Some amount was lent at 8% per annum simple interest. After one year, Rs 10,944 is withdraw and the remaining of the amount is repaid at 6% per annum at the end of second year. If the ratio of the first year interest to that of the second year is 28:9 then find the amount that was lent out initially.

Options

A. Rs 17400
B. Rs 16800
C. Rs 16400
D. Rs 16000

Answer

To solve the problem step by step, we will break down the information given and apply the formula for simple interest.

### Step 1: Define the variables
Let the principal amount (the amount lent initially) be \( P \).

### Step 2: Calculate the interest for the first year
The interest for the first year at 8% per annum is given by:
\[ \text{Interest}_1 = P \times \frac{8}{100} = 0.08P \]

### Step 3: Calculate the total amount after the first year
The total amount after the first year will be:
\[ \text{Total Amount}_1 = P + \text{Interest}_1 = P + 0.08P = 1.08P \]

### Step 4: Withdraw Rs 10,944 after the first year
After one year, Rs 10,944 is withdrawn. Therefore, the remaining amount after withdrawal is:
\[ \text{Remaining Amount} = 1.08P - 10,944 \]

### Step 5: Calculate the interest for the second year
The interest for the second year at 6% per annum on the remaining amount is:
\[ \text{Interest}_2 = \left(1.08P - 10,944\right) \times \frac{6}{100} = 0.06(1.08P - 10,944) \]

### Step 6: Set up the ratio of the first year interest to the second year interest
According to the problem, the ratio of the first year interest to the second year interest is given as:
\[ \frac{\text{Interest}_1}{\text{Interest}_2} = \frac{28}{9} \]
Substituting the values we have:
\[ \frac{0.08P}{0.06(1.08P - 10,944)} = \frac{28}{9} \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 9 \times 0.08P = 28 \times 0.06(1.08P - 10,944) \]
This simplifies to:
\[ 0.72P = 1.68(1.08P - 10,944) \]

### Step 8: Distribute on the right side
Expanding the right side:
\[ 0.72P = 1.8P - 18,353.92 \]

### Step 9: Rearrange the equation to isolate \( P \)
Rearranging gives:
\[ 0.72P - 1.8P = -18,353.92 \]
\[ -1.08P = -18,353.92 \]
Dividing both sides by -1.08:
\[ P = \frac{18,353.92}{1.08} \approx 16,800 \]

### Conclusion
The amount that was lent out initially is:
\[ \boxed{16,800} \]