A sum is lent at 20% per annum compound interest. What is the ratio of the increase in amount (that is, the interest earned) in the 4th year to the increase in amount in the 5th year?

Introduction:
This question explores how the yearly interest amounts behave under compound interest at a constant rate. It asks for the ratio of the interest earned in the 4th year to the interest earned in the 5th year, without even needing the actual principal value.

Given Data / Assumptions:

• Rate of interest r = 20% per annum.
• Compounding is annual.
• We are interested in the increase in amount (interest) during the 4th year and the 5th year.
• The principal P is not specified, but we will see it cancels out.

Concept / Approach:
Under compound interest, the amount at the start of year n is P * (1 + r)^(n−1). The interest earned in year n is this amount multiplied by r. Therefore, yearly interests form a geometric sequence. We use this to find the ratio of the interest in the 4th and 5th years.

Step-by-Step Solution:
Let P be the initial principal and r = 20% = 0.20.Amount at the start of 4th year = P * (1.20)^3Interest in 4th year I4 = P * (1.20)^3 * 0.20Amount at the start of 5th year = P * (1.20)^4Interest in 5th year I5 = P * (1.20)^4 * 0.20Now I4 : I5 = [P * (1.20)^3 * 0.20] : [P * (1.20)^4 * 0.20]Cancel P and 0.20: I4 : I5 = (1.20)^3 : (1.20)^4 = 1 : 1.201 : 1.20 = 5 : 6

Verification / Alternative Check:
Assume an easy principal, say P = Rs 100. Compute actual interest values. Interest in 4th year is 100 * (1.2)^3 * 0.2 = 100 * 1.728 * 0.2 = Rs 34.56. Interest in 5th year is 100 * (1.2)^4 * 0.2 = 100 * 2.0736 * 0.2 = Rs 41.472. Ratio 34.56 : 41.472 simplifies to 5 : 6, confirming our earlier result.

Why Other Options Are Wrong:
4 : 5 and 5 : 4: These ratios do not match the actual growth pattern under 20% compound interest.Cannot be determined: Incorrect because the principal cancels out, so the ratio can be determined.1 : 1: This would imply equal interest in both years, which is not true under compound interest.

Common Pitfalls:
Some learners mistakenly calculate simple interest for each year or assume that the interest added each year is constant. Under compound interest, the amount grows, so the interest in later years is always higher than in earlier years at the same rate. Recognizing the geometric pattern makes such ratio questions straightforward.

Final Answer:
The ratio of the increase in amount in the 4th year to that in the 5th year is 5 : 6.