If the compound interest received on a certain principal during the 3rd year is Rs. 1,240 at a rate of 9% per annum, what will be the compound interest for the 4th year on the same principal at the same rate?

Introduction / Context:
This question uses the pattern of yearly compound interest amounts to find interest in a later year when the interest in an earlier year is known. It highlights the fact that under compound interest, the interest itself grows from year to year at the same rate as the principal.

Given Data / Assumptions:

• Compound interest in the 3rd year, I3 = Rs. 1,240.
• Annual interest rate r = 9% per annum.
• Interest is compounded annually.
• We must find compound interest in the 4th year, I4.

Concept / Approach:
For a principal P under annual compounding at rate r, the interest in year n is P * r * (1 + r)^(n - 1). This means each year's interest is obtained by multiplying the previous year's interest by (1 + r). Therefore I4 = I3 * (1 + r). Using this relation avoids solving for the principal directly and provides a clean shortcut for this type of problem.

Step-by-Step Solution:
Step 1: Recognize that interest each year forms a geometric sequence under compound interest.Step 2: If I3 is the interest in year 3, then interest in year 4 is I4 = I3 * (1 + r).Step 3: Here I3 = 1,240 and r = 9% = 0.09.Step 4: Compute 1 + r = 1 + 0.09 = 1.09.Step 5: Multiply: I4 = 1,240 * 1.09.Step 6: I4 = 1,240 * 1.09 = 1,351.6 rupees.

Verification / Alternative check:
If needed, we can reconstruct the principal P from the third year interest. Since I3 = P * r * (1 + r)^2, we can solve for P and then compute interest in the 4th year as P * r * (1 + r)^3. However, because multiplying I3 by (1 + r) produces exactly the same result, doing the longer method will give I4 = 1,351.6 as well, confirming that our direct shortcut and the more detailed method agree.

Why Other Options Are Wrong:

• 1245.6: This is only slightly greater than the third year interest and corresponds to a much lower effective growth than 9%.
• 1521.6: This is too large; it would imply a higher rate than 9% when compared to the third year interest.
• 1220.6: This is actually less than 1,240 and would suggest a negative or very low effective growth, which contradicts the 9% interest rate.

Common Pitfalls:
Students sometimes mistakenly add 9% of the original principal to the third year interest instead of applying 9% to the third year amount itself. Others confuse simple and compound interest and treat all yearly interest amounts as equal. It is important to remember that under compound interest, each year's interest grows by the same factor (1 + r) as the amount, so interest values form a geometric progression.

Final Answer:
The compound interest for the 4th year is 1351.6 rupees.