Rs 1900 was divided into two partsand first part was lent out at 8% per annum for 2 years and the second at 4% per annum for 3 years. If the simple interest received in both the conditions is in the ratio 16:7, then find the amount lent out at 4%?

Options

A. Rs 900
B. Rs 1000
C. Rs 1200
D. Rs 700

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI), which is:

\[ \text{SI} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount (the initial amount of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Define the variables
Let:
- \( x \) = the amount lent out at 8% per annum
- \( y \) = the amount lent out at 4% per annum

From the problem, we know:
\[ x + y = 1900 \]

### Step 2: Express the simple interest for both parts
The simple interest for the first part (lent at 8% for 2 years):
\[ \text{SI}_1 = \frac{x \times 8 \times 2}{100} = \frac{16x}{100} = 0.16x \]

The simple interest for the second part (lent at 4% for 3 years):
\[ \text{SI}_2 = \frac{y \times 4 \times 3}{100} = \frac{12y}{100} = 0.12y \]

### Step 3: Set up the ratio of the interests
According to the problem, the ratio of the simple interests is given as:
\[ \frac{\text{SI}_1}{\text{SI}_2} = \frac{16}{7} \]

Substituting the expressions for SI:
\[ \frac{0.16x}{0.12y} = \frac{16}{7} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 0.16x \cdot 7 = 0.12y \cdot 16 \]
\[ 1.12x = 1.92y \]

### Step 5: Express \( x \) in terms of \( y \)
From the equation \( 1.12x = 1.92y \), we can express \( x \) as:
\[ x = \frac{1.92y}{1.12} \]

### Step 6: Substitute \( x \) in the total amount equation
Now substitute \( x \) in the equation \( x + y = 1900 \):
\[ \frac{1.92y}{1.12} + y = 1900 \]

### Step 7: Solve for \( y \)
To solve for \( y \), first find a common denominator:
\[ \frac{1.92y + 1.12y}{1.12} = 1900 \]
\[ \frac{3.04y}{1.12} = 1900 \]

Now multiply both sides by \( 1.12 \):
\[ 3.04y = 1900 \times 1.12 \]
\[ 3.04y = 2128 \]

Now divide by \( 3.04 \):
\[ y = \frac{2128}{3.04} \]
\[ y = 700 \]

### Step 8: Find the amount lent out at 4%
Thus, the amount lent out at 4% is:
\[ y = 700 \]

### Final Answer
The amount lent out at 4% is **Rs. 700**.

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