A sum of Rs 2000 amounts to Rs 4000 in 2 years at compound interest. In how many years from the start will the same principal amount become Rs 8000 at the same rate of interest?

Introduction / Context:
This question explores how money grows under compound interest when the amount is doubling and then doubling again. The key idea is that, for a fixed compound interest rate, the growth factor over any equal time interval remains the same. Here, we know that Rs 2000 becomes Rs 4000 in 2 years, and we need to find how long it will take to become Rs 8000 at the same rate. This situation gives a nice opportunity to use ratios and powers rather than complicated calculations.

Given Data / Assumptions:

• Principal P = Rs 2000
• Amount after 2 years A2 = Rs 4000
• Future target amount A = Rs 8000
• Same compound interest rate throughout
• Interest is compounded annually

Concept / Approach:
If Rs 2000 grows to Rs 4000 in 2 years, the amount has doubled in that period, so the growth factor over 2 years is 4000 / 2000 = 2. Under compound interest, this factor over the same interval will remain constant. To go from Rs 2000 to Rs 8000, the money must become four times the original principal. Since a factor of 2 is achieved in 2 years, a factor of 4 will be achieved in twice that time, because 4 = 2^2. Therefore, the required time is simply 2 intervals of 2 years each, totaling 4 years.

Step-by-Step Solution:
Given: 2000 becomes 4000 in 2 years Growth factor for 2 years = 4000 / 2000 = 2 We want to know when 2000 becomes 8000 Required total growth factor = 8000 / 2000 = 4 Notice that 4 = 2^2 Each 2 year period multiplies the amount by 2 Two such periods yield factor 2 * 2 = 4 over 4 years Therefore, required time = 2 years + 2 years = 4 years

Verification / Alternative check:
We can also express the situation with the compound interest formula. Let r be the annual rate. Then 2000 * (1 + r)^2 = 4000, so (1 + r)^2 = 2. Therefore, 1 + r = square root of 2. For time t, we need 2000 * (1 + r)^t = 8000, so (1 + r)^t = 4. Since (1 + r)^2 = 2, we have (1 + r)^4 = 4, so t = 4. This matches our simple reasoning based on doubling factors.

Why Other Options Are Wrong:
2 years would only double the principal once, taking it to Rs 4000, not Rs 8000. 6 years would imply one and a half doubling intervals, which does not fit the neat doubling pattern derived from the given data. 8 years would be four doubling periods, which would produce a growth factor of 2^4 = 16 times the original amount. Only 4 years yields exactly four times the principal.

Common Pitfalls:
Some learners mistakenly treat this as simple interest and assume a linear growth, which leads to an incorrect time. Others forget that the same factor applies over equal intervals when interest is compounded at a constant rate. Recognizing the doubling pattern and linking it to powers of 2 is the fastest and most reliable method here.

Final Answer:
The principal will become Rs 8000 in 4 years at the same compound interest rate.