A sum of Rs. 1000 is lent partly at 10% and 15% at simple interest per annum. If total interest received after 3 years is Rs. 390. Find the amount lent at 10% rate.

Options

A. A)100
B. B)900
C. C)600
D. D)400

Answer

To solve the problem step by step, we need to find the amount lent at the 10% interest rate from a total sum of Rs. 1000, which is lent partly at 10% and partly at 15%. The total interest received after 3 years is Rs. 390.

### Step 1: Define Variables
Let:
- \( x \) = amount lent at 10%
- \( y \) = amount lent at 15%

From the problem, we know:
\[ x + y = 1000 \] (Equation 1)

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

For the amount lent at 10%:
\[ \text{Interest at 10%} = \frac{x \times 10 \times 3}{100} = \frac{30x}{100} = 0.3x \]

For the amount lent at 15%:
\[ \text{Interest at 15%} = \frac{y \times 15 \times 3}{100} = \frac{45y}{100} = 0.45y \]

### Step 3: Set Up the Total Interest Equation
According to the problem, the total interest received is Rs. 390:
\[ 0.3x + 0.45y = 390 \] (Equation 2)

### Step 4: Substitute \( y \) from Equation 1 into Equation 2
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 1000 - x \]

Substituting \( y \) in Equation 2:
\[ 0.3x + 0.45(1000 - x) = 390 \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ 0.3x + 450 - 0.45x = 390 \]

Combining like terms:
\[ -0.15x + 450 = 390 \]

### Step 6: Solve for \( x \)
Subtract 450 from both sides:
\[ -0.15x = 390 - 450 \]
\[ -0.15x = -60 \]

Dividing both sides by -0.15:
\[ x = \frac{-60}{-0.15} = 400 \]

### Step 7: Conclusion
The amount lent at the 10% rate is Rs. 400.