There is `80%` increase in an amount in 8 years at simple interest. What will be the compound interest of ₹ 14,000 after 3 years at the same rate?

Options

A. ₹ 4634
B. ₹ 3714
C. ₹ 3794
D. ₹ 4612

Answer

To solve the problem, we first need to determine the rate of interest based on the information given about the simple interest.

### Step 1: Calculate the Simple Interest Rate
Given that there is an 80% increase in the amount in 8 years at simple interest, we can express this mathematically. If the principal amount is \( P \), the total amount after 8 years is:

\[ A = P + \text{SI} \]

Where SI (Simple Interest) is given by:

\[ \text{SI} = \frac{P \times R \times T}{100} \]

In this case, the increase is 80% of \( P \), so:

\[ A = P + 0.8P = 1.8P \]

Now, we can set up the equation:

\[ 1.8P = P + \frac{P \times R \times 8}{100} \]

Subtracting \( P \) from both sides gives:

\[ 0.8P = \frac{P \times R \times 8}{100} \]

Dividing both sides by \( P \) (assuming \( P \neq 0 \)):

\[ 0.8 = \frac{R \times 8}{100} \]

Now, multiplying both sides by 100:

\[ 80 = R \times 8 \]

Finally, dividing by 8:

\[ R = 10\% \]

### Step 2: Calculate the Compound Interest for ₹ 14,000 after 3 years
Now that we have the rate of interest \( R = 10\% \), we can calculate the compound interest for the principal amount \( P = ₹ 14,000 \) after 3 years.

The formula for compound interest is:

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

Where:
- \( A \) is the amount after \( n \) years,
- \( P \) is the principal amount,
- \( R \) is the rate of interest,
- \( n \) is the number of years.

Substituting the values:

\[ A = 14000 \left(1 + \frac{10}{100}\right)^3 \]

Calculating inside the parentheses:

\[ A = 14000 \left(1 + 0.1\right)^3 = 14000 \left(1.1\right)^3 \]

Calculating \( (1.1)^3 \):

\[ (1.1)^3 = 1.331 \]

Now substituting back:

\[ A = 14000 \times 1.331 = 18634 \]

### Step 3: Calculate the Compound Interest
The compound interest (CI) is given by:

\[ \text{CI} = A - P \]

Substituting the values we found:

\[ \text{CI} = 18634 - 14000 = 4634 \]

### Final Answer
The compound interest of ₹ 14,000 after 3 years at the same rate is **₹ 4634**.

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