A man borrows a total of Rs. 24,000 from two moneylenders. On one part of the loan he pays 15% per annum simple interest and on the remaining part he pays 18% per annum simple interest. At the end of 1 year he pays Rs. 4,050 as interest in all. How much did he borrow at 18% per annum?

Introduction / Context:
This is a typical mixture of rates problem in simple interest. A total principal is split into two parts, each invested at a different rate. We know the total interest and must find how much was invested at one of the rates. It is a direct application of linear equations with simple interest.

Given Data / Assumptions:

• Total principal = Rs. 24,000
• Part of the amount is at 15% per annum
• Remaining amount is at 18% per annum
• Time = 1 year
• Total simple interest paid = Rs. 4,050
• We must find the amount at 18% per annum

Concept / Approach:
Let the amount borrowed at 15% be x rupees. Then the amount at 18% is 24000 - x. We compute the interest for each part using the simple interest formula for 1 year and equate the sum to 4050. Solving the resulting linear equation will give x and then the amount at 18%.

Step-by-Step Solution:
Let amount at 15% = xThen amount at 18% = 24000 - xInterest on x at 15% for 1 year = (x * 15 * 1) / 100 = 0.15xInterest on (24000 - x) at 18% for 1 year = (24000 - x) * 18 / 100 = 0.18(24000 - x)Total interest = 0.15x + 0.18(24000 - x)Given total interest = 4050So 0.15x + 0.18(24000 - x) = 40500.15x + 4320 - 0.18x = 40504320 - 0.03x = 40500.03x = 4320 - 4050 = 270x = 270 / 0.03 = 9000Amount at 15% = Rs. 9,000Amount at 18% = 24000 - 9000 = Rs. 15,000
Verification / Alternative check:
Compute interest using the found values. Interest on Rs. 9,000 at 15% is 9000 * 15 / 100 = Rs. 1,350. Interest on Rs. 15,000 at 18% is 15000 * 18 / 100 = Rs. 2,700. Total interest = 1,350 + 2,700 = Rs. 4,050, which matches the given total interest. This confirms that the amounts are correct.

Why Other Options Are Wrong:
If the amount at 18% were 16,000, interest at 18% on that part plus interest on 8,000 at 15% would not sum to 4,050. Similar checks show that 12,000 or 13,000 at 18% also do not satisfy the total interest condition. Only Rs. 15,000 gives the correct total interest.

Common Pitfalls:
Students may mistakenly assume that the 4,050 is divided equally between the two rates or that the principal is split in the same ratio as the rates. The correct method is always to form and solve the linear equation based on the simple interest formula.

Final Answer:
The man borrowed Rs. 15,000 at 18% per annum.