GCSE Maths: Ratio & Proportion — Deep Revision Workbook

Inside GCSE Maths: Ratio & Proportion — Deep Revision Workbook

simplify and share ratios and link ratio to fractions, solve direct and inverse proportion with the unitary method and compare best buys, work with scale factors, maps and scale drawings, calculate percentage change and compound growth and decay, use compound measures — speed, density and pressure — and convert units and rates, then finish with an extended multi-step exam-style challenge. About 45 minutes.

Learning objectives

• simplifying ratios (including three-part ratios) to lowest terms, sharing amounts in a given ratio, and linking ratio to fractions and to a known part;
• direct and inverse proportion using the unitary method, and comparing best buys by cost per unit;
• scale factors, maps and scale drawings, converting between drawing and real-life lengths in cm, m and km;
• percentage change, increase and decrease, and compound growth and decay such as compound interest and depreciation over several years;
• compound measures — speed, density and pressure — rearranged both ways, plus rate conversion such as km/h to m/s.

Activities in this book

1. Section 1 — Simplifying, Sharing & Ratio to Fractions

   to simplify a ratio, divide every part by the same number (the highest common factor). To share an amount, add the parts to find the number of equal shares, work out one share, then multiply. In the ratio a:b the first part is the fraction a/(a+b) of the whole. If you know one part in real life, find the value of one share first.

2. Section 2 — Direct & Inverse Proportion, Best Buys

   use the unitary method: find the value of ONE item first, then scale up or down. In direct proportion both amounts grow together. In inverse proportion (workers and time, taps and time) more of one means less of the other — multiply the two numbers to find the fixed total, then divide. For best buys, compare the cost of one unit and pick the smaller.

3. Section 3 — Scale, Maps, Percentage Change & Compound

   a map ratio like 1:25000 means 1 unit on paper is 25000 units in real life — convert carefully (100 cm = 1 m, 100000 cm = 1 km). For percentage change, divide the actual change by the ORIGINAL amount, then multiply by 100. For compound growth or decay, multiply by the same decimal multiplier once for each year (1.04 grows by 4%, 0.9 falls by 10%).

4. Section 4 — Compound Measures & Rate Conversion

   compound measures combine two quantities: speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area. Rearrange for what you need (distance = speed × time, mass = density × volume, force = pressure × area). Check units match first, and to change km/h to m/s multiply by 1000 and divide by 3600.

5. Section 5 — Extended Exam Challenge

   multi-step exam questions weave several tools together — a ratio, a proportion, a percentage and maybe a compound measure — inside one real context. Read carefully, plan the order of steps, convert units early, and show every line of working so an examiner can award method marks even if a final figure slips.

Suggestions for teaching

• Unitary method first. Model finding the value of one item before scaling — it works for cost, ingredients and best buys alike.
• Ratio bar models. Draw the parts of a ratio as equal bars so sharing becomes “find one part, then multiply”, and extend to three-part ratios.
• Original amount for percentages. Stress that percentage change is always divided by the original, and that compound change multiplies once per year.
• Formula triangles. Teach the speed, density and pressure triangles and rearrange each to find the missing quantity.
• Convert units early. Insist on matching units (km to m, hours to minutes, km/h to m/s) before substituting into any formula.