If 20% of (a + b) = 50% of (ab) and 40% of (a − b) = 70% of (ab), then what…

Concept: When two given conditions relate (a + b) and (a − b) to a common multiple of the product ab, treat the two percentage statements as a pair of simultaneous linear equations in a and b, with ab acting as a common factor. Adding and subtracting the pair isolates a and b individually, and dividing the two results gives any required ratio between them — the ab factor cancels, so the individual values need not be pinned down first.

Working:

1. Convert both percentage statements into equations: 20% of (a + b) = 50% of (ab) gives 0.2(a + b) = 0.5ab, i.e. a + b = 2.5ab. Similarly, 40% of (a − b) = 70% of (ab) gives 0.4(a − b) = 0.7ab, i.e. a − b = 1.75ab.

2. Add the two equations: (a + b) + (a − b) = 2.5ab + 1.75ab, so 2a = 4.25ab. Since a ≠ 0, divide both sides by a: 2 = 4.25b, giving b = 8/17.

3. Subtract the two equations: (a + b) − (a − b) = 2.5ab − 1.75ab, so 2b = 0.75ab. Since b ≠ 0, divide both sides by b: 2 = 0.75a, giving a = 8/3.

4. The required fraction is b ÷ a = (8/17) ÷ (8/3) = (8/17) × (3/8) = 3/17.

Cross-check: Divide the two simplified equations directly instead of solving a and b individually: (a + b)/(a − b) = 2.5ab/1.75ab = 10/7. Applying componendo-dividendo, a/b = (10 + 7)/(10 − 7) = 17/3, so b/a = 3/17 — the same result, confirming the answer without needing the individual values of a and b.

Hence, b is 3/17 of a.