Rs. 8000 have been divided into two parts such that if one part is lent a `7 1/2`% per annum for 2 years and the other at 8% per annum for 3 years, then the total interest received is Rs. 1416. Find the two parts.

Options

A. Rs. 5550,Rs. 2450
B. Rs. 5650, Rs. 2350
C. Rs. 5625, Rs. 2375
D. Rs. 5600, Rs. 2400

Answer

To solve the problem, we need to divide Rs. 8000 into two parts such that the interest earned from both parts equals Rs. 1416. Let's denote the two parts as follows:

Let:
- Part 1 = x (the amount lent at 7.5% per annum for 2 years)
- Part 2 = 8000 - x (the amount lent at 8% per annum for 3 years)

### Step-by-Step Solution:

1. **Calculate the Interest from Part 1:**
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
- For Part 1:
- Principal (P) = x
- Rate (R) = 7.5% = \frac{15}{2}%
- Time (T) = 2 years
- Therefore, the interest (I1) from Part 1 is:
\[ I_1 = \frac{x \times \frac{15}{2} \times 2}{100} = \frac{15x}{100} = \frac{3x}{20} \]

2. **Calculate the Interest from Part 2:**
- For Part 2:
- Principal (P) = 8000 - x
- Rate (R) = 8%
- Time (T) = 3 years
- Therefore, the interest (I2) from Part 2 is:
\[ I_2 = \frac{(8000 - x) \times 8 \times 3}{100} = \frac{24(8000 - x)}{100} = \frac{192000 - 24x}{100} \]

3. **Set Up the Equation for Total Interest:**
- According to the problem, the total interest from both parts is Rs. 1416:
\[ I_1 + I_2 = 1416 \]
- Substituting the expressions for I1 and I2:
\[ \frac{3x}{20} + \frac{192000 - 24x}{100} = 1416 \]

4. **Clear the Denominators:**
- To eliminate the fractions, multiply the entire equation by 100:
\[ 100 \left(\frac{3x}{20}\right) + (192000 - 24x) = 141600 \]
- Simplifying gives:
\[ 15x + 192000 - 24x = 141600 \]

5. **Combine Like Terms:**
- Combine the x terms:
\[ -9x + 192000 = 141600 \]

6. **Isolate x:**
- Rearranging gives:
\[ -9x = 141600 - 192000 \]
\[ -9x = -50400 \]
\[ x = \frac{50400}{9} = 5600 \]

7. **Find the Two Parts:**
- Part 1 = x = Rs. 5600
- Part 2 = 8000 - x = 8000 - 5600 = Rs. 2400

### Final Answer:
- The two parts are Rs. 5600 and Rs. 2400.