A sum of money invested at compound interest amounts to Rs. 800 in 3 years and to Rs. 840 in 4 years. What is the annual rate of interest (per annum), assuming interest is compounded annually?

Introduction / Context:This problem is very similar to a standard textbook example of finding the compound interest rate from two successive annual amounts. You are told how the amount grows from year 3 to year 4, and from this one-year growth factor you can directly read the annual rate of interest.

Given Data / Assumptions:- Amount after 3 years, A3 = Rs. 800.- Amount after 4 years, A4 = Rs. 840.- Interest is compounded annually at a fixed rate r% per annum.- We must find r, the annual rate of compound interest.

Concept / Approach:Let P be the principal and r be the annual rate. Then:- A3 = P * (1 + r/100)^3.- A4 = P * (1 + r/100)^4.By dividing A4 by A3, the principal and the cube term cancel, leaving A4 / A3 = (1 + r/100). From this ratio, r is easily found as (A4 / A3 - 1) * 100.

Step-by-Step Solution:Step 1: Compute the ratio of the amounts: A4 / A3 = 840 / 800.Step 2: Simplify the fraction: 840 / 800 = 21 / 20 = 1.05.Step 3: Under compound interest, A4 / A3 = 1 + r/100.Step 4: So 1 + r/100 = 1.05.Step 5: Subtract 1: r/100 = 0.05.Step 6: Multiply by 100: r = 5.Step 7: Therefore, the annual interest rate is 5% per annum.

Verification / Alternative check:As a quick check, if the rate is 5%, then each year the amount multiplies by 1.05. Starting from year 3 amount 800, after one more year (year 4), the amount would be 800 * 1.05 = 840, exactly as given. This confirms that r = 5% is correct.

Why Other Options Are Wrong:At 4% interest, the ratio A4 / A3 would be 1.04, giving 832 instead of 840. At 6% interest, the ratio would be 1.06 and year 4 amount would be 848. Similarly, 7% and 3% give 856 and 824 respectively. Only 5% reproduces the given amounts.

Common Pitfalls:Some learners unnecessarily try to find the principal first using equations for A3 and A4, which complicates the algebra. Instead, notice that under compound interest the ratio of successive year-end amounts gives the yearly growth factor directly. Remember to keep the ratio in simplest form to see this easily.

Final Answer:The rate of compound interest is 5% per annum.