The price (p) of a commodity is first increased by k%; then decreased by k%; again increased by k%; and again decreased by k%. If the new price is q, then what is the relation between p and q?
Answer & explanation
Answer: (a) p(10⁴ − k²)² = q × 10⁸
Each k% rise followed by a k% fall multiplies the price by (100 + k)(100 − k)/10⁴ = (10⁴ − k²)/10⁴. Two such pairs give q = p(10⁴ − k²)²/10⁸, i.e. p(10⁴ − k²)² = q × 10⁸.
- A k% increase multiplies the price by (100 + k)/100; a k% decrease multiplies it by (100 − k)/100.
- One increase and one decrease: (100 + k)(100 − k)/10⁴ = (10⁴ − k²)/10⁴.
- This pair happens twice, so q = p × (10⁴ − k²)²/10⁸.
- Cross-multiplying: p(10⁴ − k²)² = q × 10⁸.
- Check with k = 10: q = p × 1.1 × 0.9 × 1.1 × 0.9 = 0.9801p, and (10⁴ − 100)²/10⁸ = 9900²/10⁸ = 0.9801.
Remember · Successive percentage changes multiply. A k% rise and k% fall together leave the factor (10⁴ − k²)/10⁴ — always a net loss.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·