dfrac{3}{4} \times \dfrac{8}{9} \times \dfrac{5}{7}\) | \(=\) | \(\dfrac{1}{1} \times \dfrac{2}{3} \times \dfrac{5}{7}\) |
| \(\) | \(=\) | \(\dfrac{10}{21}\) |
\(3\) and \(9\) share \(3\); \(8\) and \(4\) share \(4\).
Common pitfalls
Frequently asked questions
How do you multiply two fractions?
Multiply the tops and multiply the bottoms. \(\dfrac{3}{5} \times \dfrac{4}{7} = \dfrac{12}{35}\).
What does cancelling mean?
Dividing a top and a bottom by the same factor before multiplying, to keep the numbers small.
Do you need a common denominator to multiply?
No. Common denominators are needed for adding and subtracting, not multiplying.
How do you multiply a whole number by a fraction?
Write the whole number over \(1\). \(5 \times \dfrac{2}{9} = \dfrac{10}{9}\).
What does of mean with fractions?
Multiply. \(\dfrac{3}{4}\) of \(\dfrac{1}{2}\) is \(\dfrac{3}{8}\).