Comparing fractions: methods, examples and practice

Why a bigger number doesn't always mean a bigger fraction

The denominator tells you how many equal pieces the whole is cut into, so a bigger denominator means smaller pieces. 1/8 is smaller than 1/3, even though 8 is bigger than 3. Ask a child whether they would rather share one pizza with 2 friends or with 7 friends, and the idea usually clicks.

Comparisons only make sense when both fractions are of the same whole. Half of a large pizza is more than half of a small one, so comparing questions always assume the same whole.

Comparing fractions with the same denominator or numerator

Same denominator: the pieces are the same size, so the fraction with more pieces is bigger. 5/8 > 3/8. Picture two equal bars cut into eighths: 5 shaded parts cover more of the bar than 3.

Same numerator: you have the same number of pieces, so the fraction with bigger pieces wins, and bigger pieces come from a smaller denominator. 3/4 > 3/5, because quarters are bigger than fifths.

How to compare fractions with different denominators

When both the numerators and the denominators differ, there are a few methods. The common denominator method always works; the others are quick shortcuts for the right pair of fractions.

• Common denominator: rename both fractions so the pieces match. For 2/3 and 3/5, use fifteenths: 10/15 and 9/15, so 2/3 > 3/5.
• Benchmark 1/2: 3/8 is less than half, because half of 8 is 4. 4/7 is more than half, because half of 7 is 3 1/2. So 3/8 < 4/7.
• Common numerator: 2/5 = 4/10, and tenths are smaller than ninths, so 4/10 < 4/9. That means 2/5 < 4/9.
• Distance from 1: 7/8 is 1/8 away from 1 and 5/6 is 1/6 away. 1/8 is the smaller gap, so 7/8 > 5/6.
• Cross-multiplying: for 2/3 and 3/5, 2 × 5 = 10 and 3 × 3 = 9. Since 10 > 9, 2/3 > 3/5. This is the common denominator method with the denominator left out.

Ordering fractions

To put several fractions in order, write them all with the same denominator. For 2/3, 5/12 and 3/4, use twelfths: 8/12, 5/12 and 9/12. From smallest to greatest that is 5/12, 2/3, 3/4. A quick check against 1/2 helps too: 5/12 is the only one less than half.

Comparing fractions in 3rd and 4th grade

In the US, Common Core starts comparing in grade 3 with fractions that share a numerator or a denominator, recording the result with >, = or < and justifying it with a picture (3.NF.A.3.d). In grade 4, children compare fractions with different numerators and denominators by making common denominators or numerators, or by comparing with a benchmark such as 1/2 (4.NF.A.2).

In England, Year 3 pupils compare and order unit fractions and fractions with the same denominator. Year 5 moves on to fractions whose denominators are all multiples of the same number, and Year 6 includes fractions greater than 1.

The practice set below covers both grades: choosing <, > or =, deciding whether a fraction is more or less than 1/2, picking the greater of two unit fractions, and putting three fractions in order from smallest to greatest.

How parents can help

• Use a ruler. Is 3/8 of an inch longer or shorter than 1/2 inch? Your child can check by looking.
• Ask for a reason, not just a sign. "It's less than half" or "the pieces are smaller" shows real understanding.
• Teach the half check early. It sorts many pairs in seconds and catches careless answers.
• When your child is ready, print a fresh comparing fractions worksheet with an answer key and let them mark it themselves.