Dividing a whole in a ratio

You already know how to share a whole in a two-term ratio: add the terms, divide the whole by that sum, and multiply each term by the result. For 12 in the ratio 2 : 1: 2 + 1 = 3, then 12 ÷ 3 = 4, then 2 × 4 = 8 and 1 × 4 = 4.

Exactly the same method works with any number of terms: nothing about it cared that there were two.

A builder needs 110 units of concrete in the ratio cement : sand : gravel = 1 : 1.5 : 3.

Add the terms: 1 + 1.5 + 3 = 5.5. Divide the whole: 110 ÷ 5.5 = 20. So every term is multiplied by 20:

And 20 + 30 + 60 = 110, as it must.

In general: to divide a quantity x in the ratio a : b : c : …, the parts are x × a/(a + b + c + …), x × b/(a + b + c + …), x × c/(a + b + c + …), and so on. Each fraction is 'this term over the total of the terms'.

For 50 mL of purple paint in the ratio 2 : 3 : 5 (total 10): red is 50 × 2/10 = 10 mL, blue is 50 × 3/10 = 15 mL, white is 50 × 5/10 = 25 mL.

The whole doesn't have to be a quantity of stuff. To draw a triangle with angles in the ratio 1 : 3 : 5, remember that the angles of a triangle add to 180°. The terms add to 9, so ∠A = 180 × 1/9 = 20°, ∠B = 180 × 3/9 = 60°, ∠C = 180 × 5/9 = 100°.