A sum of Rs. 4000 is lent out in two parts, one at 8% simple interest and the other at 10% simple interest. If the total annual interest from both parts is Rs. 352, what is the amount (in rupees) lent at 8% per annum?

Introduction / Context:
This question is about splitting a total sum between two different simple interest rates and using the combined annual interest to determine how much was placed at each rate. The total principal of Rs. 4000 is divided between 8% and 10% per annum. The combined interest for 1 year is given, and we must work backwards using algebra to find the part invested at 8%. Problems of this type test linear equation setup skills and understanding of how different interest rates contribute to total earnings.

Given Data / Assumptions:

• Total sum = Rs. 4000.
• One part is lent at 8% simple interest per annum.
• The other part is lent at 10% simple interest per annum.
• Total interest for 1 year from both parts is Rs. 352.
• Interest is calculated with simple interest and a constant rate for each part.

Concept / Approach:
Let x be the amount invested at 8% and the remaining 4000 - x at 10%. The interest from the first part is x * 8 / 100, and from the second part is (4000 - x) * 10 / 100. The sum of these two interests equals the total annual interest of Rs. 352. This gives a linear equation in x. Solving this equation gives the amount at 8%. This approach is standard for mixture-type interest problems and highlights how to translate verbal conditions into algebraic form.

Step-by-Step Solution:
Step 1: Let the amount at 8% be x rupees. Step 2: Then the amount at 10% is 4000 - x rupees. Step 3: Interest from the 8% part in 1 year = x * 8 / 100 = 0.08x. Step 4: Interest from the 10% part in 1 year = (4000 - x) * 10 / 100 = 0.10 * (4000 - x). Step 5: Total interest for 1 year is 352, so 0.08x + 0.10(4000 - x) = 352. Step 6: Expand the second term: 0.08x + 400 - 0.10x = 352. Step 7: Combine like terms: (0.08x - 0.10x) + 400 = 352, so -0.02x + 400 = 352. Step 8: Subtract 400 from both sides: -0.02x = -48. Step 9: Divide by -0.02: x = -48 / -0.02 = 2400. Step 10: Thus, the amount invested at 8% is Rs. 2400.

Verification / Alternative check:
Verify by computing interest from each part. At 8% on Rs. 2400: interest = 2400 * 8 / 100 = 192. At 10% on the remaining 1600: interest = 1600 * 10 / 100 = 160. Total interest = 192 + 160 = 352, matching the given information. This confirms that the distribution of Rs. 2400 at 8% and Rs. 1600 at 10% is correct.

Why Other Options Are Wrong:
If Rs. 2900 were at 8%, the remaining 1100 at 10% would give total interest 2900 * 8 / 100 + 1100 * 10 / 100 = 232 + 110 = 342, not 352. If Rs. 2200 were at 8%, interest totals would also not equal 352. If Rs. 3100 were at 8%, the remaining 900 at 10% would produce 248 + 90 = 338. Only Rs. 2400 at 8% generates the correct total interest of 352 rupees.

Common Pitfalls:
Students frequently mix up which part is at which rate, or they incorrectly assume equal division of the principal. Another common error is forgetting to convert percentages into decimal form or mishandling the algebra when expanding and simplifying the equation. Careful step-by-step setup and checking of calculations helps avoid such mistakes.

Final Answer:
The sum lent at 8% per annum is Rs. 2400.