Equivalent fractions and comparing fractions

Step 6 · Long division and equivalent fractions topics

Make equal fractions by multiplying the top and bottom by the same number, and compare fractions using a common denominator or a benchmark such as one half.

Making equivalent fractions

Cut each part of a fraction bar in two and you have twice as many parts, each half the size. The shaded amount stays the same: $\frac{1}{3}$ = $\frac{2}{6}$ .

1/3

2/6

4/12

1/3, 2/6 and 4/12 shade the same amount of the same whole.

So multiply the numerator and the denominator by the same number: $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$ . Dividing both by the same number works too: $\frac{6}{8}$ = $\frac{3}{4}$ .

Example: An equivalent fraction

Fill the gap: $\frac{2}{5} = \frac{?}{15}$

5 × 3 = 15, so the denominator was multiplied by 3.
Do the same to the numerator: 2 × 3 = 6.

Answer $\frac{6}{15}$

Comparing with a common denominator

To compare fractions with different denominators, rewrite them with the same denominator. Then the parts are the same size and you can compare the numerators.

Example: A common denominator

Which is greater, $\frac{3}{4}$ or $\frac{5}{8}$ ?

8 is a multiple of 4, so use eighths.
$\frac{3}{4}$ = $\frac{6}{8}$ .
6 eighths is more than 5 eighths.

Answer $\frac{3}{4}$

With the same numerator, the fraction with the smaller denominator is greater: $\frac{2}{3}$ is more than $\frac{2}{7}$ , because thirds are bigger than sevenths.

Comparing with one half

One half is a handy benchmark. A fraction is more than $\frac{1}{2}$ when its numerator is more than half its denominator: $\frac{5}{8}$ is more than $\frac{1}{2}$ , because 5 is more than 4.

Example: Using one half

Which is greater, $\frac{3}{7}$ or $\frac{5}{9}$ ?

Half of 7 is 3.5, and 3 is less, so $\frac{3}{7}$ is less than $\frac{1}{2}$ .
Half of 9 is 4.5, and 5 is more, so $\frac{5}{9}$ is more than $\frac{1}{2}$ .

Answer $\frac{5}{9}$

3/7 is just below one half, and 5/9 is just above it.

Try it yourself

Work each one on paper first, then open the answer to check.

Write two fractions equal to $\frac{1}{2}$ .
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For example $\frac{2}{4}$ and $\frac{5}{10}$ .
Fill the gap: $\frac{3}{5} = \frac{?}{20}$
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12.
Write $\frac{8}{12}$ with a denominator of 3.
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$\frac{2}{3}$ . Divide the top and the bottom by 4.
Which is greater, $\frac{2}{3}$ or $\frac{7}{12}$ ?
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$\frac{2}{3}$ . It equals $\frac{8}{12}$ , which is more than $\frac{7}{12}$ .
Is $\frac{4}{10}$ more or less than one half?
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Less. Half of 10 is 5, and 4 is less than 5.