A man engaged a servant on the condition that he would pay him Rs. 90 and a turban after a service of one year. He served only for nine months and received the turban and Rs. 65. The price of the turban is :  

Options

A. Rs. 25
B. Rs. 18975
C. Rs. 10
D. Rs. 2.50

Answer

To solve the problem, we need to find the price of the turban (let's denote it as \( X \)). The man promised to pay the servant Rs. 90 and a turban after one year of service. However, the servant worked for only 9 months and received Rs. 65 and the turban.

### Step-by-step Solution:

1. **Total Payment for One Year**:
The total payment for one year is Rs. 90 + price of the turban \( X \).
\[ \text{Total payment for 1 year} = 90 + X \]

2. **Monthly Payment Calculation**:
Since the payment is for one year (12 months), we can find the monthly payment by dividing the total payment by 12.
\[ \text{Monthly payment} = \frac{90 + X}{12} \]

3. **Payment for 9 Months**:
The servant worked for 9 months, so the payment he should receive for 9 months is:
\[ \text{Payment for 9 months} = 9 \times \text{Monthly payment} = 9 \times \frac{90 + X}{12} \]
Simplifying this gives:
\[ \text{Payment for 9 months} = \frac{9(90 + X)}{12} = \frac{3(90 + X)}{4} \]

4. **Total Amount Received by the Servant**:
The servant received Rs. 65 and the turban. Therefore, we can express the total amount he received as:
\[ \text{Total amount received} = 65 + X \]

5. **Setting up the Equation**:
Since the amount he should receive for 9 months is equal to the amount he actually received, we can set up the equation:
\[ \frac{3(90 + X)}{4} = 65 + X \]

6. **Clearing the Fraction**:
To eliminate the fraction, multiply both sides by 4:
\[ 3(90 + X) = 4(65 + X) \]

7. **Expanding Both Sides**:
Expanding both sides gives:
\[ 270 + 3X = 260 + 4X \]

8. **Rearranging the Equation**:
To isolate \( X \), we can rearrange the equation:
\[ 270 - 260 = 4X - 3X \]
Simplifying this results in:
\[ 10 = X \]

9. **Conclusion**:
Thus, the price of the turban is Rs. 10.

### Final Answer:
The price of the turban is Rs. 10.