A sum of Rs. 7500 is divided into two parts. The simple interest on first part at the rate of 12 % per annum is equal to the simple interest on second part at the rate of 18 %. What is the interest (in Rs) on each part for one year ?

Options

A. 600
B. 360
C. 480
D. 540

Answer

To solve the problem, we need to divide the sum of Rs. 7500 into two parts such that the simple interest on each part at their respective rates is equal. Let's denote the first part as \( A \) and the second part as \( B \).

### Step-by-Step Solution:

1. **Define the Parts**:
Let the first part be \( A \) and the second part be \( B \).
\[ A + B = 7500 \]

2. **Set Up the Interest Equation**:
The simple interest (SI) for one year can be calculated using the formula:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

For the first part at 12%:
\[ \text{SI}_A = \frac{A \times 12 \times 1}{100} = \frac{12A}{100} \]

For the second part at 18%:
\[ \text{SI}_B = \frac{B \times 18 \times 1}{100} = \frac{18B}{100} \]

According to the problem, these two interests are equal:
\[ \frac{12A}{100} = \frac{18B}{100} \]

3. **Eliminate the Denominator**:
We can eliminate the denominator (100) from both sides:
\[ 12A = 18B \]

4. **Rearrange the Equation**:
Rearranging gives us:
\[ \frac{A}{B} = \frac{18}{12} = \frac{3}{2} \]

5. **Express in Terms of a Single Variable**:
Let \( A = 3x \) and \( B = 2x \) for some \( x \).

6. **Substitute Back into the Total**:
Substitute \( A \) and \( B \) back into the total:
\[ 3x + 2x = 7500 \]
\[ 5x = 7500 \]
\[ x = \frac{7500}{5} = 1500 \]

7. **Find the Values of A and B**:
Now, substituting \( x \) back to find \( A \) and \( B \):
\[ A = 3x = 3 \times 1500 = 4500 \]
\[ B = 2x = 2 \times 1500 = 3000 \]

8. **Calculate the Simple Interest for Each Part**:
Now, we can calculate the simple interest for each part for one year.

For the first part \( A \):
\[ \text{SI}_A = \frac{12 \times 4500}{100} = 540 \]

For the second part \( B \):
\[ \text{SI}_B = \frac{18 \times 3000}{100} = 540 \]

### Final Answer:
The interest on each part for one year is:
- Interest on first part (Rs. 4500) = Rs. 540
- Interest on second part (Rs. 3000) = Rs. 540