Rs. 1000 is lent out in two parts, one at 6% simple interest and the other at 8% simple interest. The yearly income is Rs. 75. The sum lent at 8% is:

Options

A. Rs. 250
B. Rs. 500
C. Rs. 750
D. Rs. 600

Answer

To solve the problem, we will use the concept of simple interest and the method of alligation. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the variables
Let:
- \( x \) = amount lent at 6% interest
- \( y \) = amount lent at 8% interest

According to the problem, the total amount lent is:
\[ x + y = 1000 \]

### Step 2: Set up the interest equations
The yearly income from the amounts lent can be calculated using the formula for simple interest:
- Interest from \( x \) at 6% = \( \frac{6}{100} \times x = 0.06x \)
- Interest from \( y \) at 8% = \( \frac{8}{100} \times y = 0.08y \)

The total interest earned in a year is given as Rs. 75:
\[ 0.06x + 0.08y = 75 \]

### Step 3: Substitute \( y \) in terms of \( x \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 1000 - x \]

### Step 4: Substitute \( y \) in the interest equation
Now substitute \( y \) in the interest equation:
\[ 0.06x + 0.08(1000 - x) = 75 \]

### Step 5: Simplify the equation
Distributing the 0.08:
\[ 0.06x + 80 - 0.08x = 75 \]
Combine like terms:
\[ -0.02x + 80 = 75 \]

### Step 6: Solve for \( x \)
Subtract 80 from both sides:
\[ -0.02x = 75 - 80 \]
\[ -0.02x = -5 \]
Now divide by -0.02:
\[ x = \frac{-5}{-0.02} = 250 \]

### Step 7: Find \( y \)
Now that we have \( x \), we can find \( y \):
\[ y = 1000 - x = 1000 - 250 = 750 \]

### Conclusion
The sum lent at 8% is \( y = 750 \) Rs.

### Final Answer
The sum lent at 8% is Rs. 750.

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