Comparing Fractions

Which is bigger, two thirds or three fifths? Nothing in the digits will tell you, and guessing gets it right about half the time, which is exactly why children keep doing it.

Fractions · topic 4 of 9Grades 4 to 6Number8 min read7 practice questions

Why the digits lie

For every number a child has met until now, bigger digits have meant more. Nine beats five, ninety beats fifty, and nothing has ever contradicted it. Then fractions arrive and quietly break the rule, without anyone warning them that it is about to happen.

18 carries a bigger number than 12 and is four times smaller. Until a child stops reading fractions as pairs of digits and starts seeing them as amounts, no method taught on this page will hold.

232 thirds353 fifths
Two thirds against three fifths. Same bar, cut two different ways. Once they are drawn to the same length you can see it, but you cannot see it from the numbers alone.

When the bottom numbers match

Spot this one first, because when it applies there is no work to do at all.

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 beats 38 because five pieces of a given size are more than three of the same size, which is the one comparison children never get wrong.

When the top numbers match

This second shortcut is barely taught, which is a pity, because it is quick, it is safe, and it turns up more often than you would expect.

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 beats 37, because you are taking three pieces either way and the pieces on the left are larger.

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 the tops nor the bottoms match, you make them match. Rewrite both fractions over a common bottom number and the problem collapses into the easy case from earlier.

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.
101510 fifteenths9159 fifteenths
The same two fractions, both cut into fifteenths. Now the comparison is just counting, and the difference is exactly one fifteenth.

Worked examples

1

Spotting the easy case

Which is larger, 49 or 79?
  1. Check the bottom numbers first. Both are 9, so the pieces are identical in size.
  2. That means no rewriting is needed at all. Just count the pieces.
  3. 7 pieces is more than 4 pieces.
79 is larger
2

Using a common denominator

Which is larger, 34 or 56?
  1. The denominators are 4 and 6. Both divide into 12, so use 12.
  2. 34: multiply top and bottom by 3, since 4 × 3 = 12. That gives 912.
  3. 56: multiply top and bottom by 2, since 6 × 2 = 12. That gives 1012.
  4. Compare the tops: 10 is more than 9.
56 is larger, by one twelfth
3

Ordering three fractions

Put 12, 25 and 35 in order, smallest first.
  1. Two of them already share a denominator, so deal with those first: 25 is smaller than 35.
  2. Now place the half. Rewrite everything in tenths, since 2 and 5 both divide into 10.
  3. 12 becomes 510, 25 becomes 410, and 35 becomes 610.
  4. Order the tops: 4, then 5, then 6.
25, then 12, then 35

Common mistakes

1. Comparing the bottom numbers as if they were whole numbers

15 gets chosen over 13 because five beats three. The bottom number is not counting how much you have, it is counting how many pieces the whole was cut into, and the more cuts you make the less each piece is worth. Would they rather share a cake with two friends or four?

2. Rewriting only one of the two fractions

One fraction gets rewritten and the other is left as it was, so 1015 ends up being compared against 35. The whole purpose of the method was to get both onto pieces of the same size, and doing it to only one of them puts you back where you started, except now it looks like progress.

3. Assuming a bigger gap means a bigger fraction

The gap between the two numbers gets treated as if it means something, so 110 is picked over 45. It does not mean anything. A fraction is one number, not a distance between two, and the only question worth asking is how much of the whole it represents.

Practice questions

Seven questions. One attempt each, and the explanation appears once the answer is in.

Check yourself

No sign up. Nothing is sent anywhere, the score stays on this device.

0 / 7

For parents

Before you practise any method, check the foundation. Ask your child whether they would rather have a third of a chocolate bar or a fifth of it, and take the answer seriously. If they choose the fifth, stop there: no amount of common denominator practice will stick on top of that, and the fix is ten minutes with a real bar and a knife rather than an hour of exercises.

Keep going

Maths CheckTwelve questions, five minutes, then a printable report naming what to work on next. Free, no sign up.OpenShow me the workingLong division, long multiplication and column methods, walked step by step on your own numbers.Open

Still stuck on this?

Send us the question and a teacher will answer it. No charge, and you do not have to be a student here.

Ask on WhatsAppOther ways to reach us

What to learn next

Equivalent fractionsMath · Grades 4 to 6Simplifying fractionsMath · Grades 5 to 7What is a fraction?Math · Grades 2 to 4Adding fractions, different denominatorsMath · Grades 5 to 7

Everything in the Learning Hub

Every topic is free and every one ends with practice questions. Start anywhere.

Fractions

Math · 9 topics

What Is a Fraction?Equivalent FractionsSimplifying FractionsComparing FractionsAdding and Subtracting Fractions With the Same DenominatorAdding Fractions With Different DenominatorsFinding a Fraction of a NumberMixed Numbers and Improper FractionsFractions to Decimals

Decimals and percentages

Math · 7 topics

What Are Decimals?Comparing and Ordering DecimalsAdding and Subtracting DecimalsRounding DecimalsWhat Are Percentages?Finding a Percentage of a NumberFractions, Decimals and Percentages

Science: matter and the water cycle

Science · 5 topics

What Is Matter?Solids, Liquids and GasesChanging StateThe Water CycleMixtures and Solutions

Whole numbers

Math · 10 topics

Place Value to 1,000,000Rounding Whole NumbersAdding and Subtracting With RegroupingTimes Tables That Actually StickMultiplying by 10, 100 and 1000Long Multiplication, Step by StepShort DivisionLong Division, Step by StepFactors, Multiples and PrimesOrder of Operations

Forces and energy

Science · 6 topics

Forces and MotionFrictionGravity, Mass and WeightLevers and Simple MachinesEnergy and Its FormsSimple Circuits

English: words and sentences

English · 6 topics

Nouns, Verbs and AdjectivesWhat Makes a Sentence, and What Does NotCapital Letters, Full Stops and CommasApostrophesVerb TensesSubject and Verb Agreement

Coding: first steps in Python

Coding · 1 topics

Your First Python Loop

Stuck on this topic? Get one class, free.

A tutor works through it live with your child, one on one, and sends you a short written report afterwards. No card needed.

See our Math programme