A person invests Rs. 7500 at 10% per annum on compound interest. If the money becomes Rs. 9075 on a certain time, then find how much interest he gets if the same money he invests at the same rate of interest for the same time on simple interest.

Options

A. Rs. 1200
B. Rs. 1500
C. Rs. 1800
D. Rs. 1400

Answer

To solve the problem step by step, we will first determine the time period for which the money was invested in compound interest, and then calculate the simple interest for the same amount and time.

### Step 1: Determine the amount of interest earned through compound interest.

Given:
- Principal (P) = Rs. 7500
- Rate of interest (R) = 10% per annum
- Amount after a certain time (A) = Rs. 9075

The formula for compound interest is:

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

Where:
- \( A \) = Final amount
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( t \) = Time in years

### Step 2: Substitute the known values into the formula.

\[ 9075 = 7500 \left(1 + \frac{10}{100}\right)^t \]

This simplifies to:

\[ 9075 = 7500 \left(1.1\right)^t \]

### Step 3: Solve for \( \left(1.1\right)^t \).

Divide both sides by 7500:

\[ \frac{9075}{7500} = \left(1.1\right)^t \]

Calculating the left side:

\[ 1.21 = \left(1.1\right)^t \]

### Step 4: Take logarithm to solve for \( t \).

Taking logarithm on both sides:

\[ \log(1.21) = t \cdot \log(1.1) \]

Now, solving for \( t \):

\[ t = \frac{\log(1.21)}{\log(1.1)} \]

Using a calculator, we find:

\[ \log(1.21) \approx 0.0827 \quad \text{and} \quad \log(1.1) \approx 0.0414 \]

Thus,

\[ t \approx \frac{0.0827}{0.0414} \approx 2 \]

### Step 5: Calculate the simple interest for the same principal, rate, and time.

Now that we have \( t = 2 \) years, we can calculate the simple interest (SI) using the formula:

\[ SI = \frac{P \cdot R \cdot t}{100} \]

Substituting the values:

\[ SI = \frac{7500 \cdot 10 \cdot 2}{100} \]

Calculating:

\[ SI = \frac{150000}{100} = 1500 \]

### Final Answer:

The interest earned through simple interest for the same amount, rate, and time is Rs. 1500.

---