1. Correct Answer: c: R < S
 2. Explanation:
     To solve this problem, we need to understand the difference between compounding interest half-yearly and annually.
     ● Compounding Half-Yearly: When interest is compounded half-yearly, the interest is calculated twice a year. Therefore, the effective rate of interest for each half-year is \( \frac{R}{2} \)%.  
     ● Compounding Annually: When interest is compounded annually, the interest is calculated once a year at the rate of S%.  
     Let's derive the expressions for the final amount \( Q \) in both scenarios:
     1. Compounded Half-Yearly:
     The formula for compound interest is given by:
     \[
     Q = P \left(1 + \frac{R/2}{100}\right)^2
     \]
     This is because the interest is compounded twice in a year, each time at a rate of \( \frac{R}{2} \)%.
     2. Compounded Annually:
     The formula for compound interest when compounded annually is:
     \[
     Q = P \left(1 + \frac{S}{100}\right)
     \]
     Since both scenarios result in the same final amount \( Q \), we equate the two expressions:
     \[
     P \left(1 + \frac{R/2}{100}\right)^2 = P \left(1 + \frac{S}{100}\right)
     \]
     Simplifying, we get:
     \[
     \left(1 + \frac{R/2}{100}\right)^2 = 1 + \frac{S}{100}
     \]
     Expanding the left side:
     \[
     1 + \frac{R}{100} + \left(\frac{R}{200}\right)^2 = 1 + \frac{S}{100}
     \]
     Subtracting 1 from both sides:
     \[
     \frac{R}{100} + \left(\frac{R}{200}\right)^2 = \frac{S}{100}
     \]
     Since \(\left(\frac{R}{200}\right)^2\) is a positive term, it implies that:
     \[
     \frac{R}{100} < \frac{S}{100}
     \]
     Therefore, \( R < S \).
     Hence, the correct answer is c: R < S.