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 first find the work done by each person per day, then determine how long they worked together and when they left, and finally calculate the total time taken to complete the work.

### Step-by-Step Solution:

1. **Calculate the work done by P, Q, and R in one day:**
- P can complete the work in 18 days, so P's work per day = \( \frac{1}{18} \).
- Q can complete the work in 36 days, so Q's work per day = \( \frac{1}{36} \).
- R can complete the work in 54 days, so R's work per day = \( \frac{1}{54} \).

2. **Calculate the total work done by P, Q, and R in one day when they work together:**
\[ \text{Total work per day} = \frac{1}{18} + \frac{1}{36} + \frac{1}{54} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 18, 36, and 54 is 108.
\[ \frac{1}{18} = \frac{6}{108}, \quad \frac{1}{36} = \frac{3}{108}, \quad \frac{1}{54} = \frac{2}{108} \]
Therefore,
\[ \text{Total work per day} = \frac{6 + 3 + 2}{108} = \frac{11}{108} \]

3. **Let the total number of days taken to complete the work be X days.**
- P works for X days.
- Q works for \( X - 1 \) days (since Q leaves 1 day before completion).
- R works for \( X - 5 \) days (since R leaves 5 days before completion).

4. **Set up the equation for the total work done:**
\[ \text{Total work} = \text{Work done by P} + \text{Work done by Q} + \text{Work done by R} \]
\[ 1 = \left(\frac{X}{18}\right) + \left(\frac{X - 1}{36}\right) + \left(\frac{X - 5}{54}\right) \]

5. **Multiply through by 108 to eliminate the denominators:**
\[ 108 = 6X + 3(X - 1) + 2(X - 5) \]
Expanding this gives:
\[ 108 = 6X + 3X - 3 + 2X - 10 \]
Combining like terms:
\[ 108 = 11X - 13 \]

6. **Solve for X:**
\[ 11X = 108 + 13 \]
\[ 11X = 121 \]
\[ X = \frac{121}{11} = 11 \]

7. **Conclusion:**
The total number of days taken to complete the work is **11 days**.