There is `40%` Increase in an amount in 8 years at simple interest. What will be the compound interest (In rupees) of `Rs 30000` after 2 years at the same rate ?

Options

A. `6150 `
B. `7687.5`
C. `4612.5 `
D. `3075 `

Answer

To solve the problem step by step, we will first determine the rate of interest based on the information given about simple interest, and then we will calculate the compound interest for the specified amount and time period.

### Step 1: Determine the Rate of Interest
We know that there is a 40% increase in the amount in 8 years at simple interest.

Let’s denote the principal amount as \( P \). The total amount after 8 years will be:
\[ A = P + \text{SI} = P + 0.40P = 1.40P \]

The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
where \( r \) is the rate of interest and \( t \) is the time in years.

From the information given:
\[ \text{SI} = 0.40P \]
Substituting into the SI formula:
\[ 0.40P = \frac{P \times r \times 8}{100} \]

Now, we can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 0.40 = \frac{r \times 8}{100} \]

Now, solving for \( r \):
\[ r = \frac{0.40 \times 100}{8} = 5\% \]

### Step 2: Calculate Compound Interest for Rs 30000 after 2 years
Now that we have the rate of interest \( r = 5\% \), we will calculate the compound interest for \( P = 30000 \) over \( t = 2 \) years.

The formula for Compound Interest (CI) is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]

Substituting the values:
\[ A = 30000 \left(1 + \frac{5}{100}\right)^2 \]
\[ A = 30000 \left(1 + 0.05\right)^2 \]
\[ A = 30000 \left(1.05\right)^2 \]
\[ A = 30000 \times 1.1025 = 33075 \]

Now, to find the compound interest:
\[ \text{CI} = A - P = 33075 - 30000 = 3075 \]

### Final Answer
The compound interest on Rs 30000 after 2 years at the same rate of interest is **Rs 3075**.

---