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The problem in the image is about calculating the total interest received by a boy who invested part of a sum with two different compound interest rates and durations.

### Problem Breakdown:
1. **Investment 1**:  
   - Fraction of sum: \( \frac{2}{5} \)
   - Interest rate: 10% per annum (compounded annually)
   - Duration: 3 years

2. **Investment 2**:  
   - Remaining fraction of sum: \( \frac{3}{5} \)
   - Interest rate: 8% per annum (compounded annually)
   - Duration: 2 years
   - Interest earned: ₹325.92

### Approach:
1. **Calculate the interest from Investment 1** using the compound interest formula:
   \[
   A = P \left(1 + \frac{r}{100}\right)^n
   \]
   where \( P \) is the principal, \( r \) is the interest rate, and \( n \) is the time period in years.

2. **Calculate the principal amount for Investment 2** and find the sum based on the given interest.

3. **Sum the interests** from both investments to get the total interest received.

I will now work through the calculations.

### Step 1: Calculate the principal of Investment 2
Given the interest earned from Investment 2:
\[
CI_2 = P_2 \left(\left(1 + \frac{8}{100}\right)^2 - 1\right)
\]
where \( CI_2 = 325.92 \).

### Step 2: Use the given options to find the total interest.
Given the options, we should find the correct interest value by adding the computed interest from Investment 1 and the given interest from Investment 2.

Let me calculate the values for you.The total interest received by the boy, calculated by summing the interest from both investments, is approximately **₹758.13**. This corresponds to option **A (₹758.57)**, which seems to be the closest match.

Would you like more details on how this was calculated, or do you have any further questions?

Here are five related questions to expand on this topic:
1. How does the frequency of compounding affect the total interest earned?
2. What would be the impact on the total interest if the rates of interest were swapped between the two investments?
3. How does the time period influence the compound interest for a given rate?
4. What would happen to the total interest if the entire sum was invested at 10% for 3 years?
5. How can you calculate the compound interest for non-annual compounding periods (e.g., quarterly)?

**Tip:** When calculating compound interest, the rate of interest and the time period must be consistent (e.g., annual rates with years).

A boy invested 2/5 of a sum at 10% compound interest compounded annually for 3 years and received ₹325.92 at 8% compound interest compounded annually for 2 years. Find the total interest received by the boy.

This problem involves calculating the total interest earned by a boy who invested part of his sum at different compound interest rates for different durations. Suitable for students in Grades 9-12, it covers key concepts like compound interest and fractional investments.

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