Equivalent fractions
Theory
Equivalent fractions name the same amount in different ways, such as \(\dfrac{1}{4}\) and \(\dfrac{2}{8}\). This Year 5 maths page shows how to see them on fraction bars, how to make them by multiplying the numerator and denominator by the same number, how to find a missing number, and how to simplify a fraction by dividing both parts by a common factor.
In \(\dfrac{3}{4}\), the denominator \(4\) tells how many equal parts the whole is cut into, and the numerator \(3\) tells how many of those parts we have.
Equivalent fractions are fractions that show the same amount. On two bars of the same length, \(\dfrac{1}{4}\) and \(\dfrac{2}{8}\) cover exactly the same length.
To make an equivalent fraction, multiply the numerator and the denominator by the same number: \(\dfrac{1}{4} = \dfrac{1 \times 3}{4 \times 3} = \dfrac{3}{12}\). Cutting every part into \(3\) smaller parts gives \(3\) times as many parts, and \(3\) times as many shaded.
A fraction is in its simplest form when the only number that divides both the numerator and the denominator is \(1\). To simplify, divide both by a common factor: \(\dfrac{8}{10} = \dfrac{4}{5}\).
Multiply or divide the numerator and denominator by the same number:
How to find a missing number
- Compare the two denominators (or the two numerators) that you know.
- Find the multiplier: what was the first number multiplied by to get the second?
- Do the same to the other part of the fraction.
- Check: both parts were multiplied by the same number.
How to simplify a fraction
- Find a common factor of the numerator and denominator.
- Divide both by it.
- Repeat until the only common factor is \(1\).
| \(\dfrac{1}{4}\) | \(=\) | \(\dfrac{2}{8} = \dfrac{3}{12}\) |
\(\dfrac{2}{8}\) and \(\dfrac{3}{12}\).
The denominator \(4\) was multiplied by \(3\) to make \(12\).
| \(3 \times 3\) | \(=\) | \(9\) |
\(\dfrac{3}{4} = \dfrac{9}{12}\), so the missing numerator is \(9\).
The numerator \(2\) was multiplied by \(3\) to make \(6\).
| \(5 \times 3\) | \(=\) | \(15\) |
\(\dfrac{2}{5} = \dfrac{6}{15}\), so the missing denominator is \(15\).
| \(2 \times 3\) | \(=\) | \(6\) |
| \(7 \times 3\) | \(=\) | \(21\) |
Both were multiplied by \(3\).
(i) \(\dfrac{8}{10}\)
(ii) \(\dfrac{12}{16}\)
(iii) \(\dfrac{12}{18}\)
| \(\dfrac{8 \div 2}{10 \div 2}\) | \(=\) | \(\dfrac{4}{5}\) |
| \(\dfrac{12 \div 4}{16 \div 4}\) | \(=\) | \(\dfrac{3}{4}\) |
| \(\dfrac{12 \div 6}{18 \div 6}\) | \(=\) | \(\dfrac{2}{3}\) |
(i) \(\dfrac{4}{5}\)
(ii) \(\dfrac{3}{4}\)
(iii) \(\dfrac{2}{3}\) (dividing by \(2\) and then by \(3\) also works).
\(\dfrac{2}{6}\), \(\dfrac{3}{9}\), \(\dfrac{4}{10}\), \(\dfrac{5}{15}\)
For a fraction equal to \(\dfrac{1}{3}\), the denominator is \(3\) times the numerator.
| \(2 \times 3\) | \(=\) | \(6\) |
| \(3 \times 3\) | \(=\) | \(9\) |
| \(4 \times 3\) | \(=\) | \(12 \neq 10\) |
| \(5 \times 3\) | \(=\) | \(15\) |
\(\dfrac{4}{10}\) is not equivalent to \(\dfrac{1}{3}\).
| \(\dfrac{12}{16}\) | \(=\) | \(\dfrac{3}{4}\) |
\(\dfrac{3}{4}\) of the seats are taken.
| \(\dfrac{1 \times 2}{2 \times 2}\) | \(=\) | \(\dfrac{2}{4}\) |
No. Adding changes the amount: \(\dfrac{2}{3}\) is more than a half. Multiply instead: \(\dfrac{1}{2} = \dfrac{2}{4}\).
Common pitfalls
Frequently asked questions
What are equivalent fractions?
Fractions that show the same amount, such as \(\dfrac{1}{4}\), \(\dfrac{2}{8}\) and \(\dfrac{3}{12}\). They cover the same length on bars of the same size.
How do you make an equivalent fraction?
Multiply the numerator and the denominator by the same number. \(\dfrac{4}{5} = \dfrac{8}{10}\) because both parts were multiplied by \(2\).
How do you find a missing numerator?
Work out what the denominator was multiplied by, then multiply the numerator by the same number. In \(\dfrac{3}{4} = \dfrac{\Box}{12}\) the multiplier is \(3\), so the numerator is \(9\).
What does simplest form mean?
The numerator and denominator have no common factor except \(1\). \(\dfrac{4}{5}\) is in simplest form; \(\dfrac{8}{10}\) is not.
How do you simplify a fraction?
Divide the numerator and denominator by a number that goes into both. \(\dfrac{12}{16} = \dfrac{3}{4}\) after dividing both by \(4\).
Why does multiplying top and bottom keep the same value?
It is like cutting every part into smaller equal pieces. There are more pieces, but the shaded amount stays the same size.