A sum of Rs. 2000 becomes Rs. 3645 in 2 years at a certain rate of compound interest. At the same annual rate of compound interest, what will be the amount of this sum after 4 years in total?

Introduction / Context:
Here we have a compound interest question where the amount after 2 years is known and the principal is Rs. 2000. We first determine the annual compound interest rate that transforms Rs. 2000 into Rs. 3645 in 2 years. Once we know this rate, we apply it for a total of 4 years to find the new amount of the same sum at compound interest.

Given Data / Assumptions:

• Principal P = Rs. 2000.
• Amount after 2 years A2 = Rs. 3645.
• Time for first stage T1 = 2 years.
• Same annual rate R is used for compound interest.
• We want the amount A4 after total time T2 = 4 years.

Concept / Approach:
For compound interest with annual compounding, amount A after T years is given by A = P * (1 + R / 100)^T. We use A2 = 2000 * (1 + R / 100)^2 = 3645 to find the growth factor (1 + R / 100). Once we have this factor, we raise it to the power 4 to get A4 = 2000 * (1 + R / 100)^4.

Step-by-Step Solution:
Let (1 + R / 100) = k. Then A2 = P * k^2, so 3645 = 2000 * k^2. Thus, k^2 = 3645 / 2000 = 1.8225. Taking the positive square root: k = sqrt(1.8225) = 1.35. So 1 + R / 100 = 1.35, giving R / 100 = 0.35 and R = 35% per annum. Now for 4 years, amount A4 = 2000 * k^4 = 2000 * (1.35)^4. Compute (1.35)^2 = 1.8225 and then (1.35)^4 = (1.8225)^2. (1.8225)^2 = 3.32150625 approximately. So A4 ≈ 2000 * 3.32150625 = Rs. 6643.0125.
Verification / Alternative check:
We can also compute year by year: After 1 year, amount = 2000 * 1.35 = 2700. After 2 years, 2700 * 1.35 = 3645, which matches the given amount. After 3 years, 3645 * 1.35 = 4920.75. After 4 years, 4920.75 * 1.35 = 6643.0125, confirming our calculation and matching the given option.

Why Other Options Are Wrong:
The other amounts (5942.0125, 7000.0000, 7243.0125, and 7498.1250) correspond to either different rates or different time periods. None of them matches the exact growth obtained by compounding at 35% for 4 years on Rs. 2000 after passing through Rs. 3645 at year 2.

Common Pitfalls:
A common error is to treat the situation as simple interest and incorrectly scale the 2-year increase linearly to 4 years. Another mistake is using an approximate rate without solving for the precise growth factor k, which leads to rounding errors that do not match any option exactly. Keeping the compound interest formula clear prevents confusion.

Final Answer:
The amount after 4 years will be approximately Rs. 6643.0125.