To find out how long C would take to complete the work alone, we can follow these steps:
### Step 1: Determine the total work in units
We know that:
- A can complete the work in 5 days.
- B can complete the work in 4 days.
- A, B, and C together can complete the work in 2 days.
To find a common unit of work, we can use the Least Common Multiple (LCM) of the days taken by A, B, and A+B+C. The LCM of 5, 4, and 2 is 20. Therefore, we can consider the total work to be 20 units.
### Step 2: Calculate the work done by A in one day
A completes the work in 5 days, so the work done by A in one day is:
\[
\text{Work done by A in one day} = \frac{20 \text{ units}}{5 \text{ days}} = 4 \text{ units/day}
\]
### Step 3: Calculate the work done by B in one day
B completes the work in 4 days, so the work done by B in one day is:
\[
\text{Work done by B in one day} = \frac{20 \text{ units}}{4 \text{ days}} = 5 \text{ units/day}
\]
### Step 4: Calculate the work done by A, B, and C together in one day
A, B, and C together complete the work in 2 days, so the work done by A, B, and C together in one day is:
\[
\text{Work done by A, B, and C in one day} = \frac{20 \text{ units}}{2 \text{ days}} = 10 \text{ units/day}
\]
### Step 5: Calculate the work done by C in one day
To find C's efficiency, we subtract the work done by A and B from the total work done by A, B, and C:
\[
\text{Work done by C in one day} = \text{Total work by A, B, and C} - (\text{Work done by A} + \text{Work done by B})
\]
\[
= 10 \text{ units/day} - (4 \text{ units/day} + 5 \text{ units/day}) = 10 - 9 = 1 \text{ unit/day}
\]
### Step 6: Calculate the time taken by C to complete the work alone
Now that we know C can do 1 unit of work in one day, we can find out how many days it would take C to complete the total work of 20 units:
\[
\text{Time taken by C to complete the work} = \frac{20 \text{ units}}{1 \text{ unit/day}} = 20 \text{ days}
\]
### Final Answer
C alone can complete the work in **20 days**.
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