Which fraction gives a recurring decimal?

1/3. 1/3 = 0.333…; the others end.

Recurring decimals · Cambridge IGCSE Mathematics (9

In plain words

Some fractions give decimals that never end but repeat for ever: 1/3 is 0.3333… Every repeating decimal is secretly a fraction, and there is a neat trick for finding which one.

Three things to know

A recurring decimal repeats a digit or a block of digits for ever. Every recurring decimal is a rational number.
To turn one into a fraction: call it x, multiply by 10 for each repeating digit (10 for one, 100 for two), and subtract x. The repeating part cancels.
Then solve for x and simplify.

Worked example

Write 0.454545… as a fraction.

Let x = 0.4545… Two digits repeat, so multiply by 100: 100x = 45.4545…
Subtract: 100x − x = 45.4545… − 0.4545…, so 99x = 45.
x = 45/99 = 5/11.

Tips and tricks

One repeating digit: multiply by 10. Two: by 100. Three: by 1000.
Always simplify at the end. 45/99 is not finished.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Recurring decimals: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is 0.3333… as a fraction?
- 3/10
- 1/3 (the answer)
- 33/100
- 3/100
9x = 3, so x = 3/9.
What is 0.5555… as a fraction?
- 1/2
- 5/10
- 5/9 (the answer)
- 55/100
10x − x = 5.
To convert 0.363636… you first multiply x by what?
- 10
- 100 (the answer)
- 1000
- 36
Two digits repeat.
What is 0.272727… as a fraction in its simplest form?
- 27/100
- 27/99
- 3/11 (the answer)
- 2/7
27/99 cancels by 9.
Which fraction gives a recurring decimal?
- 1/4
- 1/5
- 1/8
- 1/3 (the answer)
1/3 = 0.333…; the others end.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 8 marks. Write your answers on paper, then check them.