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 ?
- (a) R = S
- (b) R > S
- (c) R < S ✓ UPSC's answer
- (d) R ≤ S
Why the answer is (c)
• Half-yearly compounding at R% a year for one year: Q = P × (1 + R/200) × (1 + R/200).
• Annual compounding at S%: Q = P × (1 + S/100).
• Equating: 1 + S/100 = 1 + R/100 + R²/40000, so S = R + R²/400.
• For any positive rate R²/400 > 0, so S is strictly greater than R — option (c), R < S.
Why the other options are wrong
- (a) R = S
- Half-yearly compounding earns more at the same rate, so S must exceed R; they cannot be equal.
- (b) R > S
- S = R + R²/400, so S is greater than R, not smaller.
- (d) R ≤ S
- R ≤ S allows equality, which is impossible for a positive rate; the exact relation is R < S.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.