Ratios, Proportions and Scaling
Ratio questions test whether you can hold the distinction between part-to-part and part-to-whole, and whether you can chain two ratios into a single three-way ratio. Almost all of them reduce to finding the size of one part.
Core rules
- Simplify by dividing every term by the highest common factor; ratios carry no units.
- Dividing a quantity in the ratio a:b:c: total parts = a + b + c; one part = quantity / (a + b + c); multiply each term by that value.
- Part-to-part versus part-to-whole: in a 3:5 ratio the first share is 3/8 of the total, not 3/5.
- Chaining: given a:b and b:c, equalise the shared b term using the LCM to obtain a:b:c.
- Direct proportion: both quantities scale by the same factor.
- Inverse proportion: their product is constant - doubling the machines halves the time.
Worked example
Analysts to associates are in the ratio 3:5, and associates to managers in the ratio 4:1. Total headcount is 148. How many of each?
Associates appear as 5 and as 4; the LCM is 20. Analysts:associates becomes 12:20 and associates:managers becomes 20:5. So analysts:associates:managers = 12:20:5, giving 37 parts. One part = 148/37 = 4, hence 48 analysts, 80 associates and 20 managers. Check: 48 + 80 + 20 = 148.
If a follow-up says the total rises to 185 with the ratio unchanged, one part becomes 5 and all three terms scale together - never just the term mentioned in the question.
Scaling and maps
A scale of 1:25,000 means 1 cm on the map represents 25,000 cm, or 250 m, on the ground. Areas scale by the square of the linear factor: a 1:50 drawing covers 1/2,500 of the real area, not 1/50.
Traps
- Treating the ratio terms as fractions of the total when the ratio is part-to-part.
- Chaining ratios without equalising the shared term first.
- Scaling one side of a ratio and not the other.
- Reading a ratio of percentages (20%:30%) as two percentages of the total.
- Mistaking inverse for direct proportion: more workers mean fewer days, not more.