P,Q and R can do a piece of work in 18 days, 36 days and 54 day respectively. They start the work together but Q and R leave 1 day and 5 days respectively, before the completion of work. In how many days has the work been completed ?

Options

A. 10
B. 11
C. `10(7)/(11)`
D. `10(9)/(11)`

Explanation

To solve the problem step by step, we will calculate the efficiency of each worker and then determine how long they worked together before Q and R left.

### Step 1: Calculate the efficiency of P, Q, and R
- P can complete the work in 18 days, so his efficiency is:
\[ \text{Efficiency of P} = \frac{1}{18} \text{ of the work per day} \]
- Q can complete the work in 36 days, so his efficiency is:
\[ \text{Efficiency of Q} = \frac{1}{36} \text{ of the work per day} \]
- R can complete the work in 54 days, so his efficiency is:
\[ \text{Efficiency of R} = \frac{1}{54} \text{ of the work per day} \]

### Step 2: Find a common total work unit
To make calculations easier, we can assume the total work is 108 units (the least common multiple of 18, 36, and 54).

### Step 3: Calculate the daily work done by each worker
- P's daily work:
\[ \text{Work done by P in one day} = \frac{108}{18} = 6 \text{ units} \]
- Q's daily work:
\[ \text{Work done by Q in one day} = \frac{108}{36} = 3 \text{ units} \]
- R's daily work:
\[ \text{Work done by R in one day} = \frac{108}{54} = 2 \text{ units} \]

### Step 4: Set up the equation for total work done
Let \( x \) be the total number of days they worked together.
- P works for \( x \) days.
- Q works for \( x - 1 \) days (leaves 1 day before completion).
- R works for \( x - 5 \) days (leaves 5 days before completion).

The total work done can be expressed as:
\[ \text{Total Work} = \text{Work by P} + \text{Work by Q} + \text{Work by R} \]
\[ 6x + 3(x - 1) + 2(x - 5) = 108 \]

### Step 5: Simplify the equation
Expanding the equation:
\[ 6x + 3x - 3 + 2x - 10 = 108 \]
Combining like terms:
\[ (6x + 3x + 2x) - 13 = 108 \]
\[ 11x - 13 = 108 \]

### Step 6: Solve for \( x \)
Adding 13 to both sides:
\[ 11x = 121 \]
Dividing by 11:
\[ x = \frac{121}{11} = 11 \]

### Conclusion
The work has been completed in **11 days**.

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