Shamma lends a total of Rs. 40,000 to two friends. She lends Rs. 24,000 to the first friend at 8 percent per annum simple interest and wants an overall return of 10 percent per annum on the entire amount. At what simple interest rate should she lend the remaining sum to the second friend?

Introduction / Context:
This question checks the concept of weighted average interest rates in simple interest problems. The investor, Shamma, invests different portions of her total capital at different rates and aims for a specific overall return. Such questions are important in banking and finance related aptitude tests.

Given Data / Assumptions:

• Total amount lent = Rs. 40,000.
• Amount lent to the first friend = Rs. 24,000 at 8 percent per annum simple interest.
• Remaining amount lent to the second friend = Rs. 40,000 - Rs. 24,000 = Rs. 16,000.
• Desired overall rate of return on the entire amount = 10 percent per annum.
• Time period for both loans is assumed to be the same, so it cancels out in calculations.

Concept / Approach:
Under simple interest, interest earned is directly proportional to principal, rate, and time. Since time is the same, the total interest can be treated as a weighted sum of interest from each part. The overall effective rate can be expressed as total interest divided by total principal. By equating desired total interest to the sum of interests from the two parts, we can solve for the unknown rate for the second friend.

Step-by-Step Solution:
Let R percent be the simple interest rate for the second friend. Total principal = 40,000 rupees. Desired overall interest for one year = 40,000 * 10 / 100 = 4,000 rupees. Interest from the first friend = 24,000 * 8 / 100 = 1,920 rupees. Interest needed from the second friend = total desired interest - first interest. Interest from second friend = 4,000 - 1,920 = 2,080 rupees. Principal given to the second friend = 16,000 rupees. So 16,000 * R / 100 = 2,080. R = (2,080 * 100) / 16,000 = 13 percent.

Verification / Alternative check:
Interest at 13 percent on Rs. 16,000 for one year = 16,000 * 13 / 100 = 2,080 rupees. Total interest from both loans = 1,920 + 2,080 = 4,000 rupees. Overall rate = 4,000 / 40,000 * 100 = 10 percent. This matches the desired overall rate, so the calculation is correct.

Why Other Options Are Wrong:
8.5% and 9.5%: These rates are too low and will not generate enough interest from the second loan to reach a 10 percent overall return. 11.25%: This is closer but still insufficient; the total interest would be less than 4,000 rupees. Only 13 percent gives exactly the overall 10 percent simple interest on the total investment.

Common Pitfalls:
Many candidates mistakenly take a simple average of 8 percent and 10 percent or misapply the weighted average concept. Some also forget that the invested amounts are different, which makes the average rate weighted, not simple. Ignoring the total desired interest and directly averaging rates always leads to incorrect answers in these types of problems.

Final Answer:
The remaining sum must be lent at a simple interest rate of 13%.