Comparing fractions and ordering fractions: which is bigger?

How do you compare fractions?

You compare fractions by asking how many equal parts each one has, and how big those parts are. A fraction has two numbers. The denominator (bottom) says how many equal parts the whole is split into, so it sets the size of each part. The numerator (top) says how many of those parts you have.

When one of those two numbers is the same in both fractions, comparing is quick. When neither is, children make the denominators the same first. The sections below go through each case in the order schools teach them.

Same denominator: the bigger numerator wins

If two fractions have the same denominator, the one with the bigger numerator is bigger. The parts are the same size, so more parts means more. Five eighths is bigger than three eighths:

Government guidance for teachers puts it like this: "If the denominators are the same, then the larger the numerator, the larger the fraction." This is the first kind of comparing children do, in Year 3.

Same numerator: which fraction is bigger?

If two fractions have the same numerator, the one with the smaller denominator is bigger. This surprises many children, because 6 is bigger than 4. But splitting a whole into more parts makes each part smaller. A quarter is bigger than a sixth:

The same goes for any matching numerators. Two fifths is bigger than two sixths, because fifths are bigger than sixths:

In Year 3 children compare unit fractions (with 1 on top) like this. A fraction wall is a good way to see it: each row is the same whole, cut into more and more pieces.

For what the numerator and denominator mean in the first place, see fractions explained.

How to order fractions with different denominators

To order fractions with different denominators, give them a common denominator, then compare the numerators. This uses equivalent fractions: fractions that look different but are worth the same.

Worked example 1: one denominator is a multiple of the other. Which is bigger, a fifth or four fifteenths? 15 is a multiple of 5, so change the fifth into fifteenths. Multiply the top and bottom by 3: ¹⁄₅ = ³⁄₁₅. Now both are in fifteenths, and 4 is more than 3, so four fifteenths is bigger.

Worked example 2: neither denominator is a multiple of the other. Which is bigger, a third or three eighths? 8 isn't a multiple of 3, so multiply the denominators: 3 × 8 = 24. Change both fractions into twenty-fourths:

• ¹⁄₃ = ⁸⁄₂₄ (top and bottom times 8)
• ³⁄₈ = ⁹⁄₂₄ (top and bottom times 3)

9 is more than 8, so three eighths is bigger. The bars agree:

Multiplying the denominators always gives a common denominator, but not always the smallest one. Government guidance says that's fine at primary school, and that a smaller one is better if your child can spot it.

Put these fractions in ascending order

Ascending order means from least to greatest, smallest first. Descending order means from greatest to least. Here's a typical question: put a half, three eighths, three quarters and five eighths in ascending order.

All the denominators (2, 8 and 4) go into 8, so write them all in eighths:

• ¹⁄₂ = ⁴⁄₈
• ³⁄₄ = ⁶⁄₈
• three eighths and five eighths stay as they are

Now order the numerators 3, 4, 5, 6. In ascending order: ³⁄₈, ¹⁄₂, ⁵⁄₈, ³⁄₄. In descending order, the same list backwards.

This kind of question, where all the denominators are multiples of the same number, is the Year 5 part of the national curriculum.

Comparing fractions greater than 1

In Year 6, children compare fractions bigger than 1 too. Turn any improper fractions into mixed numbers first, then compare the whole numbers. Only if the whole numbers match do you need to compare the fraction parts.

Which is bigger, seven quarters or 2 ¹⁄₃? ⁷⁄₄ = 1 ³⁄₄, and 1 is less than 2, so 2 ¹⁄₃ is bigger. There's no need to find a common denominator at all.

Quick ways to compare without a common denominator

Government guidance for Year 6 wants children to choose between finding a common denominator and simply reasoning. Two quick checks:

• Compare with a half. Three eighths is less than a half (half of 8 is 4), and four sevenths is more than a half. So four sevenths is bigger.
• How far from a whole? Seven eighths is one eighth short of a whole. Six sevenths is one seventh short. An eighth is smaller than a seventh, so seven eighths is closer to a whole, and bigger.

On a number line, the fraction further to the right is always the bigger one.

Common mistakes to watch for

• "Bigger denominator, bigger fraction." A sixth is smaller than a quarter, even though 6 is bigger than 4.
• Comparing numerators when the denominators differ. Three tenths has a bigger numerator than two fifths, but two fifths is four tenths, so two fifths is bigger. Match the denominators first.
• Changing only the bottom number. To turn a fifth into fifteenths, both numbers are multiplied by 3. Changing only the denominator gives one fifteenth, a different fraction.
• Mixing up ascending and descending. Ascending goes up, like climbing stairs: smallest first.

How to help at home

• Ask "would you rather?" Would you rather have a third of a pizza or a quarter? Children who picture equal pizzas get the idea fast.
• Fold paper strips. Fold one strip into halves, one into quarters and one into eighths, and lay them side by side.
• Say the fractions out loud. "Three eighths" and "five eighths" make it clear the parts are the same kind.
• Practise the times tables. Finding common denominators is multiplying, so secure times tables make it much quicker.

For the next step after comparing, see adding and subtracting fractions, which also needs common denominators.

Comparing fractions questions to try

1. Which is bigger, four ninths or seven ninths?
2. Which is bigger, a fifth or an eighth?
3. Which is bigger, two thirds or seven twelfths?
4. Put these in ascending order: three quarters, a half, five eighths.
5. Which is bigger, a quarter or two ninths? Use a common denominator.
6. Who has run further: Sabijah, who has run nine tenths of a race, or Will, who has run eight ninths?

Answers

1. Seven ninths (same denominator, bigger numerator)
2. A fifth (same numerator, smaller denominator)
3. Two thirds (²⁄₃ = ⁸⁄₁₂, and 8 twelfths is more than 7 twelfths)
4. A half, five eighths, three quarters (in eighths: 4, 5, 6)
5. A quarter (¹⁄₄ = ⁹⁄₃₆ and ²⁄₉ = ⁸⁄₃₆)
6. Sabijah. She is one tenth from the end and Will is one ninth from the end, and a tenth is smaller.