A person invests a total sum of Rs. 7900 in three different simple interest schemes offering 3%, 5% and 8% per annum. After 1 year, the interest earned from each scheme is the same. What amount (in rupees) is invested at 3% per annum?

Introduction / Context:
This question tests the concept of proportional allocation of money across different simple interest rates when the interest earned from each part is equal over the same time period. Instead of directly giving the amounts invested in each scheme, the question gives the total principal and the condition that interest from all three schemes is identical after 1 year. By using the simple interest formula and reasoning with ratios, we can determine how much was invested at the lowest rate of 3% per annum.

Given Data / Assumptions:

• Total principal invested = Rs. 7900.
• Three schemes with annual simple interest rates of 3%, 5%, and 8% respectively.
• Time period T = 1 year for all schemes.
• Simple interest earned from each scheme after 1 year is the same.
• All investments are at simple interest with no change in rates during the year.

Concept / Approach:
Simple interest for 1 year is given by SI = P * R * T / 100, which simplifies to SI = P * R / 100 when T = 1 year. If the interests from three parts are equal, then P1 * R1 = P2 * R2 = P3 * R3 (up to a common factor), meaning the amounts invested are inversely proportional to the rates of interest. Therefore, the ratio of money invested at 3%, 5%, and 8% is inversely proportional to 3, 5, and 8. Once this ratio is determined, we use the fact that the total sum is Rs. 7900 to compute the actual amount invested at 3%.

Step-by-Step Solution:
Step 1: Let the amounts invested at 3%, 5%, and 8% be A, B, and C respectively. Step 2: For 1 year, simple interest from each part is proportional to P * R, so equal interest implies A * 3 = B * 5 = C * 8. Step 3: Therefore, A : B : C is inversely proportional to 3 : 5 : 8. Step 4: Hence A : B : C = 1/3 : 1/5 : 1/8. Step 5: Take a common multiple: 1/3 : 1/5 : 1/8 = 40 : 24 : 15 (by multiplying each by 120). Step 6: So A : B : C = 40 : 24 : 15. Step 7: Total parts = 40 + 24 + 15 = 79 parts. Step 8: Total money = Rs. 7900, so each part = 7900 / 79 = 100 rupees. Step 9: Amount at 3% = A = 40 parts = 40 * 100 = Rs. 4000. Step 10: Therefore, the amount invested at 3% per annum is Rs. 4000.

Verification / Alternative check:
Using A = 4000, B = 2400, C = 1500, we can verify the interests. Interest from the 3% part in 1 year = 4000 * 3 / 100 = 120. From the 5% part = 2400 * 5 / 100 = 120. From the 8% part = 1500 * 8 / 100 = 120. All three interests are equal, and the total principal is 4000 + 2400 + 1500 = 7900, which matches the given data. This confirms our distribution and the answer.

Why Other Options Are Wrong:
If 2900 were invested at 3%, the remaining amount 5000 would not split correctly to give equal interest. Likewise, 3500 or 5600 at 3% cannot produce equal interest in combination with realistic allocations to the other two rates while still totaling Rs. 7900. Only Rs. 4000 at 3% yields a valid ratio and equal interest from all three schemes.

Common Pitfalls:
A common error is to assume the investments are directly proportional to the rates instead of inversely proportional. Another mistake is forgetting that the time period is the same for all three schemes, which is crucial for using the inverse ratio method. Some learners also try to solve the problem using trial and error instead of setting up the correct ratio, which can be time-consuming and lead to miscalculations.

Final Answer:
The amount invested at 3% simple interest is Rs. 4000.