Comparing Fractions
Which is bigger, two thirds or three fifths? You cannot tell by looking at the digits. There are three methods, and one of them always works.
Why the digits lie
With whole numbers, bigger digits mean a bigger number. 9 beats 5, every time. Fractions do not behave like that, and children carry the whole number habit straight into fractions.
18 has bigger digits than 12, and it is much smaller. So the first job is to stop reading the digits and start picturing the amount.
When the bottom numbers match
This is the easy case, and it is worth spotting quickly because it saves all the work.
If two fractions have the same denominator, the pieces are the same size, so you just count them. Whichever has more pieces is bigger.
58 is bigger than 38, because five eighth-sized pieces beat three of them.
When the top numbers match
Less obvious, and children rarely get taught it, but it is quick and it is safe.
If two fractions have the same numerator, you are taking the same number of pieces from each. So the one with the smaller bottom number wins, because its pieces are bigger.
34 is bigger than 37. Three big pieces beat three small ones.
The pizza test
Would you rather have 3 slices of a pizza cut into 4, or 3 slices of the same pizza cut into 7? Children answer this correctly in about a second, then apply the same reasoning to the numbers.
The method that always works
When neither number matches, rewrite both fractions so their bottom numbers agree. Then it becomes the easy case.
Common denominator method
Find a number that both denominators divide into. Multiplying them together always gives one, though often a smaller one exists. Rewrite both fractions with that denominator, then compare the top numbers.
Take 23 and 35. Both 3 and 5 divide into 15.
- 23: multiply top and bottom by 5, giving 1015.
- 35: multiply top and bottom by 3, giving 915.
- Same size pieces now, so count them. 10 beats 9.
Worked examples
Spotting the easy case
- Check the bottom numbers first. Both are 9, so the pieces are identical in size.
- That means no rewriting is needed at all. Just count the pieces.
- 7 pieces is more than 4 pieces.
Using a common denominator
- The denominators are 4 and 6. Both divide into 12, so use 12.
- 34: multiply top and bottom by 3, since 4 × 3 = 12. That gives 912.
- 56: multiply top and bottom by 2, since 6 × 2 = 12. That gives 1012.
- Compare the tops: 10 is more than 9.
Ordering three fractions
- Two of them already share a denominator, so deal with those first: 25 is smaller than 35.
- Now place the half. Rewrite everything in tenths, since 2 and 5 both divide into 10.
- 12 becomes 510, 25 becomes 410, and 35 becomes 610.
- Order the tops: 4, then 5, then 6.
Common mistakes
1. Comparing the bottom numbers as if they were whole numbers
Saying 15 is bigger than 13 because 5 is bigger than 3. The bottom number counts how many pieces the whole was cut into, so a bigger bottom number means smaller pieces.
2. Rewriting only one of the two fractions
Changing 23 into 1015 and then comparing it against 35 as it stands. Both fractions have to end up over the same denominator, or you are comparing pieces of different sizes again.
3. Assuming a bigger gap means a bigger fraction
Thinking 110 beats 45 because 10 and 1 are further apart. The gap between the two numbers tells you nothing useful. Only the amount matters.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
Check yourself
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For parents
Before any method, check that your child believes a bigger bottom number means smaller pieces. Ask which they would rather have, a third of a chocolate bar or a fifth of it. If they pick the fifth, no amount of common denominator practice will help until that idea is fixed, and fixing it takes about ten minutes with something real to cut up.
What to learn next
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