Mr. Duggal invested 20000 with rate of interest `"@"` 20% per annum. The interest was compounded half-yearly for the first 1year and in the next year it was compounded yearly. What will be the total interest earned at the end of 2 years?

Options

A. 8800
B. 9040
C. 8040
D. 9800

Answer

To find the total interest earned by Mr. Duggal at the end of 2 years, we will follow these steps:

### Step 1: Understand the investment details
Mr. Duggal invested ₹20,000 at an interest rate of 20% per annum. The interest is compounded half-yearly for the first year and yearly for the second year.

### Step 2: Calculate the amount after the first year (half-yearly compounding)
1. **Principal (P)** = ₹20,000
2. **Rate of interest (R)** = 20% per annum, which is 10% for each half-year.
3. **Number of compounding periods (n)** = 2 (since it's half-yearly for one year).

Using the compound interest formula:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the values:
\[ A = 20000 \left(1 + \frac{10}{100}\right)^2 \]
\[ A = 20000 \left(1 + 0.10\right)^2 = 20000 \left(1.10\right)^2 \]
Calculating \( (1.10)^2 \):
\[ (1.10)^2 = 1.21 \]
Now substituting back:
\[ A = 20000 \times 1.21 = 24200 \]

### Step 3: Calculate the amount after the second year (yearly compounding)
1. **New Principal (P)** = ₹24,200 (amount after the first year)
2. **Rate of interest (R)** = 20% per annum.
3. **Number of compounding periods (n)** = 1 (since it's yearly for the second year).

Using the compound interest formula again:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the values:
\[ A = 24200 \left(1 + \frac{20}{100}\right)^1 \]
\[ A = 24200 \left(1 + 0.20\right) = 24200 \times 1.20 \]
Calculating:
\[ A = 29040 \]

### Step 4: Calculate the total interest earned
Total interest earned (I) is given by:
\[ I = A - P \]
Where:
- \( A \) is the total amount after 2 years = ₹29,040
- \( P \) is the initial principal = ₹20,000

Calculating the interest:
\[ I = 29040 - 20000 = 9040 \]

### Final Answer
The total interest earned at the end of 2 years is ₹9,040.

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