On a principal of Rs. 3500 invested at a simple interest rate of 7% per annum, Rs. 500 is earned as interest in a certain number of years. For the same time period, what annual rate of simple interest is required on Rs. 4900 so that the interest earned becomes Rs. 800?

Introduction / Context:
This question tests the basic concept of simple interest and the relationship between principal, rate, time, and interest earned. The same time period is used for two different principals, and you are asked to find a new rate of simple interest that produces a different amount of interest.

Given Data / Assumptions:

• First principal (P1) = Rs. 3500.
• First rate (R1) = 7% per annum simple interest.
• First interest (I1) = Rs. 500.
• Second principal (P2) = Rs. 4900.
• Second interest (I2) = Rs. 800.
• The time period in both cases is the same and is measured in years.

Concept / Approach:
For simple interest, the formula is:
Simple interest (SI) = (P * R * T) / 100, where P is the principal, R is the annual rate of interest in percent, and T is the time in years. First, use the given data for the initial investment to find the time. Then, use that same time with the second principal and required interest to compute the unknown rate.

Step-by-Step Solution:
Step 1: Use the first case. I1 = (P1 * R1 * T) / 100 = 500. Step 2: Substitute P1 = 3500 and R1 = 7. So 500 = (3500 * 7 * T) / 100. Step 3: Compute 3500 * 7 / 100 = 245. Thus 500 = 245 * T, so T = 500 / 245. Step 4: Now use the second case. I2 = (P2 * R2 * T) / 100 = 800, where R2 is unknown. Step 5: Substitute P2 = 4900 and T = 500 / 245 into 800 = (4900 * R2 * T) / 100. Step 6: Replace T and simplify: 800 = (4900 * R2 * 500) / (245 * 100). Step 7: Notice that 4900 / 245 = 20. So the expression becomes 800 = (20 * R2 * 500) / 100. Step 8: Further simplify 20 * 500 / 100 = 100. So 800 = 100 * R2, which gives R2 = 8%.

Verification / Alternative check:
You can quickly verify by computing the time T as T = 500 / 245, and then checking whether applying rate 8% on Rs. 4900 for this time gives Rs. 800. Using SI = (4900 * 8 * T) / 100 and substituting T shows that the interest is indeed Rs. 800, confirming the answer.

Why Other Options Are Wrong:
7.5% gives a slightly lower interest than required when applied to Rs. 4900 for the same time, so it does not reach Rs. 800. 9% gives a higher interest than Rs. 800 for the same principal and time, so it overshoots the required interest. 8.5% similarly produces more than Rs. 800, therefore it is not correct.

Common Pitfalls:
Students sometimes assume that they can directly compare the interest amounts by proportion without carefully solving for time first. Another common mistake is to round the time too early, which can introduce rounding errors. It is safer to keep the exact fraction until the final step. Forgetting that time is the same in both cases and treating it as different is also a typical source of error.

Final Answer:
The required rate of simple interest on Rs. 4900 to earn Rs. 800 in the same period is 8%.