A person invested a sum of Rs. 25,000 partly at 10% per annum simple interset and the rest at 12% per annum simple interset At the end of 2 years, the total interset received was Rs 5,640. The difference (in Rs) between the two parts of the sum is :

Options

A. 10000
B. 7000
C. 8000
D. 9000

Answer

To solve the problem, we need to find the difference between the two parts of the sum invested at different interest rates. Let's break it down step by step.

### Step 1: Define the variables
Let:
- \( x \) = amount invested at 10% per annum
- \( 25000 - x \) = amount invested at 12% per annum

### Step 2: Calculate the interest for each part
The formula for simple interest is:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

For the amount invested at 10%:
\[ \text{Interest}_1 = \frac{x \times 10 \times 2}{100} = \frac{20x}{100} = \frac{x}{5} \]

For the amount invested at 12%:
\[ \text{Interest}_2 = \frac{(25000 - x) \times 12 \times 2}{100} = \frac{24(25000 - x)}{100} = \frac{600000 - 24x}{100} = 6000 - \frac{24x}{100} = 6000 - \frac{6x}{25} \]

### Step 3: Set up the equation
According to the problem, the total interest received after 2 years is Rs. 5640. Therefore, we can set up the equation:
\[ \frac{x}{5} + \left(6000 - \frac{6x}{25}\right) = 5640 \]

### Step 4: Simplify the equation
To eliminate the fractions, we can multiply the entire equation by 100 (the least common multiple of the denominators):
\[ 100 \left(\frac{x}{5}\right) + 100 \left(6000 - \frac{6x}{25}\right) = 100 \times 5640 \]
This simplifies to:
\[ 20x + 600000 - 24x = 564000 \]

### Step 5: Combine like terms
Combine the terms involving \( x \):
\[ -4x + 600000 = 564000 \]

### Step 6: Solve for \( x \)
Subtract 600000 from both sides:
\[ -4x = 564000 - 600000 \]
\[ -4x = -36000 \]
Now divide by -4:
\[ x = 9000 \]

### Step 7: Find the second part
The second part of the investment is:
\[ 25000 - x = 25000 - 9000 = 16000 \]

### Step 8: Calculate the difference
The difference between the two parts is:
\[ |9000 - 16000| = 7000 \]

### Final Answer
The difference between the two parts of the sum is Rs. 7000.

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