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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn8300Papers 1–3 · Higher tierNumberFractions and decimals

Number · Subtopic 2 of 5

Fractions and Decimals

To add or subtract fractions, give them a common denominator. To multiply, multiply tops and bottoms; to divide, flip the second fraction and multiply. Every fraction is a terminating or recurring decimal, and every recurring decimal can be turned back into a fraction.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Add, subtract, multiply and divide fractions and mixed numbers.
Find a fraction of an amount.
Convert between fractions and decimals, including recurring decimals.
Order fractions and decimals.

Pause and think

Why can't you add $\tfrac12+\tfrac13$ by adding tops and bottoms to get $\tfrac25$ ?

Section 2 of 14

Core idea

Start hereSharing pizzas

Two-thirds of one pizza plus three-quarters of another: how much pizza is that? You can't add thirds and quarters directly, any more than metres and inches. First cut both into the same size slices: twelfths.

The big idea

Add/subtract: common denominator. Multiply: tops × tops, bottoms × bottoms. Divide: flip the second and multiply. Recurring decimals: multiply and subtract.

1. Equivalent fractions

3/4 = 6/8: same amount, cut differently

Equivalent fractions shade the same amount. That is why you can multiply (or divide) top and bottom by the same number.

\frac ab=\frac{ka}{kb}

$k$: any non-zero number: cutting each piece into $k$ smaller ones.

2. The four operations

Why we flip to divide

$\frac34\div\frac58=\frac{3/4}{5/8}$
Division as a fraction.
$=\frac{\frac34\times\frac85}{\frac58\times\frac85}=\frac{\frac34\times\frac85}{1}$
Multiply top and bottom by $\frac85$ (equivalent fraction).

So dividing by

\frac58

is multiplying by

\frac85

Worked exampleThree calculations

$\frac23+\frac34$ ; $2\frac13\times1\frac12$ ; $\frac34\div\frac58$ .

Goal: Simplest form.
Strategy: Rules above; mixed numbers become improper first.

$\frac8{12}+\frac9{12}=\frac{17}{12}=1\tfrac5{12}$
Common denominator 12.
$\frac73\times\frac32=\frac{21}6=3\tfrac12$
Improper, then multiply.
$\frac34\times\frac85=\frac{24}{20}=\frac65$
Flip and multiply.

Answer

1\frac5{12}

3\frac12

\frac65

✓ Check

0.667+0.75=1.417=1\frac5{12}

✓.

Common misconceptionAdding tops and bottoms

\frac12+\frac13\neq\frac25

. (Half a pizza plus a third is more than half, but

\frac25

is less.)

3. Recurring decimals

Worked example $0.\overline{45}$ as a fraction

Convert.

Goal: Fraction.
Strategy: Multiply by $10^{\text{length of repeat}}$ and subtract to kill the tail.

$x=0.4545\ldots$
Name it.
$100x=45.4545\ldots$
Repeat has 2 digits.
$99x=45\Rightarrow x=\frac{45}{99}=\frac5{11}$
Subtract: tails cancel.

Answer

\frac5{11}

✓ Check

5\div11=0.4545\ldots

✓.

Pause and think

Order $\frac35, 0.58, \frac47$ .

Summary

Common denominators for $\pm$ .
Multiply straight; divide by flipping.
Recurring: multiply and subtract.

Numerator: Top number.
Denominator: Bottom number.

HistoryEgyptian fractions. Ancient Egyptians wrote almost every fraction as a sum of different unit fractions, like

\tfrac34=\tfrac12+\tfrac14

. The Rhind Papyrus (around

1650

BC) has a whole table of these decompositions.

Fun factThe magic of sevenths.

\tfrac17=0.\dot14285\dot7

\tfrac27=0.\dot28571\dot4

\tfrac37=0.\dot42857\dot1

: every seventh uses the same six digits in the same cycle, just starting at a different place.

Section 3 of 14

Key formulae

Adding

\frac ab+\frac cd=\frac{ad+bc}{bd}

Dividing

\frac ab\div\frac cd=\frac ab\times\frac dc

One recurring digit

0.\dot a=\frac a9

TrickKeep, flip, change. To divide fractions: keep the first, flip the second, change

\div

\times

1\tfrac12\div\tfrac34=\tfrac32\times\tfrac43=2

Section 4 of 14

Worked examples

Example 1 · Mixed numbers0 / 2 steps

Work out

3\tfrac14-1\tfrac23

Example 2 · Recurring decimal0 / 2 steps

Write

0.1\dot2

(

0.1222\ldots

) as a fraction.

Section 5 of 14

Explore the graph

Fractions on the number line

Loading the interactive graph…

Try this

The moving point is $\tfrac ab$ ; the value is its decimal. The fixed point is $\tfrac12$ .
Which denominators $b$ give terminating decimals? Try $b=8$ and $b=6$ .
Find two different fractions at the same point. What does that show?

