Ankita borrows Rs 7,000 at simple interest from a lender. At the end of 3 years, she borrows an additional Rs 3,000 from the same lender at the same annual rate of interest. She settles all her borrowings by paying a total of Rs 4,615 as interest after 8 years from the time of the first borrowing. What is the annual rate of simple interest?

Introduction / Context:
This problem involves two separate simple interest loans taken at different times but at the same annual rate. It is a common type of question in aptitude tests that checks whether you can track interest over different time periods for different principal amounts, then combine those interest amounts to find an unknown rate. Real world analogues include top up loans and staged borrowing from the same lender.

Given Data / Assumptions:

First loan principal P1 = Rs 7,000
Second loan principal P2 = Rs 3,000
Second loan is taken 3 years after the first loan
Total interest paid on both loans together = Rs 4,615
Total time for first loan = 8 years
Time for second loan = 8 - 3 = 5 years
Same annual simple interest rate r for both loans

Concept / Approach:
Simple interest for each loan is calculated using I = P * r * T, with r as a fraction (r over 100). Since the rate is the same, the total interest is the sum of the interests for both principals over their respective time periods. Set this total equal to 4,615 rupees and solve for r. Because the principal and time for each loan are known, this results in a simple linear equation in r.

Step-by-Step Solution:
Step 1: Express interest on the first loan. I1 = P1 * r * T1 / 100 = 7,000 * r * 8 / 100. Step 2: Express interest on the second loan. I2 = P2 * r * T2 / 100 = 3,000 * r * 5 / 100. Step 3: Express total interest as sum of I1 and I2. Total interest = I1 + I2 = 4,615. Step 4: Substitute and simplify. I1 = 7,000 * 8 * r / 100 = 56,000 * r / 100 = 560 * r. I2 = 3,000 * 5 * r / 100 = 15,000 * r / 100 = 150 * r. Total interest = 560 * r + 150 * r = 710 * r. Step 5: Set 710 * r = 4,615. r = 4,615 / 710. Step 6: Compute r: 4,615 / 710 = 6.5 percent per annum.

Verification / Alternative check:
To verify, use r = 6.5 percent explicitly and recalculate the interests. First loan interest: 7,000 * 6.5 * 8 / 100 = 7,000 * 52 / 100 = 3,640 rupees. Second loan interest: 3,000 * 6.5 * 5 / 100 = 3,000 * 32.5 / 100 = 975 rupees. Total interest = 3,640 + 975 = 4,615 rupees, which matches the given total. This confirms that the annual simple interest rate is indeed 6.5 percent.

Why Other Options Are Wrong:
At 5.5 percent, the interest would be 7,000 * 5.5 * 8 / 100 + 3,000 * 5.5 * 5 / 100, which sums to less than 4,615 rupees. Similarly, 7.5 percent and 9.5 percent produce total interests larger than 4,615 rupees. Therefore, only 6.5 percent per annum yields the exact total interest specified in the problem statement.

Common Pitfalls:
A frequent mistake is to treat both loans as if they run for the same duration, ignoring that the second loan starts 3 years later. Another common error is to forget to divide by 100 when using the percentage rate in the formula, leading to very large interest values. Some learners also attempt to average time periods or principals without forming the correct linear equation. Writing separate expressions for each loan and then adding them is the safest and most systematic approach.

Final Answer:
The annual rate of simple interest charged by the lender is 6.5% per annum.