Step 1: Understanding the Question:
This is a problem based on proportional wages. The servant's pay is a combination of cash and a physical item (a cycle). We need to determine the exact monetary value of the cycle based on the pay received for a shorter duration of work (9 months instead of 12).
Step 2: Key Formula or Approach:
Calculate the monthly wage equation based on the 12-month promise. Then, equate the expected 9-month proportional pay to the actual 9-month pay received.
Step 3: Detailed Explanation:
Let the monetary price of the cycle be Rs. $C$.
The total promised compensation for 1 year (12 months) of work is Rs. 15,000 plus the cycle.
Therefore, the total value of 12 months' work = $15000 + C$.
Assuming a uniform wage rate, the servant's true wage for 1 month of work is $\frac{15000 + C}{12}$.
The servant worked for exactly 9 months. So, the compensation he rightly earned is 9 times the monthly wage:
Earned compensation for 9 months = $9 \times \left( \frac{15000 + C}{12} \right) = \frac{3}{4} \times (15000 + C)$.
The problem states that for these 9 months, he received Rs. 11,000 and the cycle. So, the value he received is $11000 + C$.
Equating the mathematically earned compensation to the value he actually received:
\[ \frac{3}{4} \times (15000 + C) = 11000 + C \]
Multiply the entire equation by 4 to clear the fraction:
\[ 3 \times (15000 + C) = 4 \times (11000 + C) \]
Expand both sides:
\[ 45000 + 3C = 44000 + 4C \]
Rearranging the terms to solve for $C$:
\[ 45000 - 44000 = 4C - 3C \]
\[ C = 1000 \]
The price of the cycle is Rs. 1000.
Step 4: Final Answer:
The price of the cycle is Rs. 1000.