To solve the problem step by step, we will first determine the work done by A, B, and C in one day and then calculate the total work completed when they work together, considering the days they leave early.
### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 36 days, so the work done by A in one day is:
\[
\text{Work done by A in one day} = \frac{1}{36}
\]
- B can complete the work in 54 days, so the work done by B in one day is:
\[
\text{Work done by B in one day} = \frac{1}{54}
\]
- C can complete the work in 72 days, so the work done by C in one day is:
\[
\text{Work done by C in one day} = \frac{1}{72}
\]
### Step 2: Find the total work done by A, B, and C together in one day.
To find the total work done by A, B, and C in one day, we add their individual work rates:
\[
\text{Total work done in one day} = \frac{1}{36} + \frac{1}{54} + \frac{1}{72}
\]
To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 36, 54, and 72 is 216.
Now, we convert each fraction:
- \(\frac{1}{36} = \frac{6}{216}\)
- \(\frac{1}{54} = \frac{4}{216}\)
- \(\frac{1}{72} = \frac{3}{216}\)
Thus,
\[
\text{Total work done in one day} = \frac{6 + 4 + 3}{216} = \frac{13}{216}
\]
### Step 3: Set up the equation based on the work done.
Let \(X\) be the total number of days taken to complete the work.
- A works for \(X - 8\) days (leaves 8 days before completion).
- B works for \(X - 12\) days (leaves 12 days before completion).
- C works for \(X\) days.
The work done by each person can be expressed as:
- Work done by A: \(\frac{6}{216} \times (X - 8)\)
- Work done by B: \(\frac{4}{216} \times (X - 12)\)
- Work done by C: \(\frac{3}{216} \times X\)
### Step 4: Write the total work equation.
The total work done by A, B, and C together must equal 1 (the whole work):
\[
\frac{6}{216}(X - 8) + \frac{4}{216}(X - 12) + \frac{3}{216}X = 1
\]
### Step 5: Simplify the equation.
Multiplying through by 216 to eliminate the denominator:
\[
6(X - 8) + 4(X - 12) + 3X = 216
\]
Expanding the equation:
\[
6X - 48 + 4X - 48 + 3X = 216
\]
Combining like terms:
\[
(6X + 4X + 3X) - 96 = 216
\]
\[
13X - 96 = 216
\]
### Step 6: Solve for \(X\).
Adding 96 to both sides:
\[
13X = 216 + 96
\]
\[
13X = 312
\]
Dividing by 13:
\[
X = \frac{312}{13} = 24
\]
### Conclusion
The total number of days taken to complete the work is **24 days**.
