Decimal-to-fraction conversion

Edexcel tests terminating decimal conversion at Foundation and recurring decimal conversion at Higher (often as a 4-mark "show that" or QWC item on Paper 1H).

Terminating decimals

A terminating decimal stops after a finite number of digits (e.g. 0.625, 0.04). Convert by reading place value and simplifying.

Examples:

• 0.6 = 6/10 = 3/5.
• 0.25 = 25/100 = 1/4.
• 0.625 = 625/1000 = 5/8.
• 0.04 = 4/100 = 1/25.

A terminating decimal arises whenever the simplified denominator has only 2s and/or 5s as prime factors.

Recurring decimals — the algebraic method

Higher tier: convert a recurring decimal to a fraction using the "let x = ..., multiply by a power of 10" trick.

One-digit recurrence (e.g. 0.6̇ = 0.666...)

1. Let x = 0.666...
2. Then 10x = 6.666...
3. Subtract: 10x − x = 6.666... − 0.666... = 6.
4. So 9x = 6, giving x = 6/9 = 2/3.

Two-digit recurrence (e.g. 0.4̇5̇ = 0.454545...)

1. Let x = 0.454545...
2. Then 100x = 45.454545...
3. Subtract: 99x = 45.
4. So x = 45/99 = 5/11.

Three-digit recurrence (e.g. 0.1̇28̇ = 0.128128128...)

1. Let x = 0.128128...
2. 1000x = 128.128128...
3. Subtract: 999x = 128.
4. x = 128/999.

Mixed (some non-recurring digits, e.g. 0.16̇ = 0.1666...)

1. Let x = 0.1666...
2. 10x = 1.666... and 100x = 16.666...
3. Subtract: 100x − 10x = 16.666... − 1.666... = 15.
4. So 90x = 15, giving x = 15/90 = 1/6.

Edexcel exam tip

For Higher Paper 1H (QWC), write every step: define x, multiply by the correct power of 10, subtract, simplify. Mark scheme awards M1 for the correct multiplier, M1 for the subtraction, M1 for the algebraic line, A1 for the simplified fraction.