Find the compound interest on Rs. 6500 for 4 years if the rate of interest is 10 percent per annum for the first 2 years and 20 percent per annum for the next 2 years, compounded annually in each period.

Introduction / Context:
This question involves compound interest with a changing rate over different time periods. For the first 2 years the rate is 10 percent per annum, and for the next 2 years it increases to 20 percent per annum. We must compute the total compound interest on Rs. 6500 over the entire 4 year period. Such questions check understanding of how to apply different rates sequentially while compounding the amount forward.

Given Data / Assumptions:

• Principal P = Rs. 6500.
• Rate for first 2 years = 10 percent per annum.
• Rate for next 2 years = 20 percent per annum.
• Compounding is annual in each phase.
• Total time = 4 years, with different rates applied sequentially.

Concept / Approach:
When the rate changes after a certain period, we treat the process in stages. First we compute the amount after the initial period using the first rate. Then we use that amount as the new principal for the next period at the new rate. The compound interest is simply the final amount minus the original principal. The compound interest formula A = P * (1 + r/100)^n is applied separately for each stage.

Step-by-Step Solution:
Stage 1: first 2 years at 10 percent. Initial principal P = 6500 Amount after 2 years A1 = 6500 * (1 + 10/100)^2 (1 + 10/100) = 1.10, so (1.10)^2 = 1.21 A1 = 6500 * 1.21 = 7865 Stage 2: next 2 years at 20 percent. New principal for this stage = A1 = 7865 Amount after next 2 years A2 = 7865 * (1 + 20/100)^2 (1 + 20/100) = 1.20, so (1.20)^2 = 1.44 A2 = 7865 * 1.44 = 11325.6 Total amount after 4 years = Rs. 11325.60 Compound interest = A2 - original principal = 11325.60 - 6500 = 4825.60 Rounding to the nearest rupee gives Rs. 4826

Verification / Alternative check:
We can combine the two stages into a single equivalent factor: Overall factor = (1.10)^2 * (1.20)^2 = 1.21 * 1.44 = 1.7424 Amount = 6500 * 1.7424 = 11325.6 This matches our earlier calculation. Subtracting the principal again gives 4825.6, which rounds to Rs. 4826. The consistency between both methods confirms the correctness of the result.

Why Other Options Are Wrong:
Rs. 3845 and Rs. 4415 are both significantly lower than the correct compound interest and would correspond to smaller effective rates. Rs. 5142 is larger than 4826 and would require a higher overall growth factor than what the given rates provide. Rs. 4600 is somewhat close but does not match the exact detailed calculation. Only Rs. 4826 correctly represents the compound interest produced by 10 percent for 2 years followed by 20 percent for 2 years on Rs. 6500.

Common Pitfalls:
Some students incorrectly take an average rate, such as 15 percent for all 4 years, instead of treating the 10 percent and 20 percent phases separately. Others forget to use the amount from the first phase as the principal for the second phase and reuse the original principal by mistake. Rounding too early or mishandling decimals in multiplication can also cause small errors. Carefully separating the computation into stages and only rounding at the final step helps maintain accuracy.

Final Answer:
The compound interest on Rs. 6500 for 4 years, with 10 percent per annum for the first 2 years and 20 percent per annum for the next 2 years, is Rs. 4826 (approximately).