GCSE Maths Fractions, decimals & percentages revision

A terminating decimal stops, and its denominator is a power of ten: 0.375 has three decimal places, so 0.375 = 375⁄1000, which simplifies to 3⁄8. Going the other way, 7⁄8 means 7 ÷ 8 = 0.875. To reach a percentage, scale to a denominator of 100: 0.375 = 37.5%. The fraction–decimal–percentage triangle is worth memorising for the famous values: ½ = 0.5 = 50%, ¼ = 0.25 = 25%, ⅕ = 0.2 = 20%, ⅛ = 0.125 = 12.5%, ⅓ = 0.333… and ⅔ = 0.666… (those last two never terminate — that is exactly why the Higher tier converts them with algebra).

To order a mixed list, convert everything to decimals first: given 0.62, ⅝ and 61%, compare 0.62, 0.625 and 0.61, so 61% < 0.62 < ⅝. Keep fractions as fractions when a question says “give your answer as a fraction” — examiners penalise the wrong form even when the value is right.

A recurring decimal repeats forever, written with dots over the repeating block: 0.7̇ = 0.777… and 0.5̇8̇ = 0.585858… Multiply by the power of ten that shifts one full repeat to the left of the point, then subtract the original: for x = 0.7̇, 10x = 7.7̇, so 10x − x = 7 and x = 7⁄9. For x = 0.5̇8̇, 100x = 58.5̇8̇, so 100x − x = 58 and x = 58⁄99. The repeating block length tells you the multiplier and the denominator: one digit → 9, two digits → 99.

Decimals with a non-repeating head need two shifts: x = 0.16̇ (0.1666…) has one non-repeating digit, so shift past it first. 10x = 1.666…, and 100x = 16.666…, so 100x − 10x = 15, giving 90x = 15 and x = 1⁄6. Always check by dividing back: 1 ÷ 6 = 0.1666… ✓

Reading a fraction or percentage as an operator turns word problems into one multiplication. “3⁄8 of 56” means 3⁄8 × 56 = 21. “Increase £240 by 15%” means £240 × 1.15 = £276: the multiplier 1.15 keeps the whole amount plus fifteen hundredths more. “Decrease by 12%” keeps 88 parts in every 100, so multiply by 0.88: 85 kg × 0.88 = 74.8 kg. Multipliers compose by multiplication, not addition — a 10% rise then a 10% fall multiplies by 1.1 × 0.9 = 0.99, a net 1% fall, not no change.

Two-stage problems chain the operators: “spends ⅖ of £90, then 25% of the rest” is 90 × ⅖ = £36 spent, £54 left, then 54 × 0.75 = £40.50 kept. Draw the two arrows when the wording stacks — each operator acts on the result of the last.

“What percentage is 15 of 48?” builds the fraction 15⁄48, then scales to hundredths: 15⁄48 = 0.3125 = 31.25%. The order matters: “63 as a percentage of 45” is 63⁄45 = 1.4 = 140% — values over 100% say the first quantity is bigger than the second, and they are perfectly good answers. Reversing the fraction answers a different question, so underline which quantity is the “of”.

Comparisons use percentages when the bases differ: Team A wins 21 of 30 matches (70%) and Team B wins 26 of 40 (65%), so A has the better win rate despite fewer wins. Change questions use the same construction with the original value underneath: a price moving from £48 to £54 has changed by 6⁄48 = 12.5%. Dividing by the new price instead is the most common percentage-change error there is.

A reverse percentage question gives the after value and asks for the before. Undo the multiplier with division: after a 20% increase, £90 is 1.2 × original, so original = 90 ÷ 1.2 = £75. After a 15% discount, £102 is 0.85 × original, so original = 102 ÷ 0.85 = £120. The classic error adds or subtracts the percentage from the given value — that measures the wrong base and cannot be marked correct, because adding 20% to 90 gives 108, not 75.

Simple interest pays the same slice every period: interest = P × r × n, with r the rate per period and n the number of periods. £600 at 4% per year for 3 years earns 600 × 0.04 × 3 = £72 interest, so the balance is £672. Simple interest is linear — the year-3 interest equals the year-1 interest — which is exactly what distinguishes it from the compound case at Higher.

Higher. When a percentage change repeats, the multipliers multiply: £600 at 4% compound interest for 3 years is 600 × 1.04³ = 600 × 1.124864 = £674.92 (to the nearest penny). Growth and decay use the same shape: a population growing 5% per year multiplies by 1.05 each year; a car losing 15% per year multiplies by 0.85, so £12 000 after two years is 12 000 × 0.85² = £8670. Fractional years follow the same rule — half a year of 4% interest is ×1.04^0.5 on some Higher questions.

The comparison examiners love: £600 at 4% simple interest for 3 years earns 600 × 0.04 × 3 = £72 (balance £672), while compound gives £674.92. Compound wins because each year’s interest itself earns interest. To find how long a value takes to pass a target, test successive powers: 600 × 1.04⁴ = £701.92 but 600 × 1.04⁵ = £730.00 (to 2 d.p.), so a £720 target is passed during the fifth year. Reverse questions — “what rate doubles a value in 10 years?” — solve x¹⁰ = 2 for x, which needs the later indices work.