Equivalent fractions sit at the same point on the number line.

In the real worldRecipes and music. Doubling a recipe that needs

\tfrac34

cup of flour, or reading a musical score full of half-, quarter- and eighth-notes, is fraction arithmetic in everyday life.

Section 6 of 14

Common mistakes

Incorrect

\tfrac12+\tfrac13=\tfrac25

Correct

\tfrac56

Use a common denominator.

IncorrectDividing by flipping the first fraction

CorrectFlip the second

Keep, flip, change.

Incorrect

1\tfrac12\times2=2\tfrac12\ldots

CorrectConvert to

\tfrac32

first

Mixed numbers break the rules if left as they are.

Incorrect

0.\dot3=\tfrac3{10}

Correct

\tfrac39=\tfrac13

0.333\ldots

is not

0.3

$\tfrac12+\tfrac13=\tfrac25$

$\tfrac36+\tfrac26=\tfrac56$

Section 7 of 14

Exam tip

Exam tipShow the working on Paper 1. Without a calculator, write the common denominator or the flipped fraction as a line of working. A correct method with a slip still earns marks; a bare wrong answer earns none.

Section 8 of 14

Guided practice

SA-4024-NU2-0012 marksL1 FoundationCalculatorFractions

Work out

\tfrac23+\tfrac34

. Give your answer as a fraction in its simplest form.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-4024-NU2-0091 markL1 FoundationNon-calculatorFractions

Find

\tfrac25

45

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0111 markL1 FoundationNon-calculatorFractions

Work out

\tfrac58\times\tfrac4{15}

, giving your answer in its simplest form.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0022 marksL1 FoundationCalculatorFractions

Work out

1\tfrac12\div\tfrac34

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0102 marksL2 StandardNon-calculatorFractions

Work out

3\tfrac14-1\tfrac23

, giving your answer as an improper fraction.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0071 markL1 FoundationCalculatorDecimals

Write

\tfrac38

as a decimal.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0121 markL1 FoundationNon-calculatorDecimals

Write

0.35

as a fraction in its simplest form.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0032 marksL2 StandardCalculatorDecimals

Write the recurring decimal

0.\dot4

(

0.444\ldots

) as a fraction.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU2-0132 marksL3 ExamNon-calculatorDecimals

Write the recurring decimal

0.1\dot2

(

0.1222\ldots

) as a fraction.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-NU-0012 marksL2 StandardNon-calculatorDecimals

Write the recurring decimal

0.\dot2\dot7

as a fraction in its simplest form.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-4024-NU2-0152 marksL3 ExamNon-calculatorFractions

Work out

\dfrac{1}{\tfrac13+\tfrac16}

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Add and subtract with a common denominator.
Multiply across; divide by flipping the second fraction.
Only denominators made of $2$ s and $5$ s terminate.
Recurring decimals: multiply by $10^n$ and subtract.

Section 12 of 14

Quick quiz

Q1 · SA-4024-NU2-0141 markL2 StandardCalculatorDecimals

Which is the smallest?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: GCSE Mathematics 8300
Where: Number
Expected: The same; recurring decimal proofs are Higher tier.
Calculator: Paper 1 non-calculator; Papers 2 and 3 calculator.
Typical style: Problem-solving questions often combine topics.
Open in: 8300 · Papers 1–3 8300 · Papers 1–3

Section 14 of 14

Past paper questions

9 questions (32 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

4024/12 · Oct/Nov 2025Question 22 marks
Two short calculations without a calculator.
Question paper, page 4 Mark scheme
4024/12 · Oct/Nov 2025Question 103 marks
Calculate with fractions, giving a mixed number.
Question paper, page 7 Mark scheme
4024/12 · May/June 2025Question 204 marks
Calculations with fractions.
Question paper, page 18 Mark scheme
4024/12 · Oct/Nov 2024Question 32 marks
Order fractions, decimals and percentages.
Question paper, page 2 Mark scheme
4024/12 · Oct/Nov 2024Question 72 marks
Adding fractions and mixed numbers.
Question paper, page 4 Mark scheme
4024/22 · Oct/Nov 2024Question 48 marks
Money problems, currency conversion and compound interest.
Question paper, page 8 Mark scheme

Show 3 more

4024/12 · May/June 2024Question 12 marks
Order numbers given as fractions, decimals and percentages.
Question paper, page 3 Mark scheme
4024/12 · May/June 2024Question 73 marks
Calculating with fractions and mixed numbers.
Question paper, page 6 Mark scheme
4024/22 · May/June 2024Question 36 marks
Exchange rates, simple and compound interest.
Question paper, page 5 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).