Comparing and Ordering Fractions

What you will learn

• Compare fractions with the same denominator or the same numerator.
• Use a common denominator to compare any two fractions.
• Put a list of fractions in order.

Before you start: you should be able to find equivalent fractions and a common multiple. See Equivalent Fractions and Simplifying.

The idea

Which is bigger, $\frac{3}{4}$ or $\frac{2}{3}$? You cannot just look at the numbers, because the parts are different sizes. The trick is to cut both into parts of the same size, then count.

Same denominator

When the bottoms match, the parts are the same size. The bigger numerator wins: $\frac{5}{8} > \frac{3}{8}$.

Same numerator

When the tops match, you have the same number of parts, so the bigger parts win. Bigger parts come from a smaller denominator. So $\frac{3}{4} > \frac{3}{5} > \frac{3}{8}$.

Different denominators: use a common denominator

Write each fraction over the same bottom number. The lowest common multiple of the denominators is the neatest choice. See HCF and LCM.

For $\frac{1}{2}$, $\frac{2}{3}$ and $\frac{3}{4}$, the lowest common multiple of $2, 3, 4$ is $12$:

$\frac{1}{2} = \frac{6}{12}, \qquad \frac{2}{3} = \frac{8}{12}, \qquad \frac{3}{4} = \frac{9}{12}$

Worked examples

Example 1: Two fractions

Which is bigger, $\frac{2}{5}$ or $\frac{3}{7}$?

1. The lowest common multiple of $5$ and $7$ is $35$.
2. $\frac{2}{5} = \frac{14}{35}$ and $\frac{3}{7} = \frac{15}{35}$.
3. Fifteen is more than fourteen.

Answer: $\frac{3}{7}$ is bigger.

Example 2: Put four in order

Put $\frac{3}{4}$, $\frac{2}{3}$, $\frac{5}{6}$ and $\frac{7}{12}$ in order, smallest first.

1. The common denominator is $12$.
2. $\frac{3}{4} = \frac{9}{12}$, $\frac{2}{3} = \frac{8}{12}$, $\frac{5}{6} = \frac{10}{12}$, $\frac{7}{12}$.
3. In order of the tops: $7, 8, 9, 10$.

Answer: $\frac{7}{12}, \frac{2}{3}, \frac{3}{4}, \frac{5}{6}$

Example 3: A quick check against half

Is $\frac{5}{9}$ more or less than $\frac{1}{2}$?

1. Half of $9$ is $4.5$.
2. Five parts is more than $4.5$ parts.

Answer: $\frac{5}{9}$ is a little more than $\frac{1}{2}$.

Tip: comparing with $\frac{1}{2}$ or with $1$ first often settles it without any working.

Why it matters

Comparing fractions is how you decide which deal is bigger, which test score is better or which jug holds more. It is also the first step to adding and subtracting fractions with different denominators, since both need a common denominator.