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+100R/2)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+100S)1
Since both scenarios yield the same amount Q from the same principal P, we can equate the two expressions:
P(1+200R)2=P(1+100S)
Assuming P is not zero, we can divide both sides by P:
(1+200R)2=1+100S
Expanding the left side of the equation:
1+2(200R)+(200R)2=1+100S
1+100R+40000R2=1+100S
Subtracting 1 from both sides and then multiplying the entire equation by 100:
R+400R2=S
Given that R represents an annual rate of interest, it is typically a positive value (R > 0). Consequently, the term 400R2 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:
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Option (1) R = S: This equality would only hold if 400R2=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.
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Option (2) R > S: This contradicts the derived relationship S=R+400R2. For any positive R, S is explicitly shown to be greater than R.
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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.