Dividing a whole in a multi-term ratio
Cut the whole into equal parts: one part for every unit of every term.
How much red in 50 ml of purple?
A special purple paint is made by mixing red, blue and white in the ratio 2 : 3 : 5. You need 50 ml of this paint in all.
How many ml of red paint do you need?
What this lesson covers
The idea
A quantity x divided in the ratio a : b : c : … has parts x × a/(a + b + c + …), x × b/(a + b + c + …), and so on.
How much red in 50 ml of purple?
A special purple paint is made by mixing red, blue and white in the ratio 2 : 3 : 5. You need 50 ml of this paint in all.
How many ml of red paint do you need?
- 2 ml
- 10 ml
- 20 ml
Find the size of one part
The ratio 2 : 3 : 5 makes 2 + 3 + 5 equal parts. Change the size of one part until the whole bar ends exactly on the flag. There are three jobs.
Parts out of all the parts
A quantity x divided in the ratio a : b : c : … has parts x × a/(a + b + c + …), x × b/(a + b + c + …), and so on.
The 50 ml of purple has 2 + 3 + 5 = 10 parts, so one part is 50 ÷ 10 = 5 ml: Red = 50 × 2/10 = 10 ml Blue = 50 × 3/10 = 15 ml White = 50 × 5/10 = 25 ml
A triangle with angles in the ratio 1 : 3 : 5 has 1 + 3 + 5 = 9 parts of 180° ÷ 9 = 20°, so the angles are 20°, 60° and 100°. The parts always add up to the whole: 20° + 60° + 100° = 180°.
Notes
A quantity x divided in the ratio a : b : c : … has parts x × a/(a + b + c + …), x × b/(a + b + c + …), and so on.
Check yourself
A 600 ml juice is made from mango, orange and lemon in the ratio 3 : 2 : 1. How many ml of mango are in it?
Answer: 300 ml
3 + 2 + 1 = 6 parts, so one part is 600 ÷ 6 = 100 ml. Mango has 3 parts: 3 × 100 = 300 ml.
The angles of a triangle are in the ratio 2 : 3 : 4. How many degrees is the largest angle?
Answer: 80 °
2 + 3 + 4 = 9 parts, so one part is 180° ÷ 9 = 20°. The largest angle has 4 parts: 4 × 20° = 80°. (The angles are 40°, 60° and 80°.)
120 is divided in the ratio 3 : 4 : 5. Which calculation gives the first part?
A quantity is divided in the ratio 2 : 3 : 5. The smallest part is 12. What is the whole quantity?
Answer: 60
The smallest part is 2 parts = 12, so one part is 6. The whole has 2 + 3 + 5 = 10 parts: 10 × 6 = 60. (The parts are 12, 18 and 30.)
- 120 × 3/5. The bottom must be all the parts added: 3 + 4 + 5 = 12, not the last term.
- 120 × 3/7. That adds only two terms (3 + 4). All three terms must be added: 3 + 4 + 5 = 12.
- 120 × 3/12 — correct. Yes! 3 + 4 + 5 = 12 parts, and the first part is 3 of them: 120 × 3/12 = 30.
- 120 × 3/4. The bottom must be all the parts added: 3 + 4 + 5 = 12, not the second term.