Ratios with more than two terms

More than two quantities

Ratios are not limited to two numbers. Viswanath’s spice powder uses coriander seeds (8 spoons), red chillies (4), toor dal (2 spoons) and fenugreek seeds (1 spoon), so the ratio of the four ingredients is 8 : 4 : 2 : 1. A ratio can have as many terms as there are quantities, as long as they all change by the same factor to keep the relationship proportional.

Keeping the taste the same

Puneet wants the same powder but has only 2 red chillies, half of Viswanath’s 4. Everything else must be halved too: 4 spoons of coriander, one spoon of toor dal and 0.5 spoon of fenugreek. The new ratio 4 : 2 : 1 : 0.5 is proportional to the old one, written 8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5. In general, a : b : c : d :: p : q : r : s when a ÷ p = b ÷ q = c ÷ r = d ÷ s.

Paint and concrete

Paint. A shade of purple needs red : blue : white = 2 : 3 : 5. Yasmin has 10 litres of white. White is 5 parts, so one part is 10 ÷ 5 = 2 litres; red is 2 × 2 = 4 litres and blue 3 × 2 = 6 litres, making 20 litres of purple in all.

Concrete. Cement, sand and gravel for strong pillars, beams and roofs go in the ratio 1 : 1.5 : 3, with steel rods for strength. With 3 bags of cement, multiply every term by 3: 3 : 4.5 : 9, which is 3 + 4.5 + 9 = 16.5 bags of concrete mix.

Dividing a whole in a ratio

To split 12 in the ratio 2 : 1, add the terms (2 + 1 = 3), divide the whole by the sum (12 ÷ 3 = 4) and multiply each term by that: 8 and 4. It works the same way with more terms.

For 110 units of concrete in the ratio 1 : 1.5 : 3, one set (1 + 1.5 + 3) makes 5.5 units, and 110 ÷ 5.5 = 20 sets: 20 units of cement, 30 of sand and 60 of gravel. As a formula, dividing x in the ratio a : b : c : … gives parts x × a ÷ (a + b + c + …), x × b ÷ (a + b + c + …), and so on. So 50 mL of the purple paint needs 50 × 2 ÷ 10 = 10 mL of red, 15 mL of blue and 25 mL of white.

Ratios in triangles

The three angles of a triangle add up to 180°, so angles in the ratio 1 : 3 : 5 make 9 parts of 20°: 20°, 60° and 100°. Sides can be in a ratio too. Triangles with sides in the ratio 3 : 4 : 5, such as 3, 4, 5 cm or 6, 8, 10 cm, all have the same right-angled shape but different sizes, so they are not all congruent. Sides in the ratio 1 : 3 : 5 cannot form a triangle at all, because 1 + 3 is less than 5.

Where you will meet them

Recipes, medicine doses, fertiliser mixes, school timetables and budgets all use ratios of several terms. The same idea draws pie charts, where 360° is divided in the ratio of the data. See pie charts, inverse proportion, the chapter notes and the NCERT solutions.