There is `100%` increase to an amount in 8 years, at simple interest. Find the compound interest of `Rs. 8000` after 2 years at the same rate of interest.

Options

A. `Rs. 2500 `
B. `Rs. 2000`
C. `Rs. 2250 `
D. `Rs. 2125 `

Answer

To solve the problem step by step, we need to follow these steps:

### Step 1: Determine the Rate of Interest
We know that there is a 100% increase in the amount in 8 years at simple interest. This means the total interest earned is equal to the principal amount.

Let the principal amount (P) be Rs. 8000. Therefore, the total interest (I) earned in 8 years is also Rs. 8000.

Using the formula for simple interest:
\[ I = P \times r \times t \]
Where:
- \( I \) = Interest
- \( P \) = Principal
- \( r \) = Rate of interest (in decimal)
- \( t \) = Time in years

Substituting the known values:
\[ 8000 = 8000 \times r \times 8 \]

Now, we can solve for \( r \):
\[ r = \frac{8000}{8000 \times 8} = \frac{1}{8} \]

To convert this to a percentage, we multiply by 100:
\[ r = \frac{1}{8} \times 100 = 12.5\% \]

### Step 2: Calculate Compound Interest for Rs. 8000 after 2 Years
Now that we have the rate of interest, we can calculate the compound interest for Rs. 8000 after 2 years at the same rate of interest.

The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Amount after n years
- \( P \) = Principal
- \( r \) = Rate of interest
- \( n \) = Number of years

Substituting the values:
\[ A = 8000 \left(1 + \frac{12.5}{100}\right)^2 \]
\[ A = 8000 \left(1 + 0.125\right)^2 \]
\[ A = 8000 \left(1.125\right)^2 \]
\[ A = 8000 \times 1.265625 \]
\[ A = 10125 \]

### Step 3: Calculate Compound Interest
Now, we can find the compound interest (CI) by subtracting the principal from the total amount:
\[ CI = A - P \]
\[ CI = 10125 - 8000 \]
\[ CI = 2125 \]

### Final Answer
The compound interest of Rs. 8000 after 2 years at the same rate of interest is Rs. 2125.

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