The difference between compound interest and simple interest on a certain sum for 2 years at 20 percent per annum is Rs. 200. If, instead, the interest is compounded half yearly, what is the difference between compound interest and simple interest for the first year on the same principal?

Introduction / Context:
This problem explores how the difference between compound interest (CI) and simple interest (SI) behaves under different compounding frequencies. First, we are told the difference between CI and SI for 2 years at 20 percent per annum with annual compounding. From this, we can deduce the principal. Then, the question asks us to switch to half yearly compounding and compute the difference between CI and SI for just the first year. Understanding the standard formulas for the CI–SI difference makes this type of question much faster to solve.

Given Data / Assumptions:

• Annual rate r = 20 percent.
• Time t₁ = 2 years for the first scenario with annual compounding.
• Difference between CI and SI for 2 years is Rs. 200.
• Same principal P is used in both scenarios.
• In the second scenario, interest is compounded half yearly for 1 year.
• We need the difference between CI and SI for just the first year in the half yearly case.

Concept / Approach:
For 2 years at rate r percent, the difference between compound and simple interest is:
D₂ = CI₂ − SI₂ = P * (r/100)^2.
This formula is a standard shortcut. Using D₂ = 200 and r = 20, we can find P. For the second part, with half yearly compounding in 1 year, the rate per half year is r/2 = 10 percent and there are 2 periods. We compute SI for 1 year at 20 percent and CI for 1 year with two half yearly periods at 10 percent, then take the difference CI − SI for that year.

Step-by-Step Solution:
Step 1: Use the formula D₂ = P * (r/100)^2 for 2 years. Here, D₂ = 200 and r = 20. Step 2: Compute (r/100)^2 = (20/100)^2 = (0.2)^2 = 0.04. Step 3: So 200 = P * 0.04, giving P = 200 / 0.04 = 5000. Step 4: Now consider the second scenario: 1 year with half yearly compounding at a nominal rate of 20 percent per annum. Step 5: Rate per half year = 20/2 = 10 percent. Number of half years in 1 year = 2. Step 6: Simple interest for 1 year at 20 percent on P = 5000 is SI₁ = (5000 * 20 * 1) / 100 = 1000. Step 7: Compound interest for 1 year with two half yearly periods is computed from the amount: A = P * (1 + 10/100)^2 = 5000 * (1.1)^2 = 5000 * 1.21 = 6050. Step 8: CI₁ = A − P = 6050 − 5000 = 1050. Step 9: The required difference for the first year is CI₁ − SI₁ = 1050 − 1000 = 50. Step 10: Therefore, the difference between compound interest and simple interest for the first year with half yearly compounding is Rs. 50.

Verification / Alternative check:
We can verify by observing that the extra amount due to compounding more frequently within the same year can be approximated as P * (r/2/100)^2 times the number of periods. Here, P = 5000, half yearly rate = 10 percent, so per period factor squared is (0.1)^2 = 0.01. Roughly, the extra due to intra-year compounding is about 5000 * 0.01 = 50, which matches our exact calculation. This quick consistency check reassures us that our more detailed steps are correct.

Why Other Options Are Wrong:
Rs. 75, Rs. 100, Rs. 150, and Rs. 25 correspond to incorrect assumptions about the principal or about how many periods are compounding within the year. For example, Rs. 100 would result if we mistakenly doubled the extra interest or misapplied the 20 percent rate directly in a compound interest formula for the full year without recognizing the half yearly structure correctly. Only Rs. 50 fits the accurate calculation with P = 5000, SI = 1000, and CI = 1050 for the first year.

Common Pitfalls:
Many learners forget the standard shortcut D₂ = P * (r/100)^2 and instead perform longer calculations that are more prone to arithmetic mistakes. Another error is to think that the difference for 1 year with half yearly compounding is the same as the 2 year difference with annual compounding divided by 2, which is not true. Confusing nominal annual rate (20 percent) with per period rate (10 percent) is another common source of error. Careful attention to compounding frequency is essential.

Final Answer:
When interest is compounded half yearly, the difference between compound interest and simple interest for the first year on the same principal is Rs. 50.