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?

The core concept here is the effective annual rate of interest.

1. Scenario 1 (Half-yearly compounding with R% annual rate): When interest is compounded half-yearly, the interest is calculated twice a year. For an annual rate of R%, the rate applied per half-year is R/2%. The amount Q after 1 year is given by: Q = P * (1 + (R/2)/100)^2 = P * (1 + R/200)^2

2. Scenario 2 (Annually compounding with S% annual rate): When interest is compounded annually, the interest is calculated once a year. For an annual rate of S%, the rate applied per year is S%. The amount Q after 1 year is given by: Q = P * (1 + S/100)^1 = P * (1 + S/100)

3. Equating the amounts: Since P becomes Q in 1 year in both cases, the final amounts are equal: P * (1 + R/200)^2 = P * (1 + S/100) (1 + R/200)^2 = (1 + S/100)

4. Analysis:

   • The term (1 + S/100) represents the growth factor for one year with annual compounding at rate S. This is the effective annual growth factor.
   • The term (1 + R/200)^2 represents the growth factor for one year with half-yearly compounding at an annual rate R. This is also the effective annual growth factor.
   • For any positive interest rate, compounding more frequently (half-yearly) results in a higher effective annual rate than compounding less frequently (annually) for the same nominal annual rate.
   • However, in this problem, the effective annual growth (P becoming Q) is the same for both scenarios.
   • To achieve the same effective annual growth with more frequent compounding (half-yearly for R), the nominal annual rate (R) must be lower than the nominal annual rate (S) for less frequent compounding (annually for S).
   • Mathematically, expanding (1 + R/200)^2 gives 1 + 2(R/200) + (R/200)^2 = 1 + R/100 + R^2/40000.
   • So, 1 + R/100 + R^2/40000 = 1 + S/100
   • R/100 + R^2/40000 = S/100
   • S = R + R^2/400 (multiplying by 100)
   • Since R is a positive interest rate, R^2/400 will be a positive value.
   • Therefore, S must be greater than R (S > R), which means R < S.

Conclusion: For the same principal to grow to the same amount in the same time, if one scenario involves more frequent compounding, its nominal annual rate must be lower than the nominal annual rate of the less frequently compounded scenario.

The final answer is C) R < S.