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# Recurring Decimals to Fractions
- URL: https://www.esheets.io/recurring-decimals-to-fractions/
- Published: 2026-09-28T20:09:57.000Z
- Updated: 2026-09-28T20:09:57.000Z
- Author: Richard Linnington
- Tags: Maths, Number

Practise converting recurring decimals to fractions in simplest form. This Grade 6 worksheet moves from single recurring digits to longer recurring blocks, mixed recurring decimals and values greater than 1.

Use place value, powers of 10 and algebraic subtraction to find an exact fraction, then simplify it using the highest common factor. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Eight questions progress from pure recurring decimals with one, two and three recurring digits to decimals with a non-recurring prefix.

Later questions include values greater than 1 and a final decimal with two non-recurring decimal digits. Enter every answer as a fraction in simplest form, using a/b.

#### What you’ll practise

- Reading dots that identify the recurring block.
- Choosing powers of 10 to align recurring parts before subtracting.
- Finding exact fractions and simplifying fully.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Recurring decimals to fractions

Convert each decimal to a fraction in simplest form, entering a/b. Dots mark the first and last recurring digits (one dot for a single recurring digit).

## Topic guide

### Reading a recurring decimal

A recurring decimal has a digit or block of digits that repeats forever. Dots above the first and last digits mark the recurring block; one recurring digit has just one dot. For example, 0.•1•5 repeats the block 15, while 0.2•7 has a non-recurring 2 followed by recurring 7s.

### The method

1. Let x equal the recurring decimal.
2. Multiply by a power of 10 to move all non-recurring decimal digits before the decimal point.
3. Multiply again by 10 for each digit in the recurring block.
4. Subtract the two equations with matching recurring parts.
5. Divide by the coefficient of x and simplify the fraction fully.

### Pure recurring example

Let x = 0.•1•5. The recurring block has two digits, so 100x = 15.•1•5.

Subtract x from 100x: 99x = 15\. Therefore x = 15/99 = 5/33, so 0.•1•5 \= **5/33**.

### Mixed recurring example

Let x = 0.2•7. First move the non-recurring digit before the decimal point:

10x = 2.•7  
100x = 27.•7

Subtract the aligned equations: 100x − 10x = 27 − 2, so 90x = 25\. Therefore x = 25/90 = 5/18, and 0.2•7 \= **5/18**.

Other algebraic routes can work, but aligning the recurring tails before subtracting keeps the calculation clear. With two non-recurring decimal digits, start with 100x; then use 1000x for a one-digit block or 10000x for a two-digit block.

### Values greater than 1

Keep the whole-number part in each equation. For x = 1.•4, 10x = 14.•4, so 9x = 13\. Hence 1.•4 \= **13/9**. An improper fraction is a valid answer.

### Common mistakes to avoid

- Choose powers of 10 from the lengths of the prefix and recurring block. Check that the decimal tails match before subtracting.
- Subtract both sides in the same order. For the mixed example, the coefficient is 100 − 10 = 90, not 99.
- Use whole numbers for the numerator and denominator; do not leave a decimal inside the fraction.
- Divide the numerator and denominator by their highest common factor. An equivalent fraction that is not simplified is not a finished answer.

Finish by stating the original decimal equals your simplified fraction. Enter it as a/b with a positive denominator.