10.4 Dividing a Whole in a Given Ratio

Earlier you learned to split a whole in a two-term ratio. To divide 15 in the ratio 3 : 2: add the terms (3 + 2 = 5), divide the whole by this sum (15 \div 5 = 3), then multiply each term by the quotient, giving 3 \times 3 = 9 and 2 \times 3 = 6. So 15 split in 3 : 2 is 9 : 6.

The same recipe stretches to ratios with many terms.

Worked Example

88 units of mortar are required, mixed cement : sand : gravel = 1 : 1.5 : 3. How much of each?

The terms add to 1 + 1.5 + 3 = 5.5 units. We must repeat this 88 \div 5.5 = 16 times, so multiply each term by 16: 1 \times 16 = 16 \text{ units cement}, \quad 1.5 \times 16 = 24 \text{ units sand}, \quad 3 \times 16 = 48 \text{ units gravel}.

The Sharing Formula When a quantity x is divided in the ratio a : b : c : \dots, the parts are x \times \frac{a}{a + b + c + \dots}, \quad x \times \frac{b}{a + b + c + \dots}, \quad x \times \frac{c}{a + b + c + \dots}, \quad \dots

Worked Example

Make 80 ml of leaf-green paint from Yellow : Blue : White = 3 : 2 : 5.

The terms add to 3 + 2 + 5 = 10, so \text{Yellow} = 80 \times \tfrac{3}{10} = 24 \text{ ml}, \quad \text{Blue} = 80 \times \tfrac{2}{10} = 16 \text{ ml}, \quad \text{White} = 80 \times \tfrac{5}{10} = 40 \text{ ml}.

Worked Example

Construct a triangle with angles in the ratio 2 : 3 : 4.

The angles of a triangle add to 180°, and the ratio terms add to 2 + 3 + 4 = 9. So \angle A = 180° \times \tfrac{2}{9} = 40°, \quad \angle B = 180° \times \tfrac{3}{9} = 60°, \quad \angle C = 180° \times \tfrac{4}{9} = 80°. A quick check: 40° + 60° + 80° = 180°.

Worked Example

A scholarship fund of ₹72{,}000 is shared among three students in the ratio 5 : 4 : 3. How much does each receive?

The terms add to 5 + 4 + 3 = 12, so one part = 72{,}000 \div 12 = ₹6{,}000. Therefore ₹5 \times 6{,}000 = ₹30{,}000, \quad ₹4 \times 6{,}000 = ₹24{,}000, \quad ₹3 \times 6{,}000 = ₹18{,}000. Check: 30{,}000 + 24{,}000 + 18{,}000 = ₹72{,}000.

Figure it Out: Sharing in a Ratio