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. What is the annual rate of interest, in percent?

Introduction / Context:
This question again uses both simple interest and compound interest values over 2 years to determine the annual rate of interest. The principal is the same, and the time period is the same, but the simple interest and compound interest amounts differ slightly. By using the formula for the difference between compound and simple interest, we can find the rate without needing the principal explicitly at first.

Given Data / Assumptions:
- Simple interest for 2 years, SI = Rs 1500.
- Compound interest for 2 years (annual compounding), CI = Rs 1575.
- Rate of interest R is the same for both types of interest.
- The principal P is the same in both cases.
- We must determine R in percent per annum.

Concept / Approach:
For 2 years, the difference between CI and SI on the same principal P at annual rate R is:
D = CI − SI = P * (R/100)^2We can compute D easily as 1575 − 1500. Also, from the simple interest formula for 2 years:
SI = (P * R * 2) / 100we obtain a relation between P and R. Combining these two equations allows us to solve for R.

Step-by-Step Solution:
First, compute the difference: D = CI − SI = 1575 − 1500 = 75.Let R be the annual rate in percent and P be the principal.From simple interest for 2 years: 1500 = (P * R * 2) / 100.So, P * R = 1500 * 100 / 2 = 75000.From the difference formula for 2 years: D = P * (R/100)^2.So, 75 = P * (R/100)^2.Write (R/100)^2 as R^2 / 10000, so 75 = P * R^2 / 10000.Multiply both sides by 10000: 75 * 10000 = P * R^2.Thus, 750000 = P * R^2.But from earlier, P * R = 75000, so multiply this by R: P * R^2 = 75000 * R.Set them equal: 750000 = 75000 * R.Divide both sides by 75000: R = 750000 / 75000 = 10.Therefore, the annual rate of interest is 10 %.

Verification / Alternative check:
We can verify by first finding the principal. Since P * R = 75000 and R = 10, we get P = 75000 / 10 = 7500. Simple interest for 2 years at 10 %: SI = (7500 * 10 * 2) / 100 = 1500, which matches the given SI. Compound amount after 2 years: A = 7500 * (1.10)^2 = 7500 * 1.21 = 9075. Compound interest CI = 9075 − 7500 = 1575, matching the given CI. This confirms that R = 10 % is correct.

Why Other Options Are Wrong:
At 8 %, the differences between CI and SI for 2 years would be smaller than 75. At 12 % or 15 %, the differences would be larger. At 5 %, the interest amounts would be much lower and could not match the given 1500 and 1575. Only an annual rate of 10 % fits both the simple and compound interest values given.

Common Pitfalls:
Common mistakes include forgetting that the time is 2 years when applying the simple interest formula, or ignoring the helpful difference formula D = P * (R/100)^2. Some students also mismanage the algebra with R^2 and the factor of 10000. Carefully setting up both equations and solving systematically avoids these issues.

Final Answer:
The annual rate of interest is 10 %.