Decimal ↔ fraction conversion

Every OCR J560 paper rewards confident conversion between decimal and fraction form. Foundation focuses on terminating decimals; Higher requires the algebraic method for recurring decimals.

Terminating decimals → fractions

Read the place value of the last digit, then simplify.

• 0.7 = 7/10
• 0.25 = 25/100 = 1/4
• 0.125 = 125/1000 = 1/8
• 0.36 = 36/100 = 9/25
• 1.04 = 104/100 = 26/25 (improper) or 1 1/25 (mixed)

Method: count the digits after the decimal point — that gives the power of 10 in the denominator. Then cancel.

Recurring decimals → fractions (Higher)

Use the dot/bar notation: 0.3̇ means 0.333..., and 0.1̇2̇ means 0.121212...

The standard algebraic method:

Convert 0.4̇5̇ (= 0.454545...) to a fraction.

Step 1: Let x = 0.454545... Step 2: Multiply by 10ⁿ where n is the length of the repeat (here n = 2). So 100x = 45.454545... Step 3: Subtract: 100x − x = 45.454545... − 0.454545... → 99x = 45 Step 4: Solve: x = 45/99 = 5/11.

Repeat length 1 (e.g. 0.3̇): multiply by 10. 10x − x = 3 → x = 3/9 = 1/3. Repeat length 3 (e.g. 0.1̇23̇): multiply by 1000. 1000x − x = 123 → x = 123/999 = 41/333.

Mixed: non-recurring then recurring

Convert 0.16̇ (= 0.1666...) to a fraction.

Multiply by 10 to shift the non-recurring part: 10x = 1.666... Multiply again by 10 to align the recurring: 100x = 16.666... Subtract: 100x − 10x = 16.666... − 1.666... → 90x = 15 → x = 15/90 = 1/6.

OCR mark scheme conventions

• M1 for setting up the algebraic equation (let x = ...).
• M1 for the multiplication and subtraction step that eliminates the recurring part.
• A1 for the final simplified fraction (cao for full marks).
• "Show that" demands the working — answer alone scores zero on Higher.