Hemant borrows ₹4000 at a rate of compund interest 5% per annum. After 2 years he pays back ₹ 2210 and after an other two years he pays all his dues with interest. Find the total interest he gave ?

Options

A. 2420
B. 682
C. 210
D. `635.5`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the amount due after the first 2 years.

The formula for compound interest is given by:

\[ A = P \left(1 + \frac{R}{100}\right)^N \]

Where:
- \( A \) = Total amount after \( N \) years
- \( P \) = Principal amount (initial amount)
- \( R \) = Rate of interest per annum
- \( N \) = Number of years

Here, \( P = 4000 \), \( R = 5\% \), and \( N = 2 \).

Substituting the values into the formula:

\[ A = 4000 \left(1 + \frac{5}{100}\right)^2 \]

Calculating \( \left(1 + \frac{5}{100}\right) = 1.05 \):

\[ A = 4000 \times (1.05)^2 \]

Calculating \( (1.05)^2 = 1.1025 \):

\[ A = 4000 \times 1.1025 = 4410 \]

So, after 2 years, the total amount due is ₹4410.

### Step 2: Calculate the remaining amount after Hemant pays back ₹2210.

After paying back ₹2210, the remaining amount is:

\[ \text{Remaining Amount} = 4410 - 2210 = 2200 \]

### Step 3: Calculate the amount due after the next 2 years on the remaining amount.

Now, we will calculate the amount due on the remaining ₹2200 for another 2 years at the same interest rate of 5%.

Using the same compound interest formula:

\[ A = P \left(1 + \frac{R}{100}\right)^N \]

Here, \( P = 2200 \), \( R = 5\% \), and \( N = 2 \).

Substituting the values:

\[ A = 2200 \left(1 + \frac{5}{100}\right)^2 \]

Calculating \( (1.05)^2 = 1.1025 \):

\[ A = 2200 \times 1.1025 = 2425.5 \]

So, after another 2 years, the total amount due is ₹2425.5.

### Step 4: Calculate the total interest paid by Hemant.

To find the total interest paid, we need to subtract the principal amount from the total amount paid back.

The total amount paid back by Hemant is:

\[ \text{Total Amount Paid} = 2210 + 2425.5 = 4635.5 \]

The total interest paid is:

\[ \text{Total Interest} = \text{Total Amount Paid} - \text{Original Principal} \]

\[ \text{Total Interest} = 4635.5 - 4000 = 635.5 \]

### Final Answer:
The total interest Hemant paid is ₹635.5.

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