Decimals to fractions — terminating and recurring

A WJEC Higher Unit 1 staple: convert a recurring decimal to a fraction in lowest terms.

Terminating decimals

A terminating decimal stops after a finite number of digits.

Method:

1. Read the place value: 0.45 → 45/100.
2. Simplify: 45/100 = 9/20.

Examples:

• 0.7 = 7/10.
• 0.125 = 125/1000 = 1/8.
• 0.04 = 4/100 = 1/25.

Recurring decimals

A recurring decimal has one or more digits repeating forever. Common notations:

• 0.333… or 0.3̇ (one dot, "three recurring").
• 0.272727… or 0.2̇7̇ (dots over the 2 and the 7, the recurring block is "27").
• 0.1666… (only the 6 recurs).

Method: recurring → fraction

The trick: multiply by a power of 10 to shift the recurring block, then subtract.

Example: convert 0.4̇ (= 0.444…) to a fraction.

• Let x = 0.444…
• Then 10x = 4.444…
• Subtract: 10x − x = 4.444… − 0.444… = 4.
• 9x = 4 → x = 4/9.

Example: convert 0.2̇7̇ (= 0.272727…) to a fraction.

• Block of 2 digits → multiply by 100.
• Let x = 0.272727…
• Then 100x = 27.272727…
• Subtract: 99x = 27 → x = 27/99 = 3/11.

Example with non-recurring start: 0.16̇ (= 0.1666…).

• Multiply by 10 to push the recurring part: 10x = 1.666… (let this be A).
• Multiply by 100: 100x = 16.666… (let this be B).
• Subtract B − A: 90x = 15 → x = 15/90 = 1/6.

When does a fraction give a recurring decimal?

A fraction in its lowest terms gives:

• A terminating decimal iff its denominator's prime factors are only 2 and/or 5.
• A recurring decimal otherwise.

So 1/8 (denom = 2³) terminates; 1/3, 1/7, 1/11 recur.

WJEC exam tip

Always start a recurring-decimal conversion with "Let x = …" — this is the M1 communication mark. The subtraction is the second M1 and simplifying to lowest terms is the A1.