A man invested a sum of money at 10% per annum on compound interest. After 2 years, he invested an additional amount equal to half of the initial principal at the same rate. If his total interest after 3 years is Rs 457.20, what amount would he have received after 3 years if, instead, he had invested only the initial principal at 15% per annum on simple interest for 3 years?

Introduction / Context:
This question combines compound interest and simple interest in two different scenarios. In the first scenario, the investor uses compound interest at 10% with an additional investment after 2 years, and the total interest after 3 years is given. In the second scenario, you must determine what amount he would have after 3 years if he instead invested only the original principal at 15% simple interest. The problem tests the ability to handle compounding over different periods and then compare with an equivalent simple interest investment.

Given Data / Assumptions:

Initial principal in the compound interest scenario is P rupees.
Rate under compound interest scenario = 10% per annum, compounded yearly.
After 2 years, an additional amount equal to half of P (that is, P / 2) is invested at the same rate.
Total interest earned in the 3-year compound interest scenario = Rs 457.20.
In the comparison scenario, only P is invested at 15% simple interest for 3 years.
We must find the final amount in the simple interest scenario.

Concept / Approach:
First, express total interest in the compound interest scenario in terms of P. The original principal P grows for 3 years at 10% compound interest, while the additional P / 2 grows for only 1 year at 10%. Summing the interest from these parts and equating to 457.20 allows us to find P. Once P is known, we compute the amount in the alternative simple interest scenario using A = P + SI with SI = (P * 15 * 3) / 100.

Step-by-Step Solution:
Step 1: For the original principal P at 10% compound interest, amount after 3 years is A1 = P * (1.1)^3. Step 2: Interest on this part is I1 = P * (1.1^3 - 1) = P * (1.331 - 1) = 0.331P. Step 3: After 2 years, an additional amount equal to P / 2 is invested. This extra amount earns compound interest only for the last year, so its interest is I2 = (P / 2) * 10 / 100 = 0.05P. Step 4: Total interest in the compound interest scenario is I_total = I1 + I2 = 0.331P + 0.05P = 0.381P. Step 5: Given I_total = 457.20, we have 0.381P = 457.20. Step 6: Therefore, P = 457.20 / 0.381 = 1200. Step 7: In the comparison scenario, only P = 1200 is invested at 15% simple interest for 3 years. Step 8: Compute simple interest: SI_simple = (P * 15 * 3) / 100 = (1200 * 45) / 100 = 540. Step 9: Amount in the simple interest scenario A_simple = P + SI_simple = 1200 + 540 = 1740.

Verification / Alternative check:
Reconfirm the compound interest side. For P = 1200, interest on P for 3 years is 1200 * (1.1^3 - 1) = 1200 * 0.331 = 397.2. Extra principal P / 2 = 600 earns interest for 1 year: 600 * 10 / 100 = 60. Total interest = 397.2 + 60 = 457.2, matching the given 457.20. This confirms P = 1200. The computed simple interest amount 1740 is therefore reliable.

Why Other Options Are Wrong:
Amounts such as Rs 1,720, Rs 1,760, Rs 1,780, and Rs 1,800 correspond to incorrect calculations of either the compound interest part or the simple interest part. They may arise from using 10% instead of 15%, using 2 years instead of 3 years, or misapplying the formula for compound interest. Only Rs 1,740 is consistent with the correctly derived principal and rate.

Common Pitfalls:
A common mistake is to assume that the additional amount P / 2 was invested from the beginning, which would cause it to compound for 3 years instead of 1. Another error is to confuse simple interest and compound interest formulas. Always carefully separate each time segment and ensure the correct formula is applied to each principal for the correct duration.

Final Answer:
Under the simple interest alternative at 15% for 3 years, he would have received Rs 1,740.