A sum of money amounts to Rs 9,800 after 5 years and to Rs 12,005 after 8 years at the same rate of simple interest. What is the rate of interest per annum implied by these two amounts?

Introduction / Context:
Here, two amounts of the same principal at the same simple interest rate but for different times are given. From the change in amount over the extra years, you can find the yearly interest and then compute the rate of interest. This is a standard technique in simple interest questions where principal is unknown but the rate is required.

Given Data / Assumptions:

Amount after 5 years, A1 = Rs 9,800.
Amount after 8 years, A2 = Rs 12,005.
Principal P is the same in both cases.
Rate of interest per annum is r%.
Simple interest: SI = (P * r * T) / 100 and A = P + SI.

Concept / Approach:
Just as in similar problems, the difference in amounts over the extra 3 years equals the simple interest for those 3 years. From that difference we find yearly interest and then relate it to the principal. The principal can be found from either amount, and then the rate is obtained by using SI for a known number of years.

Step-by-Step Solution:
Step 1: Let principal be P. Then A1 = P + SI_5 and A2 = P + SI_8. Step 2: The difference A2 - A1 = SI_8 - SI_5 = interest for 3 extra years. Step 3: Compute A2 - A1 = 12005 - 9800 = 2205. Step 4: Thus, interest for 3 years is Rs 2205, so yearly interest = 2205 / 3 = Rs 735. Step 5: Interest for 5 years is 5 * 735 = Rs 3675. Step 6: Since A1 = P + interest for 5 years, we have 9800 = P + 3675. Step 7: So, P = 9800 - 3675 = 6125. Step 8: Now find the rate using yearly interest: SI_1year = (P * r * 1) / 100 = P * r / 100 = 735. Step 9: Therefore, P * r = 735 * 100 = 73500, and r = 73500 / 6125. Step 10: Compute r = 73500 / 6125 = 12. Step 11: Hence, the rate of interest per annum is 12%.

Verification / Alternative check:
Check with the 8-year amount. Interest for 8 years at 12% on principal 6125 is SI_8 = (6125 * 12 * 8) / 100 = 6125 * 0.96 = 5880. Amount after 8 years is 6125 + 5880 = 12005, which matches A2. This confirms that r = 12% is correct.

Why Other Options Are Wrong:
Rates like 5%, 8%, 10%, or 15% give yearly interests that, when applied for 5 and 8 years, cannot produce the given pair of amounts 9800 and 12005. Only 12% produces the correct annual interest and hence the correct amounts.

Common Pitfalls:
A frequent mistake is to compute the rate directly from only one of the amounts without first finding the principal, which often leads to algebraic confusion. Another mistake is to take the difference in amounts and divide by 8 instead of by 3, forgetting that the extra period is 3 years. Always focus on the extra time when taking the difference between two simple interest amounts.

Final Answer:
The rate of interest per annum is 12%.