If 20% of A = 30% of B = 1/6 of C, then A : B : C is:

Concept — when several different multiples of different quantities are all set equal to one another, give that common value a single name, say k. Each quantity is then k divided by its own multiplier, so the quantities come out inversely proportional to those multipliers: the larger the fraction taken of a quantity, the smaller that quantity must be to reach the same k. Percentages and fractions therefore have to be rewritten as plain multiplying factors before the rule can be used.

Application — convert every expression to a multiplier, equate them, then solve for each letter.

1. Rewrite each expression as a plain multiplier of its own letter: 20% of A = A/5, 30% of B = 3B/10, and one-sixth of C = C/6.

2. All three are equal, so set them to one common value k: A/5 = 3B/10 = C/6 = k.

3. Solve each equation for its own letter: A = 5k, B = 10k/3, and C = 6k.

4. Write the ratio and cancel the common k: A : B : C = 5 : 10/3 : 6.

5. Multiply every term by 3 to clear the fraction: A : B : C = 15 : 10 : 18.

Cross-check — substitute a value back, and test the inverse-proportion reasoning independently.

• Take k = 30. Then A = 150, B = 100 and C = 180. Checking: 20% of 150 = 30, 30% of 100 = 30, and 1/6 of 180 = 30, so all three expressions do agree, and 150 : 100 : 180 reduces to 15 : 10 : 18.

• Inverse-proportion sanity check: the multipliers are 0.30 for B, 0.20 for A and about 0.167 for C; since a larger multiplier needs a smaller quantity to reach the same k, the sizes must run C > A > B, and 18 > 15 > 10 follows that order.

Result — A : B : C = 15 : 10 : 18.