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Recurring decimals

Free Printable Recurring decimals PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 6)

Download free recurring decimals worksheets (PDF), GCSE maths past paper questions and worked solutions for Recurring decimals (6A). This is a Grade 6 number topic from the GCSE questions by topic collection for Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Maths4Everyone, Corbett Maths, Maths Made Easy, Dr Frost, 1st Class Maths, Metatutor, Maths Takeaway, Maths DIY, Dr Austin, Mr Chapman, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our recurring decimals revision notes and worked examples with diagrams, try real GCSE past paper recurring decimals questions with step-by-step solutions, or download a PDF of all recurring decimals worksheet links.

Free Recurring decimals PDF Worksheets & Answers

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GCSE Maths Grade 6 Revision Notes & Method

Guide to Recurring decimals: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To convert a recurring decimal into a fraction algebraically, let x equal the recurring decimal written out with several repeating digits. Multiply x by powers of 10 (10x, 100x, 1000x) so that two equations have the exact same recurring tail after the decimal point.

Subtract the smaller equation from the larger equation so that all the recurring digits after the decimal point cancel out to zero, then divide and simplify the fraction.

Example 1: Prove algebraically that 0.̇4̇5 = 511.

Solution:

Let x = 0.454545...
Multiply by 100 (2 repeating digits): 100x = 45.454545...
Subtract (100x − x): 99x = 45
x = 4599 = 511 (divide top and bottom by 9)
Answer: 511

Example 2: Convert 0.2̇7̇3 (where 73 recurs) into a fraction in its simplest form.

Solution:

Let x = 0.2737373... → 10x = 2.737373...
Multiply by 1000: 1000x = 273.737373...
Subtract (1000x − 10x): 990x = 271
Divide by 990: x = 271990
Answer: 271990
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Recurring decimals Questions & Answers

Past Paper Example 1 OCR Jun 2024 5H Q17 5 marks
The diagram shows a number line.
PQR
P = 1.2̇ and Q = 123.
Q is the midpoint of PR.
Find the value of R.
Give your answer as a mixed number in its simplest form.
You must show your working.
Worked Solution & Mark Scheme (5 marks)
P = 1.2̇ = 1 2/9 = 11/9. M2
Q = 1 2/3 = 1 6/9 = 15/9. M1
Step from P to Q = 15/9 − 11/9 = 4/9. M1
R = Q + 4/9 = 15/9 + 4/9 = 19/9 = 2 1/9. A1
Answer: 219
Past Paper Example 2 GCSE Nov 2022 2H Q14 3 marks
Using algebra, prove that 1.062̇ can be written as 114225
Worked Solution & Mark Scheme (3 marks)
1000x – 100x = 956 M2
⇒ x = 956/900 = 239/225 = 1 14/225 C1
Answer: Let x = 1.06222..., 100x = 106.222..., 1000x = 1062.222...; 900x = 956 ⇒ x = 956/900 = 1 56/900 = 1 14/225
Past Paper Example 3 GCSE May 2022 1H Q12 3 marks
Express 0.11̇7̇ as a fraction.
You must show all your working.
Worked Solution & Mark Scheme (3 marks)
Let x = 0.1171717... M1
10x = 1.171717... and 1000x = 117.171717... M1
990x = 116 ⇒ x = 116/990 (or 58/495) A1
Answer: 116990

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