A certain sum of money becomes double of its original value in 15 years at a fixed rate of simple interest. At the same rate of simple interest, in how many years will this sum become five times its original value?

Introduction / Context:
This question explores how the time required to reach different multiples of the principal changes under simple interest. Since simple interest grows linearly with time, once we know the time taken for the sum to double, we can determine the rate and then find the time required for the sum to become any other multiple, such as five times the original principal.

Given Data / Assumptions:

• The principal is some sum P (unknown but constant).
• The sum becomes 2P (double) in 15 years.
• Interest is simple interest at rate R% per annum.
• We are asked for the time required for the amount to become 5P.

Concept / Approach:
Under simple interest, amount A after T years is A = P + SI = P + (P * R * T) / 100 = P * (1 + R * T / 100). We first use the doubling condition (A = 2P in 15 years) to find R. Then we apply the same rate R with A = 5P to calculate the new time T needed for the amount to become five times the principal.

Step-by-Step Solution:
For doubling: 2P = P * (1 + R * 15 / 100). Divide both sides by P: 2 = 1 + (R * 15 / 100). So, R * 15 / 100 = 1, which gives R = 100 / 15 = 6.666...% per annum. Now we want A = 5P: 5P = P * (1 + R * T / 100). Divide both sides by P: 5 = 1 + R * T / 100. So R * T / 100 = 4, therefore T = 4 * 100 / R. Substitute R = 100 / 15: T = 4 * 100 / (100 / 15) = 4 * 15 = 60 years.
Verification / Alternative check:
Once we know that 15 years of interest equals the principal P (because the sum doubled), each block of 15 years adds another P. To go from P to 5P, we need an increase of 4P in interest. Each P takes 15 years, so time required is 4 * 15 = 60 years. This short reasoning confirms the detailed calculation.

Why Other Options Are Wrong:
37.5, 45, 75, and 30 years arise from miscounting how many times the principal needs to be added or from incorrectly scaling the time period. For example, 45 years corresponds to tripling, not quintupling, the principal, and 75 years would imply 5P plus an extra half P, which is not required here.

Common Pitfalls:
Many students assume that if doubling takes 15 years, then five times might be obtained by simply multiplying 15 by 2 or 3 without carefully thinking about how much interest is needed. Others mistakenly apply compound interest reasoning. Remember that under simple interest, the increase is linear, so each additional P of interest always takes the same amount of time.

Final Answer:
The sum will become five times its original value in 60 years.