Rs. 1000 is invested at 50 percent per annum compound interest. What is the interest earned in the third year alone under compound interest?

Introduction / Context:
This problem focuses on interest earned in a specific year under compound interest, rather than the total interest over the whole period. The principal of Rs. 1000 is invested at a high rate of 50 percent per annum, and we are asked for the interest earned in the third year alone. Understanding how compound interest grows year by year is important for answering such questions correctly.

Given Data / Assumptions:

• Principal P = Rs. 1000.
• Rate r = 50 percent per annum.
• We are interested in the third year's interest, not the total.
• Compounding is annual.

Concept / Approach:
Under compound interest, the interest for each year is calculated on the amount at the beginning of that year. Therefore, the interest in the third year is equal to the amount at the end of the second year multiplied by the rate. We can find the amount after each year by repeatedly multiplying by (1 + r/100). Alternatively, we can use the idea that the third year interest equals P * (1 + r/100)^2 * r/100.

Step-by-Step Solution:
P = 1000, r = 50 percent Year 1 amount A1 = 1000 * (1 + 50/100) = 1000 * 1.5 = 1500 Year 2 amount A2 = A1 * 1.5 = 1500 * 1.5 = 2250 The amount at the start of year 3 is Rs. 2250 Interest for year 3 = A2 * r / 100 = 2250 * 50 / 100 Interest in third year = 2250 * 0.5 = 1125 Therefore, the interest for the third year alone is Rs. 1125

Verification / Alternative check:
We can also check by computing total amounts and subtracting earlier year interest. After 3 years, the amount A3 is: A3 = 1000 * (1.5)^3 (1.5)^2 = 2.25, so (1.5)^3 = 2.25 * 1.5 = 3.375 A3 = 1000 * 3.375 = 3375 Total compound interest for 3 years is 3375 - 1000 = 2375. Interest in year 1 is 500, interest in year 2 is 750 (since 1500 * 0.5), and interest in year 3 must be 2375 - (500 + 750) = 1125. This confirms the earlier result.

Why Other Options Are Wrong:
625 is actually the square of 25 and might distract some students, but the interest for the first year is 500, not 625. 1025 and 1122 are close numbers but do not match the exact calculation of 2250 * 0.5. 750 is the second year's interest, not the third year's. Only 1125 is consistent with the compounding process and the total growth to Rs. 3375 by the end of the third year.

Common Pitfalls:
A common mistake is to simply compute 1000 * 50/100 and assume that each year earns the same interest, which is true only for simple interest, not compound interest. Others may forget that the base for calculation in the third year is the amount at the end of the second year, not the original principal. Miscomputing powers of 1.5 can also lead to errors. Understanding that compound interest applies to the updated amount each year is crucial here.

Final Answer:
The interest earned in the third year alone is Rs. 1125.