IGCSE Maths Fractions, decimals, percentages notes

A proper fraction (3/8) is less than one whole; an improper fraction (13/5) is greater than one; a mixed number (2⅗) pairs the whole parts with a proper fraction. A percentage is a fraction with denominator 100. The conversions run through division and multiplication by 100: 0.375 = 37.5% = 3/8, and 7/8 = 7 ÷ 8 = 0.875 = 87.5%. Always give fractions in their simplest form — 6/14 must become 3/7.

Extended. A recurring decimal repeats forever and is written with dots over the repeating block: 0.2̇7̇ means 0.272727…. To convert one to a fraction, subtract the shifted decimal: let x = 0.2̇7̇, then 100x − x = 27.2727… − 0.2727… = 27, so 99x = 27 and x = 27/99 = 3/11. Terminating decimals are easier: 0.375 = 375/1000 = 3/8.

To add or subtract fractions, build a common denominator first: 3/8 + 1/4 = 3/8 + 2/8 = 5/8. To multiply, multiply the numerators and denominators straight across, cancelling early where you can. To divide, flip the second fraction and multiply: (2/3) ÷ (4/9) = (2/3) × (9/4) = 18/12 = 3/2. Mixed numbers should be converted to improper fractions before multiplying or dividing: 2⅗ × 1½ = (13/5) × (3/2) = 39/10 = 3.9.

Order of operations matters: brackets, then powers, then multiplication and division left to right, then addition and subtraction left to right. So 2 + 3 × 4 = 14, not 20; and (2 + 3) × 4 = 20. Negative numbers obey the same rules with sign care: (−5) + 12 = 7, 3 − (−4) = 7, and (−6) ÷ (−2) = +3.

To find a percentage of a quantity, use a multiplier: 15% of 48 = 0.15 × 48 = 7.2. To express one quantity as a percentage of another, build the fraction: 9 out of 24 = 9/24 = 37.5%. Percentage change is change ÷ original × 100 — a price moving from 80 to 92 has risen by 12/80 × 100 = 15%. Increase 240 by 12.5% with the multiplier 1.125: 240 × 1.125 = 270.

Extended — reverse percentages. If a sale price is known *after* a change, divide by the multiplier. A shirt costing 48 after a 20% discount has original price 48 ÷ 0.8 = 60 — *not* 48 + 20%, because the 20% came off the original, not the sale price. The same logic finds a cost price from a selling price with a known profit percentage: sold at 207 with 15% profit means 207 = 1.15 × cost, so cost = 180.

Simple interest pays the same amount every year: interest = P × r × n ÷ 100. So 450 invested at 4% per year for 3 years earns 450 × 0.04 × 3 = 54 in interest (total 504). Compound interest multiplies each year by (1 + rate): 800 at 5% compounded for 3 years becomes 800 × 1.05³ = 926.10, so the interest is 126.10. Neither formula is printed on the paper — recall the multiplier method.

Extended. Repeated percentage change chains multipliers: rise 15% then fall 10% gives × 1.15 × 0.90 = × 1.035, a net 3.5% increase. Exponential growth and decay reuse the compound idea on any quantity: a population 12 000 growing at 3% per year reaches 12 000 × 1.03ⁿ after n years, while a car worth 12 500 depreciating 18% per year is worth 12 500 × 0.82² = 8405 after two years.

Calculate time in the 24-hour clock: 3.15 p.m. is 15 15, and 07 45 is 7.45 a.m. Durations cross midnight cleanly once you work in hours and minutes as separate columns: from 07 45 to 15 15 is 7 h 30 min. Timetables are read down the columns; time zones shift local time by whole hours (or fractions) — moving east adds, moving west subtracts. A year is taken as 365 days.

Money follows the conventions of the display: in money, 4.8 means 4.80; in time, 3.25 hours means 3 h 15 min (0.25 × 60 minutes — not a quarter past three on the clock face unless converted). Currency conversion multiplies or divides by the rate depending on direction: if £1 = $1.25, then £80 = $100, and $50 = £40. Decide the direction by checking that the answer is sensible (a bigger dollar figure for the same £).