The simple interest on a sum of money for 10 years is Rs. 3130 at a certain rate per annum. If, after 5 years, the principal is increased so that it becomes 5 times the original principal for the remaining 5 years at the same rate of simple interest, what will be the total interest obtained after 10 years?

Introduction / Context:
This question explores how total simple interest changes when the principal itself changes during the investment period, while the rate remains the same. Initially, we are told that the simple interest on a certain sum for 10 years is Rs. 3130, which lets us determine the product of principal and rate. Then the scenario is modified: after 5 years, the principal becomes 5 times its original value for the remaining 5 years. We must use the same rate and compute the new total interest over the full 10 year period. This requires careful use of proportional reasoning under simple interest.

Given Data / Assumptions:
Simple interest on the original principal P for 10 years at rate r is 3130 rupees.

So I_10 = P * r * 10 / 100 = 3130.
The interest type is simple interest for all calculations.
In the modified scenario, for the first 5 years the principal is P and for the next 5 years the principal is 5P.
The rate of interest r remains unchanged throughout the 10 years.

Concept / Approach:
From the original information, P * r * 10 / 100 = 3130, so P * r is fixed. In the new situation, total interest is the sum of interest for the first 5 years on P and interest for the next 5 years on 5P. Because simple interest is linear in both time and principal, we can express the new total interest directly in terms of P * r and then relate that to the known value of 3130. This avoids solving separately for P and r and simplifies the calculation.

Step-by-Step Solution:
From the original statement, simple interest for 10 years is I_10 = P * r * 10 / 100 = 3130.Therefore, P * r = 3130 * 100 / 10 = 31300.In the new scenario, interest for the first 5 years on principal P is I_1 = P * r * 5 / 100.Substitute P * r = 31300 to get I_1 = 31300 * 5 / 100.Compute I_1 = 31300 * 0.05 = 1565 rupees.For the last 5 years, principal becomes 5P, so interest I_2 = 5P * r * 5 / 100.Rewrite I_2 as I_2 = P * r * 25 / 100 because 5 * 5 = 25.Substitute P * r = 31300 again to get I_2 = 31300 * 25 / 100.Compute I_2 = 31300 * 0.25 = 7825 rupees.Total interest in the modified scenario is I_total = I_1 + I_2 = 1565 + 7825 = 9390 rupees.

Verification / Alternative check:
Notice that the original interest for 10 years is 3130 rupees. In the new scenario, the number of effective principal years has changed. For the first 5 years we have 1P, and for the next 5 years we have 5P. This is equivalent to 1P for 5 years plus 5P for 5 years, or a total of (1 * 5 + 5 * 5) = 30 principal years at the same rate. Originally, 10 years on P gave 10 principal years. Because interest is proportional to principal years, the new total interest must be 3 times the original 10 year interest, that is 3 * 3130 = 9390 rupees, which matches the computed result.

Why Other Options Are Wrong:
Rs. 6260 equals exactly double 3130 and would correspond to only doubling the effective principal years, not tripling them, which does not match the scenario with 5P in the last 5 years.
Rs. 7825 represents only the interest from the larger principal over 5 years and ignores the interest from the first 5 years.
Rs. 15650 is far too high and would require a much higher effective principal or rate.
Rs. 5000 is not a clear multiple of the original interest and does not follow from the proportional reasoning described above.

Common Pitfalls:
Some learners try to compute principal and rate separately, which introduces unnecessary complexity and potential arithmetic errors. Others forget that the increased principal applies only for the second half of the period, not the entire 10 years. Confusing simple interest with compound interest is another risk. Thinking in terms of total principal years and using the fixed product P * r can greatly simplify logical reasoning for such problems.

Final Answer:
The total simple interest obtained after 10 years when the principal becomes 5 times after 5 years is Rs. 9390.