GCSE Maths · Fractions · Foundation · Lesson 3.3

GCSE Maths: Ordering and comparing fractions (Foundation)

Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

Download worksheet (PDF)Answers (PDF)

3.3

Ordering and comparing fractions

Edexcel 1MA1: N1, N8
Grade 2–3

A fraction wall helps us see which fractions are bigger. Each row is the same length, split into different equal parts.

A fraction wall. Compare columns from left to right.

Key ideas

If the denominators are the same, the bigger numerator means the bigger fraction.
If the numerators are the same (for unit fractions especially), the bigger denominator means the smaller fraction: 18 < 15.
Otherwise, write the fractions with a common denominator. Use the LCM of the denominators.

Worked example 3.3.1

Which is bigger, 23 or 35?

Method. To compare fractions with different denominators, rewrite them with the same denominator (the lowest common multiple of the denominators). Then compare the numerators.

Find the lowest common multiple (LCM) of 3 and 5. The LCM is the smallest number that both denominators divide into. It will be our common denominator.
Multiples of 3: 3, 6, 9, 12, 15
Multiples of 5: 5, 10, 15
LCM = 15
Rewrite the first fraction with denominator 15. Multiply the top and the bottom by the same number: 15 ÷ 3 = 5.
23 = 1015 (× 5)
Rewrite the second fraction with denominator 15. 15 ÷ 5 = 3, so multiply the top and the bottom by 3.
35 = 915 (× 3)
Compare the numerators. The denominators now match, so the pieces are the same size and the bigger numerator is the bigger fraction.
10 > 9
Write the answer using the original fractions. The first fraction is bigger.
23 > 35

Check. A different route, as decimals: 23 = 0.6667 and 35 = 0.6, so 0.6667 > 0.6. ✓

Watch out. Do not compare the denominators alone: for fractions with the same top, a bigger denominator means a smaller fraction.

Worked example 3.3.2

Put these in order, smallest first: 12, 25, 710, 34

Method. To put fractions in order, give them all the same denominator, then order the numerators.

Find the LCM of the denominators 2, 5, 10, 4. This is the common denominator.
LCM = 20
Rewrite every fraction with denominator 20. Multiply the top and bottom of each by the same number.
12 = 1020
25 = 820
710 = 1420
34 = 1520
Put the numerators in order. The order is smallest first.
8, 10, 14, 15
Write the original fractions in that order. The question asks for the original fractions, not the converted ones.
25 , 12 , 710 , 34

Check. As decimals: 25 = 0.4, 12 = 0.5, 710 = 0.7, 34 = 0.75.

Watch out. Always give your final answer using the original fractions, not the ones with the common denominator. And never order fractions by their denominators alone.

Worked example 3.3.3

Which is bigger, 37 or 35?

Method. When two fractions have the same numerator, compare the size of the pieces. A bigger denominator means smaller pieces.

Look at the numerators and denominators. Both fractions have the same top number, 3. So we are comparing 3 pieces of one size with 3 pieces of another size.
37 and 35: both have numerator 3
Compare the size of one piece of each. If a whole is cut into 7 equal pieces, each piece is smaller than if the whole is cut into only 5 equal pieces.
17 < 15 (sevenths are smaller than fifths)
Take 3 pieces of each. Three small pieces make less than three bigger pieces.
3 sevenths < 3 fifths
Write the comparison. Put the correct sign between the original fractions.
37 < 35, so 35 is bigger.

Check. Use a common denominator of 35: 37 = 1535 and 35 = 2135. 15 < 21, so 35 is bigger. ✓

Watch out. A bigger denominator does not mean a bigger fraction. When the numerators match, the fraction with the bigger denominator is smaller.

Worked example 3.3.4

Write down a fraction that lies between 12 and 34.

Method. Rewrite both fractions with a common denominator that is big enough to leave a gap, then choose a numerator between the two numerators.

Say what “between” means. We need a fraction that is bigger than one of them and smaller than the other.
12 < ? < 34
Try the lowest common denominator, 4. 12 = 24, and 34 stays the same. But there is no whole number between 2 and 3, so there is no room for a fourth.
24 and 34: no whole number between 2 and 3
Use a bigger common denominator, 8. Doubling the top and bottom of both fractions makes more room (8 is a multiple of 4).
12 = 48 and 34 = 68
Pick a numerator between 4 and 6. The only whole number between 4 and 6 is 5.
48 < 58 < 68
Write the answer. Any correct fraction is allowed, for example this one.
58

Check. As decimals: 12 = 0.5, 58 = 0.625, 34 = 0.75, and 0.5 < 0.625 < 0.75. ✓

Watch out. The lowest common denominator might leave no room. If so, use a bigger one such as double.

Exam tip

Always give your final answer using the original fractions, not the ones with the common denominator, unless the question says otherwise.

Exercise 3C Ordering and comparing fractions

Non-calculator

FCopy each pair and write < or > between them.

(a)13 ___ 15

(b)38 ___ 58

(d)56 ___ 79

(e)45 ___ 710

(f)37 ___ 12

Done

FWrite the larger fraction in each pair.

(a)35 and 710

(b)49 and 12

(d)712 and 35

Done

FWrite each set in order, smallest first.

(a)12 , 38 , 14 , 58

(b)23 , 56 , 12 , 34

(d)45 , 78 , 34 , 910

Done

RWrite a fraction that lies between 12 and 34.

Done

PSSam ran 34 km and Zara ran 710 km. Who ran further?

Done

RJo says “18 is bigger than 15 because 8 is bigger than 5.” Is Jo correct? Explain.

Done

PSAmy eats 23 of a pizza, Ben eats 58 of a pizza and Cal eats 712 of a pizza. Who ate the most? Who ate the least?

Done

RFind the fraction with denominator 12 that lies between 12 and 23.

Done

RChallenge. Write down three fractions, each between 13 and 12.

Done