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 under this changed arrangement?

Introduction / Context:
This problem involves a change in principal during the investment period under simple interest. Initially, a sum is kept for 10 years at a certain rate, and the simple interest for the full 10 years is known to be Rs 3130. In a modified scenario, after 5 years the principal is increased so that it becomes 5 times the original principal for the remaining 5 years. The rate remains the same throughout. The question asks for the total interest that will be obtained in this modified arrangement. Such questions test a deeper understanding of simple interest, especially how changes in principal affect total interest over different segments of time.

Given Data / Assumptions:

• Original principal = P rupees.
• Rate of simple interest per annum = R% (unknown).
• Simple interest for 10 years at rate R on principal P is Rs 3130.
• In the modified case, principal is P for the first 5 years and 5P for the next 5 years.
• The rate of interest R remains the same throughout.
• We must find the total interest in the modified case.

Concept / Approach:
Under simple interest, interest depends linearly on principal and time. From the original information, we can express Rs 3130 in terms of P, R, and 10 years:
3130 = (P * R * 10) / 100 This allows us to find the product P * R as a single combined constant. In the modified scenario, interest accumulates in two segments:
Segment 1: P at rate R for 5 years. Segment 2: 5P at rate R for 5 years. We compute the interest in each segment using the simple interest formula and then add them. By expressing both in terms of P * R, we can relate the new total interest to the original 10 year interest.

Step-by-Step Solution:
From the original 10 year scenario: 3130 = (P * R * 10) / 100. So (P * R) / 100 = 3130 / 10 = 313. Thus, P * R = 313 * 100 = 31300. Now consider the modified scenario. Segment 1 (first 5 years): principal = P, time = 5 years. Interest I1 = (P * R * 5) / 100. But (P * R) / 100 = 313, so I1 = 313 * 5 = 1565. Segment 2 (next 5 years): principal = 5P, time = 5 years. Interest I2 = (5P * R * 5) / 100 = 25 * (P * R) / 100. Using (P * R) / 100 = 313, I2 = 25 * 313 = 7825. Total interest in the modified scenario = I1 + I2. Total interest = 1565 + 7825 = 9390. Therefore, the total interest obtained after 10 years in the changed arrangement is Rs 9390.

Verification / Alternative check:
We can verify the proportional reasoning. In the original case, interest for 10 years on P is 3130. This implies interest for 5 years on P is half of 3130, that is 1565. In the modified case, we still have 5 years on P (giving 1565), plus 5 years on 5P. Interest for 5 years on P is 1565, so on 5P it is 5 times that, which is 7825. Total = 1565 + 7825 = 9390. This checks out and matches the previous calculation exactly.

Why Other Options Are Wrong:
Rs 6260 would correspond to only doubling the principal for the second half rather than multiplying it by 5.
Rs 7825 is just the interest on 5P for 5 years and ignores the first 5 years on P.
Rs 15650 or Rs 5000 do not match the structure of the two segment calculation and would require different relationships between principal, rate, and time that are not consistent with the given data. Only Rs 9390 matches the correct segment wise interest computation.

Common Pitfalls:
A common mistake is to treat the principal as 5P for the entire 10 years, which vastly overestimates the interest. Another error is to forget that the first 5 years still earn interest on the original P and to count that part incorrectly. Some students also try to find the numeric values of P and R separately, which is unnecessary because the combinations P * R and (P * R) / 100 are sufficient. Working with these combined terms keeps the algebra simple and helps avoid arithmetic overload.

Final Answer:
The total simple interest obtained after 10 years in the changed arrangement is Rs 9390.