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

Introduction / Context:
This problem focuses on how interest grows under compound interest. Instead of asking for a specific amount, it compares the interest earned in the 4th year with the interest earned in the 5th year, both at the same compound interest rate. Understanding how yearly interest forms a geometric progression is the key idea.

Given Data / Assumptions:

• The sum is invested at 20% per annum compound interest.
• We compare the increase in the amount in the 4th year with that in the 5th year.
• The principal is not specified, but for ratios the exact value of the principal is not needed.

Concept / Approach:

Under compound interest at rate r, the amount after n years is A_n = P * (1 + r)^n (where r is expressed as a decimal). The interest earned in year n is A_n - A_(n-1). These yearly interest amounts form a geometric progression with common ratio 1 + r. Therefore, the ratio of interest in consecutive years depends only on 1 + r, not on the principal itself.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

Ratios 4 : 5 and 5 : 4 do not reflect the multiplicative factor 1.20 between consecutive years. The option Cannot be determined is incorrect, because the ratio depends only on the rate and not on the actual principal. Once the rate is known, the ratio can be obtained exactly.

Common Pitfalls:

One pitfall is to think that interest is the same every year under compound interest, which is false. Another mistake is to try to compute the exact interest values with an arbitrary principal but then mis-handle rounding. Recognizing the geometric progression nature of yearly interest simplifies the problem greatly.

Final Answer:

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