A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct?
Answer & explanation
Answer: (c) R < S
At the same rate, half-yearly compounding earns interest on interest and so grows faster than annual compounding. To reach the same amount Q in one year, the half-yearly rate R must therefore be smaller than the annual rate S.
- Half-yearly at R% a year: Q = P(1 + R/200)². Annually at S%: Q = P(1 + S/100).
- So 1 + S/100 = (1 + R/200)² = 1 + R/100 + R²/40000.
- S = R + R²/400, which is greater than R for any positive rate.
- Example: R = 10 gives (1.05)² = 1.1025, so S = 10.25. Hence R < S.
- Option (d), R ≤ S, is not false as a statement, but equality never occurs for a positive rate; the exact relation is R < S.
Remember · More frequent compounding at the same rate gives more; to give the same amount, the more frequent rate must be lower.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·