Find the compound interest on Rs. 6,500 for 4 years if the rate of interest is 10% per annum for the first 2 years and 20% per annum for the next 2 years, with interest compounded annually throughout.

Introduction / Context:
This question deals with compound interest at changing rates over different time intervals. The principal earns 10% per annum for the first two years and then 20% per annum for the following two years. You must apply the compound interest process step by step rather than using a single rate. Such mixed-rate problems are common in aptitude exams to test careful handling of successive growth periods.

Given Data / Assumptions:
- Principal P = Rs. 6,500.
- First 2 years: interest rate = 10% per annum, compounded annually.
- Next 2 years: interest rate = 20% per annum, compounded annually.
- Total time = 4 years.
- We must find the total compound interest earned over 4 years.

Concept / Approach:
We cannot use a single compound interest formula with a uniform rate, because the rate changes after 2 years. Instead, we compound year by year (or block by block):
- First compound the principal for 2 years at 10%.
- Then take the amount after 2 years as a new principal and compound it for the next 2 years at 20%.
The final amount minus the original principal gives the required compound interest.

Step-by-Step Solution:
Step 1: Start with principal P = 6500.Step 2: After first year at 10%: A1 = 6500 * 1.10 = 7150.Step 3: After second year at 10%: A2 = 7150 * 1.10 = 7865.Step 4: Now rate becomes 20% for the third year: A3 = 7865 * 1.20.Step 5: Compute A3: 7865 * 1.20 = 9438.Step 6: Fourth year also at 20%: A4 = 9438 * 1.20 = 11325.6 (approximately).Step 7: Total amount after 4 years ≈ Rs. 11,325.60.Step 8: Compound interest = A4 - P ≈ 11325.6 - 6500 = Rs. 4,825.6.Step 9: Rounding to the nearest rupee gives Rs. 4,826.

Verification / Alternative check:
You can group the first two years and the last two years using factors: after 2 years at 10%, amount = 6500 * (1.10)^2 = 6500 * 1.21 = 7865; after 2 years at 20%, multiply by (1.20)^2 = 1.44, giving 7865 * 1.44 = 11325.6. This reconfirms the same amount and thus the same compound interest of about Rs. 4,826.

Why Other Options Are Wrong:
Rs. 3,845, Rs. 4,415, Rs. 4,250 and Rs. 5,142 arise from mistakes such as using simple interest, applying 10% or 20% for all four years, or mis-multiplying the growth factors. Only Rs. 4,826 correctly reflects successive compounding with different rates.

Common Pitfalls:
A frequent mistake is to average the rates and use a single compound interest formula, which is incorrect when rates change over time. Another mistake is forgetting that the amount after the first interval becomes the principal for the next interval. Carefully compounding stepwise according to each rate prevents these errors.

Final Answer:
The compound interest on Rs. 6,500 over the 4 years is approximately Rs. 4,826.