Arun invested Rs. 10,000 for three years at CI at the rate of 20% per annum. If in `1^(st)` and `3^(rd)` year interest is calculated annually and in `2^(nd)` year it was calculated half-yearly, then find the total interest received by Arun in three years?

Options

A. A)Rs 7554
B. B)Rs 7424
C. C)Rs 7868
D. D)Rs 7262

Answer

To solve the problem step by step, we will calculate the compound interest (CI) for each year based on the given conditions.

### Step 1: Calculate the amount after the 1st year
- **Principal (P)** = Rs. 10,000
- **Rate of Interest (R)** = 20% per annum
- **Time (T)** = 1 year

The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

For the 1st year:
\[ A_1 = 10,000 \left(1 + \frac{20}{100}\right)^1 \]
\[ A_1 = 10,000 \left(1 + 0.2\right) \]
\[ A_1 = 10,000 \times 1.2 \]
\[ A_1 = 12,000 \]

### Step 2: Calculate the amount after the 2nd year (half-yearly)
For the 2nd year, the interest is calculated half-yearly. Therefore, we will split the year into two halves.

- **Rate for half-year** = 10% (half of 20%)
- **Time** = 2 half-years

Using the formula for compound interest:
\[ A_2 = A_1 \left(1 + \frac{10}{100}\right)^2 \]
\[ A_2 = 12,000 \left(1 + 0.1\right)^2 \]
\[ A_2 = 12,000 \times (1.1)^2 \]
\[ A_2 = 12,000 \times 1.21 \]
\[ A_2 = 14,520 \]

### Step 3: Calculate the amount after the 3rd year
Now, we will calculate the amount for the 3rd year using the amount from the 2nd year as the principal.

- **Principal for 3rd year** = Rs. 14,520
- **Rate of Interest (R)** = 20% per annum
- **Time (T)** = 1 year

Using the compound interest formula again:
\[ A_3 = A_2 \left(1 + \frac{20}{100}\right)^1 \]
\[ A_3 = 14,520 \left(1 + 0.2\right) \]
\[ A_3 = 14,520 \times 1.2 \]
\[ A_3 = 17,424 \]

### Step 4: Calculate the total interest received
To find the total interest received by Arun over the three years, we subtract the principal from the final amount after the 3rd year.

Total Interest (I) = Final Amount - Principal
\[ I = A_3 - P \]
\[ I = 17,424 - 10,000 \]
\[ I = 7,424 \]

### Final Answer
The total interest received by Arun in three years is **Rs. 7,424**.

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