The work has been completed in 108/11 days, which is approximately 9.82 days.
To determine the number of days it takes to complete the work, we can calculate the combined efficiency of P, Q, and R when they work together.
First, let's find the individual efficiencies of P, Q, and R by dividing the total work by the number of days each person takes to complete the work.
- P completes the work in 18 days, so their efficiency is 1/18 of the total work per day.
- Q completes the work in 36 days, so their efficiency is 1/36 of the total work per day.
- R completes the work in 54 days, so their efficiency is 1/54 of the total work per day.
Now, let's calculate the combined efficiency of P, Q, and R when they work together. We can add their individual efficiencies to get the combined efficiency.
Combined efficiency = 1/18 + 1/36 + 1/54
To add these fractions, we need to find the least common multiple (LCM) of the denominators, which is 108.
So, the combined efficiency is (6/108) + (3/108) + (2/108) = 11/108.
Now, let's find the total work that needs to be done. Since P takes 18 days to complete the work, the total work can be calculated as 18 times P's efficiency:
Total work = 18 * (1/18) = 1
Now, let's calculate the work done by P, Q, and R in the given number of days.
When Q and R leave the work:
- P completes 1/18 of the work in 1 day.
- Q completes 1/36 of the work in 1 day.
- R completes 1/54 of the work in 1 day.
So, in 1 day, the total work completed is (1/18) + (1/36) + (1/54) = 11/108.
To find the number of days it takes to complete the work, we can divide the total work (1) by the work completed in 1 day (11/108).
Number of days to complete the work = 1 / (11/108)
To divide by a fraction, we multiply by its reciprocal:
Number of days to complete the work = 1 * (108/11) = 108/11
Therefore, the work has been completed in 108/11 days, which is approximately 9.82 days