2100 is divided into two parts such that the simple interest on the one part at 4.5% for 3.5 years be the same as that on the other at 5.25% for 4 years. Find out the second part ?

Options

A. Rs. 800
B. Rs. 1400
C. Rs. 1200
D. None of these

Answer

To solve the problem, we need to find the second part of the amount when ₹2100 is divided into two parts such that the simple interest on the first part at 4.5% for 3.5 years is equal to the simple interest on the second part at 5.25% for 4 years.

### Step-by-Step Solution:

1. **Define the Parts**:
Let the first part be \( P_1 \) and the second part be \( P_2 \).
We know that:
\[ P_1 + P_2 = 2100 \]

2. **Set Up the Simple Interest Formula**:
The formula for simple interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

3. **Calculate Simple Interest for Both Parts**:
For the first part \( P_1 \):
\[ SI_1 = \frac{P_1 \times 4.5 \times 3.5}{100} \]
For the second part \( P_2 \):
\[ SI_2 = \frac{P_2 \times 5.25 \times 4}{100} \]

4. **Set the Simple Interests Equal**:
According to the problem, the simple interests are equal:
\[ SI_1 = SI_2 \]
Substituting the expressions for \( SI_1 \) and \( SI_2 \):
\[ \frac{P_1 \times 4.5 \times 3.5}{100} = \frac{P_2 \times 5.25 \times 4}{100} \]
We can eliminate the denominator (100) from both sides:
\[ P_1 \times 4.5 \times 3.5 = P_2 \times 5.25 \times 4 \]

5. **Express \( P_2 \) in Terms of \( P_1 \)**:
From the first equation \( P_2 = 2100 - P_1 \). Substitute this into the interest equation:
\[ P_1 \times 4.5 \times 3.5 = (2100 - P_1) \times 5.25 \times 4 \]

6. **Expand and Rearrange**:
Expanding the right-hand side:
\[ P_1 \times 4.5 \times 3.5 = 2100 \times 5.25 \times 4 - P_1 \times 5.25 \times 4 \]
Combine like terms:
\[ P_1 \times (4.5 \times 3.5 + 5.25 \times 4) = 2100 \times 5.25 \times 4 \]

7. **Calculate the Coefficients**:
Calculate \( 4.5 \times 3.5 \) and \( 5.25 \times 4 \):
\[ 4.5 \times 3.5 = 15.75 \]
\[ 5.25 \times 4 = 21 \]
Therefore:
\[ P_1 \times (15.75 + 21) = 2100 \times 21 \]
\[ P_1 \times 36.75 = 2100 \times 21 \]

8. **Solve for \( P_1 \)**:
\[ P_1 = \frac{2100 \times 21}{36.75} \]
Calculate \( P_1 \):
\[ P_1 = \frac{44100}{36.75} = 1200 \]

9. **Find \( P_2 \)**:
Now substitute back to find \( P_2 \):
\[ P_2 = 2100 - P_1 = 2100 - 1200 = 900 \]

### Final Answer:
The second part \( P_2 \) is ₹900.