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

Introduction / Context:
This problem is similar to an earlier one and again uses consecutive yearly amounts under compound interest to infer the annual rate of interest. The amount after 3 years and 4 years is given for the same principal and unknown rate. Because compounding is annual, the ratio between the amounts for consecutive years directly indicates the growth factor. We must determine the rate from this ratio.

Given Data / Assumptions:

• Amount after 3 years A3 = Rs. 800.
• Amount after 4 years A4 = Rs. 840.
• Compounding is annual at rate r percent.
• The principal is the same in both cases.

Concept / Approach:
Under annual compounding at rate r percent, each year multiplies the amount by: 1 + r/100 Therefore the ratio: A4 / A3 = 1 + r/100 We can compute A4 / A3 and then convert the result into a percentage rate. This approach avoids any need to know the principal explicitly.

Step-by-Step Solution:
A3 = 800, A4 = 840 Ratio A4 / A3 = 840 / 800 840 / 800 = 1.05 So 1 + r/100 = 1.05 r/100 = 1.05 - 1 = 0.05 r = 0.05 * 100 = 5 percent Thus the rate of interest per annum is 5 percent

Verification / Alternative check:
We could pick an arbitrary principal P and check consistency. If the factor is 1.05 per year, then: A3 = P * (1.05)^3 A4 = P * (1.05)^4 = A3 * 1.05 Given A3 = 800, then A4 must be 800 * 1.05 = 840, which matches the data. This confirms that the growth factor and therefore the rate are correctly determined.

Why Other Options Are Wrong:
A rate of 4 percent gives a yearly factor of 1.04, which would yield A4 = 800 * 1.04 = 832, not 840. A rate of 6 percent would give a factor of 1.06, producing A4 = 848, and 7 percent would give 856. A rate of 3 percent would give 824. None of these match the given A4 of 840. Only a rate of 5 percent gives the correct ratio between A3 and A4.

Common Pitfalls:
A typical error is to overcomplicate the problem by trying to work back to the principal using long algebra instead of using the simple ratio method. Some students also misread the amounts or accidentally compute 800 / 840 instead of 840 / 800, which leads to an incorrect rate less than 1. Recognizing that the fourth year amount is simply the third year amount grown by one more year at the same rate is the key insight here.

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