A sum of Rs 1300 is lent into 2 parts so that interest on the first part for 3 Years at 5 % may be equal to the interest on the second part for 4 Years at 6 %. First part is equal to

Options

A. a. Rs. 800
B. b. Rs. 1200
C. c. Rs. 1600
D. d. Rs. 2000

Answer

To solve the problem, we need to find the first part of the sum of Rs 1300 that is lent out. Let's denote the first part as \( P_1 \) and the second part as \( P_2 \).

Given:
- Total amount = Rs 1300
- Interest rate for the first part = 5% for 3 years
- Interest rate for the second part = 6% for 4 years
- The interest earned on both parts is equal.

### Step 1: Set up the equations for the interest
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

For the first part:
\[ SI_1 = \frac{P_1 \times 5 \times 3}{100} = \frac{15P_1}{100} = 0.15P_1 \]

For the second part:
\[ SI_2 = \frac{P_2 \times 6 \times 4}{100} = \frac{24P_2}{100} = 0.24P_2 \]

### Step 2: Set the interests equal to each other
Since the interest on both parts is equal:
\[ 0.15P_1 = 0.24P_2 \]

### Step 3: Express \( P_2 \) in terms of \( P_1 \)
We know that:
\[ P_2 = 1300 - P_1 \]
Substituting this into the interest equation gives:
\[ 0.15P_1 = 0.24(1300 - P_1) \]

### Step 4: Solve for \( P_1 \)
Expanding the equation:
\[ 0.15P_1 = 0.24 \times 1300 - 0.24P_1 \]
\[ 0.15P_1 + 0.24P_1 = 0.24 \times 1300 \]
\[ 0.39P_1 = 0.24 \times 1300 \]
Calculating \( 0.24 \times 1300 \):
\[ 0.24 \times 1300 = 312 \]
So we have:
\[ 0.39P_1 = 312 \]
Now, divide both sides by 0.39:
\[ P_1 = \frac{312}{0.39} \approx 800 \]

### Step 5: Conclusion
Thus, the first part \( P_1 \) is approximately Rs 800.