A sum of Rs. 2540 is lent out into two parts. One at `12%` and another one at `12 1/ 2 %`. If the total annual income is Rs. 311.60, the money lent at 12% is-

Options

A. Rs. 1180
B. Rs. 1360
C. Rs. 1240
D. Rs. 1340

Answer

To solve the problem step by step, we need to find out how much money is lent at 12% interest when a total of Rs. 2540 is divided into two parts, one part at 12% and the other at 12.5%, with a total annual income of Rs. 311.60.

### Step 1: Define Variables
Let:
- \( x \) = the amount lent at 12%
- \( 2540 - x \) = the amount lent at 12.5%

### Step 2: Write the Interest Formulas
The interest earned from each part can be calculated using the formula for simple interest:
- Interest from the amount lent at 12%:
\[ \text{Interest}_1 = \frac{x \times 12 \times 1}{100} = \frac{12x}{100} = 0.12x \]
- Interest from the amount lent at 12.5%:
\[ \text{Interest}_2 = \frac{(2540 - x) \times 12.5 \times 1}{100} = \frac{12.5(2540 - x)}{100} = 0.125(2540 - x) \]

### Step 3: Set Up the Equation
According to the problem, the total interest earned from both parts is Rs. 311.60:
\[ 0.12x + 0.125(2540 - x) = 311.60 \]

### Step 4: Simplify the Equation
Expanding the equation:
\[ 0.12x + 0.125 \times 2540 - 0.125x = 311.60 \]
Calculating \( 0.125 \times 2540 \):
\[ 0.125 \times 2540 = 317.5 \]
So, the equation becomes:
\[ 0.12x - 0.125x + 317.5 = 311.60 \]
Combining like terms:
\[ -0.005x + 317.5 = 311.60 \]

### Step 5: Isolate \( x \)
Subtract 317.5 from both sides:
\[ -0.005x = 311.60 - 317.5 \]
Calculating the right side:
\[ 311.60 - 317.5 = -5.90 \]
So, we have:
\[ -0.005x = -5.90 \]
Dividing both sides by -0.005:
\[ x = \frac{-5.90}{-0.005} = 1180 \]

### Step 6: Conclusion
The amount lent at 12% is Rs. 1180.