Anurag borrows a total sum of Rs. 8,000 from two banks, A and B. Bank A lends…

Concept: Simple Interest for a principal P at rate R% per annum over T years is SI = (P × R × T) / 100. When a total sum is split between two lenders at different rates for the same time period, the total interest paid is the sum of each portion's individual simple interest. This can be solved either by setting up a system of two linear equations, or by the alligation (weighted-average) method, which finds the average rate on the whole sum and splits the principal proportionally.

Step-by-step solution:

1. Let the amount borrowed from Bank A be x (Rs.). Then the amount borrowed from Bank B is (8000 − x) (Rs.), since the total sum is Rs. 8,000.

2. Simple interest from Bank A over 2 years at 12% per annum = (x × 12 × 2) / 100 = 0.24x.

3. Simple interest from Bank B over 2 years at 10% per annum = ((8000 − x) × 10 × 2) / 100 = 0.20(8000 − x).

4. The total interest paid is Rs. 1,816, so: 0.24x + 0.20(8000 − x) = 1816.

5. Expand: 0.24x + 1600 − 0.20x = 1816, which gives 0.04x = 216.

6. Solve for x: x = 216 / 0.04 = 5400.

Cross-check (Alligation method):

1. Average rate of interest on the whole sum = (1816 × 100) / (8000 × 2) = 11.35%.

2. By alligation, the ratio of Bank A's share to Bank B's share = (11.35 − 10) : (12 − 11.35) = 1.35 : 0.65 = 27 : 13.

3. Bank A's share = 27/40 × 8000 = 5400, which matches the algebraic result.

Verification: interest on Rs. 5,400 at 12% for 2 years = (5400 × 12 × 2)/100 = 1296; interest on Rs. 2,600 at 10% for 2 years = (2600 × 10 × 2)/100 = 520; total = 1296 + 520 = Rs. 1,816, which matches the total interest given in the question.

Answer: Anurag borrowed Rs. 5,400 from Bank A.