Multiplicative relationships as ratio or fraction
When two quantities are connected by multiplication (one is a constant multiple of the other), we can describe the link as a ratio, a fraction, or a decimal/percentage. All four are different views of the same relationship.
The core idea
If A is k times B (A = kB), then:
- Ratio: A : B = k : 1.
- A as a fraction of B: k.
- B as a fraction of A: 1/k.
Worked example: A = 30, B = 12. A is 2.5 times B.
- Ratio A : B = 30 : 12 = 5 : 2.
- A as fraction of B = 30/12 = 5/2 (= 2.5).
- B as fraction of A = 12/30 = 2/5.
Going from ratio to fraction
If A : B = 3 : 5, then:
- A is 3/5 of B (because for every 5 units of B, A has 3).
- B is 5/3 of A.
- The TOTAL split: A is 3/8 of total; B is 5/8.
Worded contexts
Worked example: in a school the ratio of girls to boys is 4 : 3. What fraction of pupils are girls?
- Total parts = 7. Girls = 4/7.
Worked example: a shop sells small and large t-shirts in the ratio 5 : 2. What fraction more small are sold than large?
- Difference = 3 parts; large = 2 parts. So small are 3/2 = 1½ times more.
When converting fraction → ratio
Worked example: 2/3 of a class are girls. What is the ratio of girls to boys?
- Girls : total = 2 : 3, so boys : total = 1 : 3.
- Girls : boys = 2 : 1.
Multiplicative scaling problems
Worked example: A is 1.4 times B. Express the relationship as a ratio in form a : b with integers.
- A : B = 1.4 : 1 = 14 : 10 = 7 : 5.
⚠Common mistakes
- Confusing "of total" with "of the other" — re-read what's being asked.
- Saying ratio 3 : 5 means 3/5 are X, when really X is 3 of 8 parts (3/8 of total).
- Decimal-fraction slip — A = 1.5B as A : B = 3 : 2, not 1 : 1.5 (always integerise).
- Reciprocal direction error — A is k times B is NOT B is k times A.
- Mixing additive with multiplicative — "5 more" is additive; "5 times" is multiplicative.
➜Try this— Quick check
A is ⅗ of B. Express A : B in simplest form. Answer: 3 : 5.
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