Correct Option (3)

The problem requires a comparison between two annual interest rates, R and S, which result in the same final amount Q from an initial principal P over one year, under different compounding frequencies.

When interest is compounded half-yearly at an annual rate of R%, the amount Q after one year (which consists of two half-year periods) is calculated as:
$Q = P {(1 + \frac{R / 2}{100})}^{2}$
When interest is compounded annually at an annual rate of S%, the amount Q after one year (which is one annual period) is calculated as:
$Q = P {(1 + \frac{S}{100})}^{1}$

Since both scenarios yield the same amount Q from the same principal P, we can equate the two expressions:

$P {(1 + \frac{R}{200})}^{2} = P (1 + \frac{S}{100})$

Assuming P is not zero, we can divide both sides by P:

${(1 + \frac{R}{200})}^{2} = 1 + \frac{S}{100}$

Expanding the left side of the equation:

$1 + 2 (\frac{R}{200}) + {(\frac{R}{200})}^{2} = 1 + \frac{S}{100}$

$1 + \frac{R}{100} + \frac{R^{2}}{40000} = 1 + \frac{S}{100}$

Subtracting 1 from both sides and then multiplying the entire equation by 100:

$R + \frac{R^{2}}{400} = S$

Given that R represents an annual rate of interest, it is typically a positive value (R > 0). Consequently, the term $\frac{R^{2}}{400}$ will also be a positive value. This mathematical relationship demonstrates that S is equal to R plus a positive quantity. Therefore, S must be greater than R.

Thus, $R < S$ .

Incorrect Options:

Option (1) R = S: This equality would only hold if $\frac{R^{2}}{400} = 0$ , which implies R = 0. For any positive interest rate R, R cannot be equal to S because of the additional compounding effect in the half-yearly calculation.
Option (2) R > S: This contradicts the derived relationship $S = R + \frac{R^{2}}{400}$ . For any positive R, S is explicitly shown to be greater than R.
Option (4) R ≤ S: While R < S is the correct relationship for positive interest rates, the inclusion of R = S makes this option less precise. If R=0, then S=0, making R=S true. However, in the context of meaningful interest rates, R is typically positive, leading to R < S. Therefore, R < S is the more accurate and specific answer.