Ratios
Comparing two quantities, and the part-to-part vs part-to-whole trap.
The explanation
A ratio compares two quantities. If a class has 12 boys and 16 girls, the ratio of boys to girls is 12:16, which simplifies to 3:4.
Order matters. 3:4 boys to girls is not the same statement as 4:3.
The trap: 3:4 is part-to-part. The total is 3 + 4 = 7 parts, so boys are 3/7 of the class, not 3/4. Any time a question mixes ratios with totals, write down the number of parts first.
A ratio a:b is an equivalence class of ordered pairs under scaling — a:b and ka:kb denote the same ratio for any k ≠ 0. That is why ratios simplify exactly like fractions.
The practical technique is the unit-of-parts method. Given a:b, introduce a scale factor k so the quantities are ak and bk. The total is k(a+b), which converts any total into k in one step and then recovers both quantities. For 3:4 with a total of 28: 7k = 28, so k = 4, giving 12 and 16.
Ratios extend to more than two terms (a:b:c), where the same scaling logic applies, and to continued comparisons where you must first rescale so the shared term matches — combining 2:3 with 3:5 on the middle term requires converting to 2:3 and 3:5 with a common 3, giving 2:3:5.
Worked example
A recipe uses flour to sugar in the ratio 5:2. If you use 350 g of flour, how much sugar?
- 5 parts flour = 350 g, so 1 part = 70 g.
- Sugar is 2 parts.
- 2 × 70 = 140.
Answer: 140 g of sugar
Common mistakes
- Reading 3:4 as three quarters. It is three sevenths of the whole.
- Reversing the order of the terms.