A man possessing Rs. 6,800 lent a part of it at 10% simple interest and the remaining at 7.5% simple interest. His total income after `3(1)/(2)` years was Rs. 1,904. Find the sum lent at 10% rate.

Options

A. Rs.1,260
B. Rs.1,360
C. Rs.1,700
D. Rs.1,800

Answer

To solve the problem, we need to break it down step by step.

### Step 1: Define Variables
Let:
- \( x \) = amount lent at 10% interest
- \( 6800 - x \) = amount lent at 7.5% interest

### Step 2: Calculate the Time Period
The time period given is \( 3 \frac{1}{2} \) years, which can be converted to an improper fraction:
\[ 3 \frac{1}{2} = \frac{7}{2} \text{ years} \]

### Step 3: Write the Interest Formulas
The formula for simple interest is given by:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
For the amount lent at 10%:
\[ \text{Interest from } x = x \times \frac{10}{100} \times \frac{7}{2} = \frac{7x}{20} \]
For the amount lent at 7.5%:
\[ \text{Interest from } (6800 - x) = (6800 - x) \times \frac{7.5}{100} \times \frac{7}{2} = \frac{7.5(6800 - x)}{200} \]

### Step 4: Set Up the Total Interest Equation
The total interest earned from both amounts is given as Rs. 1,904. Therefore, we can set up the equation:
\[ \frac{7x}{20} + \frac{7.5(6800 - x)}{200} = 1904 \]

### Step 5: Simplify the Equation
To eliminate the fractions, we can multiply the entire equation by 200:
\[ 200 \times \left(\frac{7x}{20}\right) + 200 \times \left(\frac{7.5(6800 - x)}{200}\right) = 200 \times 1904 \]
This simplifies to:
\[ 70x + 7.5(6800 - x) = 380800 \]

### Step 6: Distribute and Combine Like Terms
Distributing \( 7.5 \):
\[ 70x + 51000 - 7.5x = 380800 \]
Combine like terms:
\[ (70 - 7.5)x + 51000 = 380800 \]
\[ 62.5x + 51000 = 380800 \]

### Step 7: Isolate \( x \)
Subtract 51,000 from both sides:
\[ 62.5x = 380800 - 51000 \]
\[ 62.5x = 329800 \]
Now, divide by 62.5:
\[ x = \frac{329800}{62.5} = 5286.4 \]

### Step 8: Conclusion
The amount lent at 10% interest is approximately Rs. 5,286.40.