Recurring decimals as fractions | AQA GCSE Maths Notes

Visual model

Line up recurring digits before subtracting

x=0.333\ldots

10x=3.333\ldots

subtract to remove the recurring tail

Gold-standard guide

26 mins

What you will learn

Convert recurring decimals into exact fractions.

Use a clear step-by-step method for recurring decimals as fractions.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier number skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

The trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part

Step 1

Let x equal the recurring decimal

x = 0.272727..

Step 2

Multiply by 100 because two digits recur (27 repeats)

100x = 27.272727.

Step 3

Subtract the original equation from the new one

100x − x = 27.272727.

Watch out

Students multiply by 10 regardless of how many digits recur — this only works when one digit repeats

One recurring digit

If x = 0.333..., use 10x - x.

Two recurring digits

If x = 0.272727..., use 100x - x.

Worked example

Convert 0.2̄7̄ (i.e. 0.272727...) into a fraction in its simplest form

Let x equal the recurring decimal: x = 0.272727...

272727...We multiply by 100 so that the recurring block lines up with the original decimal, making it possible to subtract them.

Subtract the original equation from the new one: 100x − x = 27.272727... − 0.272727... so 99x = 27.

Solve and simplify: x = $2\frac{7}{9}9$ . Divide both by 9: x = $\frac{3}{1}1$ .

Final answer

$\frac{3}{1}1$

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Convert 0.2̄7̄ (i.e. 0.272727...) into a fraction in its simplest form

4 marks4 minsrecurring-decimals-worked

Show solution

Worked solution

1.Let x equal the recurring decimal: x = 0.272727...
2.272727...We multiply by 100 so that the recurring block lines up with the original decimal, making it possible to subtract them.
3.Subtract the original equation from the new one: 100x − x = 27.272727... − 0.272727... so 99x = 27.
4.Solve and simplify: x = $2\frac{7}{9}9$ . Divide both by 9: x = $\frac{3}{1}1$ .

Final answer

$\frac{3}{1}1$

Mark points

M1: let x equal the recurring decimal
M1: multiply by 100 because two digits recur (27 repeats)
M1: subtract the original equation from the new one
M1: solve and simplify
A1: $\frac{3}{1}1$

Watch out

Students multiply by 10 regardless of how many digits recur — this only works when one digit repeats. g.), you must multiply by 100 so that the recurring block aligns perfectly and cancels when you subtract.If three digits repeat, multiply by 1000.

Diagnosticrecall

Convert 0.5̄ (0.5555...) to a fraction

1 mark2 minsrecurring-decimals-q1

Show solution

Worked solution

1.Spot the skill: The trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.
2.Use the let x equal the recurring decimal stage first, then multiply by 100 because two digits recur (27 repeats).
3.Keep the final answer visible: $\frac{5}{9}$ .

Final answer

$\frac{5}{9}$

Mark points

M1: use the correct the trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.two recurring digits → multiply by 100. three recurring digits → multiply by 1000.
A1: $\frac{5}{9}$

Watch out

Easyprocedure

Convert 0.4̄ (0.4444...) to a fraction

2 marks3 minsrecurring-decimals-q2

Show solution

Worked solution

1.Spot the skill: The trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.
2.Use the multiply by 100 because two digits recur (27 repeats) stage first, then subtract the original equation from the new one.
3.Keep the final answer visible: $\frac{4}{9}$ .

Final answer

$\frac{4}{9}$

Mark points

M1: use the correct the trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.two recurring digits → multiply by 100. three recurring digits → multiply by 1000.
A1: $\frac{4}{9}$

Watch out

Mediumreasoning

Convert 0.1̄8̄ (0.181818...) to a fraction

3 marks4 minsrecurring-decimals-q3

Show solution

Worked solution

1.Spot the skill: The trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.
2.Use the subtract the original equation from the new one stage first, then solve and simplify.
3.Keep the final answer visible: $\frac{2}{1}1$ .

Final answer

$\frac{2}{1}1$

Mark points

M1: use the correct the trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.two recurring digits → multiply by 100. three recurring digits → multiply by 1000.
A1: $\frac{2}{1}1$

Watch out

Hardproblem solving

Convert 0.41̄ (0.4111...) to a fraction

3 marks5 minsrecurring-decimals-q4

Show solution

Worked solution

1.Spot the skill: The trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.
2.Use the solve and simplify stage first, then let x equal the recurring decimal.
3.Keep the final answer visible: $3\frac{7}{9}0$ .

Final answer

$3\frac{7}{9}0$

Mark points

M1: use the correct the trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.two recurring digits → multiply by 100. three recurring digits → multiply by 1000.
A1: $3\frac{7}{9}0$

Watch out

Exam-stylemulti-step

Show that 0.9̄ = 1 using the algebraic method

4 marks6 minsrecurring-decimals-q5

Show solution

Worked solution

1.Spot the skill: The trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.
2.Use the let x equal the recurring decimal stage first, then multiply by 100 because two digits recur (27 repeats).
3.Keep the final answer visible: Let x = 0.999...; 10x = 9.999...; 9x = 9; x = 1.

Final answer

Let x = 0.999...; 10x = 9.999...; 9x = 9; x = 1

Mark points

M1: use the correct the trick: multiply by 10ⁿ where n is the number of recurring digits, then subtract to eliminate the recurring part.two recurring digits → multiply by 100. three recurring digits → multiply by 1000.
A1: Let x = 0.999...; 10x = 9.999...; 9x = 9; x = 1

Watch out

Grade 9 stretchproblem solving

Write 0.1666... as a fraction in its simplest form.

4 marks7 minsrecurring-g9

Show solution

Worked solution

1.Let x = 0.1666...
2.Use 10x and 100x so the recurring digits align.
3.Subtract and simplify.

Final answer

$\frac{1}{6}$

Mark points

M1: form two aligned equations
M1: subtract correctly
A1: $\frac{1}{6}$

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Hard exam-stylemulti-step problem

Write 0.2333... as a fraction in its simplest form.

3 marks6 minsrecurring-paper

Show solution

Worked solution

1.Let x = 0.2333....
2.Use 10x and 100x so the recurring digits line up.
3.Subtract and simplify.

Final answer

$\frac{7}{3}0$

Mark points

M1: form 10x = 2.333... and 100x = 23.333...
M1: subtract to obtain 90x = 21
A1: obtain $\frac{7}{3}0$

Watch out

Read the full question before calculating. Keep each stage of your working visible.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Recurring decimals as fractions - 2 marksConvert 0.5̄ (0.5555...) to a fractionMark answer

Answer

$\frac{5}{9}$

2Calculations and order of operations - 2 marksWork out (6 + 2) × 3 − 5Mark answer

Answer

3Integers, decimals and place value - 2 marksOrder these from smallest to largest: 0.3, 0.03, 0.303, 0.033Mark answer

Answer

0.03, 0.033, 0.3, 0.303

4Fractions - 3 marksA recipe needs ¾ cup of sugar. How much sugar is needed for 2½ batches?Mark answer

Answer

1⅞ cups

Mastery check

I can explain the method for recurring decimals as fractions.
I can show clear working without skipping key steps.
I can avoid this mistake: Students multiply by 10 regardless of how many digits recur — this only works when one digit repeats.g. ), you must multiply by 100 so that the recurring block aligns perfectly and cancels when you subtract.If three digits repeat, multiply by 1000.

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