The simple interest on a certain sum of money at the rate of 5% per annum for 8 years is Rs. 840. For what annual rate of simple interest (in percent) would the same sum earn the same interest of Rs. 840 in 5 years?

Introduction / Context:
This question involves two different simple interest situations on the same principal but with different time periods and rates. In the first case, we are given the interest, rate, and time, which allows us to find the principal. In the second case, the principal remains the same, but the time and rate are different, and we require the new rate that will produce the same interest. Such questions test your ability to link multiple simple interest scenarios through the common principal or common interest amount.

Given Data / Assumptions:

• In the first scenario, rate R1 = 5% per annum.
• Time T1 = 8 years.
• Simple interest SI = Rs. 840.
• The principal is the same in both scenarios.
• In the second scenario, time T2 = 5 years.
• We want the same interest SI = Rs. 840, and must find the new rate R2.

Concept / Approach:
First, use SI = P * R1 * T1 / 100 to find the principal P from the first situation. Once P is known, apply the same simple interest formula to the second situation with SI = 840 and T2 = 5 years, then solve for the unknown rate R2. It is important to work systematically: state the formula, isolate the unknown, and substitute carefully. This approach clearly demonstrates how the same principal can produce the same interest under different combinations of rate and time.

Step-by-Step Solution:
Step 1: Use the first scenario to find principal P. SI = P * R1 * T1 / 100. Step 2: Substitute SI = 840, R1 = 5, T1 = 8: 840 = P * 5 * 8 / 100. Step 3: Simplify 5 * 8 / 100 = 40 / 100 = 0.4, so 840 = 0.4 * P. Step 4: Solve for P: P = 840 / 0.4 = 2100. Step 5: Now use the second scenario with the same principal P = 2100, time T2 = 5 years, and SI = 840. Step 6: Simple interest formula: 840 = 2100 * R2 * 5 / 100. Step 7: Simplify 2100 * 5 / 100 = 2100 * 0.05 = 105. Step 8: So 840 = 105 * R2. Step 9: Therefore, R2 = 840 / 105 = 8. Step 10: Hence, the required rate of interest is 8% per annum.

Verification / Alternative check:
Verify by computing the interest in the second scenario with R2 = 8%. SI = 2100 * 8 * 5 / 100 = 2100 * 40 / 100 = 2100 * 0.4 = 840, which matches the given interest. This confirms that the upped rate of 8% per annum generates the same interest in 5 years as 5% per annum does in 8 years on the same principal.

Why Other Options Are Wrong:
If R2 = 7%, then SI would be 2100 * 7 * 5 / 100 = 735, which is less than 840. If R2 = 9%, SI would be 945, which is too high. If R2 = 10%, SI would be 1050. Only 8% yields exactly Rs. 840 over 5 years on the same principal, so the other options are incorrect.

Common Pitfalls:
One common mistake is to misinterpret the given interest as belonging to the second scenario rather than the first, leading to incorrect equations. Another error is failing to compute the principal correctly from the first case, which then misleads all further calculations. Some students also attempt shortcuts without explicitly computing P, increasing the risk of algebra mistakes. Working in two clear stages avoids such confusion.

Final Answer:
The rate of interest for the second scenario is 8% per annum.