The compound interest on Rs. 24000 at 10% per annum for 1.5 years, when interest is compounded semiannually, is equal to how many rupees?

Introduction / Context:
This question mixes annual rates with semiannual compounding and a time period of one and a half years, which equals three half years. The goal is to test whether the learner correctly converts the annual rate into a rate per compounding period and then applies the correct number of periods. Such problems are standard in competitive exams where care with time units and compounding frequency is very important.

Given Data / Assumptions:

• Principal P = Rs. 24000.
• Nominal rate r = 10% per annum.
• Interest is compounded semiannually, that is twice a year.
• Total time = 1.5 years, which equals 3 half year periods.
• We need the compound interest CI on the principal.

Concept / Approach:
For semiannual compounding, the rate per half year is the annual rate divided by 2 and the number of compounding periods is the number of half years. Here, the rate per half year r1 is 10% / 2 = 5% and the number of periods n is 3 because 1.5 years contains three halves. The amount A after n periods is A = P * (1 + r1) ^ n. Once the amount is computed, the compound interest is CI = A - P. This framework correctly handles both the shorter time unit and the compounding structure.

Step-by-Step Solution:
Step 1: Compute the effective rate per half year: r1 = 10% / 2 = 5% = 0.05. Step 2: Compute the number of half year periods: 1.5 years * 2 = 3 periods. Step 3: Use the formula A = P * (1 + r1) ^ n, giving A = 24000 * (1.05) ^ 3. Step 4: First calculate (1.05) ^ 2 = 1.1025. Step 5: Multiply by 1.05 once more to get (1.05) ^ 3 = 1.157625. Step 6: Compute the amount: A = 24000 * 1.157625 = 27783. Step 7: Compute the compound interest: CI = A - P = 27783 - 24000 = 3783. Step 8: Therefore, the compound interest is Rs. 3783.

Verification / Alternative check:
We can quickly check the reasonableness of the answer. A simple interest approximation at 10% per year for 1.5 years on 24000 is SI = 24000 * 0.10 * 1.5 = 3600. Under compounding every half year, the interest should be slightly higher than this simple interest figure. Our result 3783 is close to 3600 and slightly larger, which fits the expectation for compound interest relative to simple interest over the same time and rate. This provides a good consistency check.

Why Other Options Are Wrong:
The options 3777, 3780, and 3781 are all close to the true value but slightly off, which suggests either rounding errors or incorrect intermediate computations such as using fewer decimal places or approximating (1.05) ^ 3 inaccurately. The option 3790 is a bit higher than the correct answer and represents a situation where the growth factor is slightly overstated. Only Rs. 3783 matches the correct result when the arithmetic is done precisely with the correct rate per period and the correct number of periods.

Common Pitfalls:
A typical mistake is to treat 1.5 years as 1.5 periods rather than 3 half year periods. Another error is to continue to use 10% as the rate per period instead of halving it for semiannual compounding, which leads to a much larger interest than appropriate. Learners may also incorrectly switch to simple interest for the fractional part of the year, which does not follow the wording of the problem. Accurate identification of the period, the rate per period, and consistent use of the compounding formula are crucial steps.

Final Answer:
The compound interest on Rs. 24000 at 10% per annum for 1.5 years, when interest is compounded semiannually, is Rs. 3783, which corresponds to option A.