Rs. 100 doubles to Rs. 200 in 5 years when invested at compound interest, with interest compounded annually. Assuming the same rate of compound interest continues, how many more years will it take for the investment to earn another Rs. 200 in interest (that is, to grow from Rs. 200 to Rs. 400)?

Introduction:
This compound interest problem stresses the idea that if an amount doubles in a certain period at a fixed rate, then any amount will also double in the same period at that rate. We are told that Rs. 100 doubles in 5 years, and we must find how much longer it will take for the already doubled amount to double once more and thus gain another Rs. 200 in interest.

Given Data / Assumptions:
Initial principal P₁ = Rs. 100. After 5 years at some compound rate, amount becomes Rs. 200. So (1 + r/100)^5 = 2. After 5 years, new principal for next phase is P₂ = Rs. 200. We want P₂ to become Rs. 400 at the same rate.

Concept / Approach:
Under compound interest, growth is multiplicative. If a certain rate doubles any amount in 5 years, it doubles every amount in 5 years. Therefore, the time needed to go from Rs. 200 to Rs. 400 at the same rate is the same as the time needed to go from Rs. 100 to Rs. 200, which is 5 years.

Step-by-Step Solution:
Step 1: From the first period, we know (1 + r/100)^5 = 2. Step 2: For the second period, the amount starts at 200 and must become 400. This again requires a multiplication by 2. So we need (1 + r/100)^n = 2 for the second stage as well. Step 3: Since (1 + r/100)^5 = 2, the smallest n that satisfies the equation again is 5 years. Thus, it takes another 5 years for Rs. 200 to become Rs. 400 at the same rate.

Verification / Alternative check:
We can assume any realistic rate that doubles in 5 years, such as r that satisfies the given condition, but the exact numerical value of r is not needed. The key observation is that the doubling time depends only on the rate, not on the starting amount. So the second doubling must also take 5 years.

Why Other Options Are Wrong:
Options like 3 years, 4 years, 6 years, or 7 years ignore the fact that the growth factor has to reach exactly 2 again, which is tied to the same 5 year period at this rate. These values do not correspond to a full doubling at the same fixed interest rate.

Common Pitfalls:
Students often think that earning another Rs. 200 on Rs. 200 takes less time than earning the first Rs. 100 on Rs. 100, confusing simple and compound interest. Under compound interest with a fixed rate, percentage growth is constant, so doubling time stays the same regardless of the starting principal.

Final Answer:
It will take an additional 5 years to earn another Rs. 200 interest and reach Rs. 400.