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% annual rate gives Q = P(1 + R/200)^2 after 1 year.

• Annual compounding at S% gives Q = P(1 + S/100) after 1 year.

• Equating the two amounts: (1 + R/200)^2 = 1 + S/100.

• Expanding: 1 + R/100 + R^2/40000 = 1 + S/100, so S = R + R^2/400.

• Since R is a positive interest rate, R^2/400 > 0, hence S > R, i.e. R < S, which is option (c).

Why the other options are wrong

(a) R = S: R = S is false because half-yearly compounding at the same nominal annual rate would produce a larger amount, so the annual rate S must be higher to match Q.
(b) R > S: R > S is false because the equation gives S = R + R^2/400, making S greater than R.
(d) R ≤ S: R ≤ S is not the exact relation because equality R = S is impossible for a positive interest rate; the strict relation R < S is the correct one.

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.