Ratio, proportion and scaling
Maths · GCSE · The School
One part first
Almost every ratio question falls out once you know what one part is worth. Add the numbers in the ratio to get the number of parts. Divide the total by that. Then multiply back up for each share. Share 45 in the ratio 4:5 — nine parts, so one part is 5, giving 20 and 25. Check by adding: 20 plus 25 is 45.
When you are not given the total
Exams often give you one side instead. Concrete is cement to sand in the ratio 1:4, and you have 12 kg of cement. Do not divide anything by five — the total was never mentioned. One part is 12 kg, so the sand is 48 kg and the total happens to be 60 kg. Ask which number you have been handed before you touch it.
Direct and inverse proportion
In DIRECT proportion, doubling one doubles the other: three books cost 12, so six cost 24. In INVERSE proportion, doubling one halves the other: three painters take 8 hours, so six painters take 4. The test is what happens when you double the first quantity. Getting these the wrong way round turns a sensible answer into an absurd one, which is your warning.
A recipe for 4 people uses 200 g rice, 300 ml stock and 2 onions. FOR 6 PEOPLE, the scale factor is 6 divided by 4, which is 1.5: multiply everything by 1.5 to get 300 g, 450 ml and 3 onions. FOR 10 PEOPLE, the factor is 2.5: 500 g, 750 ml, 5 onions. NOW BACKWARDS — you have only 150 g of rice. That is 150 divided by 200, a factor of 0.75, so it serves 3 people and needs 225 ml of stock and 1.5 onions. Every one of those used the same move: find the scale factor, then apply it to everything. Notice the last one is the same skill run in reverse, which is what a harder exam question does to you.