The simple interest and compound interest that can be earned in 2 years at the same annual rate are Rs. 1500 and Rs. 1575, respectively. For this investment, what is the annual rate of interest in percent per annum?

Introduction / Context:
This question provides both the simple interest (SI) and the compound interest (CI) for 2 years at the same rate and on the same principal, and asks us to determine the rate of interest per annum. The key idea is that the difference between CI and SI over 2 years is related to the principal and the square of the rate. By combining the facts SI = 1500 and CI = 1575, we can solve for both the principal–rate product and the rate itself.

Given Data / Assumptions:

• Simple interest SI over 2 years = Rs. 1500.
• Compound interest CI over 2 years, compounded annually = Rs. 1575.
• Principal P is the same in both cases.
• Rate of interest r percent per annum is the same in both cases.
• Time t = 2 years.

Concept / Approach:
Under simple interest for 2 years, SI = (P * r * 2) / 100. Under compound interest, the difference between CI and SI for 2 years is D₂ = P * (r/100)^2. We know SI and CI, so the difference D₂ = 1575 − 1500 = 75. From SI we get P * r, and from D₂ we get P * r^2. Dividing P * r^2 by P * r yields r directly, which is the annual rate of interest.

Step-by-Step Solution:
Step 1: Compute the difference between CI and SI: D₂ = 1575 − 1500 = 75. Step 2: From simple interest, SI = (P * r * 2) / 100 = 1500. Step 3: Rearranging gives P * r = (1500 * 100) / 2 = 75000. Step 4: From the difference formula, D₂ = P * (r/100)^2 = 75. Step 5: Multiply both sides by 10000: P * r^2 = 75 * 10000 = 750000. Step 6: Now we have P * r = 75000 and P * r^2 = 750000. Step 7: Divide P * r^2 by P * r to isolate r: (P * r^2) / (P * r) = r = 750000 / 75000. Step 8: Compute 750000 / 75000 = 10. Step 9: Thus, the annual rate of interest is 10 percent per annum.

Verification / Alternative check:
To verify, we can find P using P * r = 75000 and r = 10. Then P = 75000 / 10 = 7500. For simple interest: SI = (7500 * 10 * 2) / 100 = 1500, which matches the given SI. For compound interest: amount A = 7500 * (1.10)^2. Compute (1.10)^2 = 1.21, so A = 7500 * 1.21 = 9075. CI = A − P = 9075 − 7500 = 1575, matching the given CI. This confirms r = 10 percent is correct.

Why Other Options Are Wrong:
At 5 %, SI over 2 years would be much smaller and CI − SI would also be much smaller than 75. At 8 % or 12 %, the product P * r and the difference P * (r/100)^2 would not match the given SI and CI values simultaneously. A rate as high as 15 % would produce a much larger difference between CI and SI than Rs. 75 for realistic principal values. Only 10 % satisfies both conditions exactly.

Common Pitfalls:
Some candidates misapply the CI formula directly without using the neat P * r and P * r^2 relationships, leading to more complicated algebra and potential mistakes. Others forget to multiply by 100 or 10000 at the right stage and end up with fractional or unrealistic rates. Being systematic in using SI to get P * r and the CI–SI difference to get P * r^2 makes this question straightforward.

Final Answer:
The annual rate of interest that leads to simple interest of Rs. 1500 and compound interest of Rs. 1575 in 2 years on the same principal is 10 % per annum.