Ratio and Proportion: Cheat Sheet with Worked Examples

A ratio compares the sizes of two or more quantities. A bag with 12 red beads and 8 blue beads has a red-to-blue ratio of 12 : 8. The order of the words gives the order of the numbers, so blue to red is 8 : 12.

A ratio is not a fraction. The ratio 3 : 2 compares one part with another; the matching fractions compare each part with the whole. With 3 red beads for every 2 blue, there are 5 beads in each group, so red beads are 3/5 of the total and blue beads 2/5.

To write a ratio in its simplest form, divide every part by the highest common factor (HCF). 12 : 8 divides by 4 to give 3 : 2, and 18 : 24 divides by 6 to give 3 : 4.

Two extra checks: put every part in the same unit before simplifying, so 40 cm : 2 m becomes 40 : 200 = 1 : 5; and clear decimals by multiplying, so 1.5 : 2 becomes 3 : 4.

Use three steps: add the parts, find the value of one part, then multiply.

1. Share EUR 350 in the ratio 2 : 5. Total parts: 2 + 5 = 7.
2. One part: 350 ÷ 7 = EUR 50.
3. Shares: 2 × 50 = EUR 100 and 5 × 50 = EUR 250. Check: 100 + 250 = 350.

The method works for three-part ratios too: 72 kg in the ratio 1 : 3 : 5 is 9 parts of 8 kg, giving 8 kg, 24 kg and 40 kg. If you know one share instead of the total, find one part from that share: with boys to girls 4 : 5 and 20 boys, one part is 5, so there are 25 girls.

A unit rate is the amount for one unit of something: kilometres per hour, price per kilogram, pay per hour. Find it by dividing by the quantity you want one of. A car covering 150 km in 2.5 hours averages 150 ÷ 2.5 = 60 km/h.

To find the best buy, compare the price of the same amount. A 500 g bag of rice for GBP 2.40 costs GBP 0.48 per 100 g; a 750 g bag for GBP 3.45 costs GBP 0.46 per 100 g. The larger bag is better value, as long as you use it all.

Two quantities are in direct proportion when they change by the same factor: double one and the other doubles. The ratio between them stays the same, so they fit y = kx, where k is the unit rate, and their graph is a straight line through the origin.

If 250 g of flour makes 10 biscuits, one biscuit needs 25 g, so 18 biscuits need 450 g. Notebooks at USD 1.50 each are proportional: 3 cost USD 4.50 and 5 cost USD 7.50. A taxi with a fixed EUR 3 charge plus EUR 2 per km is not: 1 km costs EUR 5, but 2 km cost EUR 7, not EUR 10.

A scale is a ratio between a length on a drawing and the real length, in the same unit. On a plan at 1 : 50, a 3.5 m wall (350 cm) is drawn 350 ÷ 50 = 7 cm long. On a map at 1 : 25,000, 4 cm stands for 4 × 25,000 = 100,000 cm, which is 1,000 m or 1 km. Most mistakes happen in the final unit conversion, so write each step.