A, B and C can complete a piece of worik in 4, 28 and 56 days respectively. Working together, they can complete the same work in how many days?

Options

A. `5 (5)/(17)`
B. `3 1/(17)`
C. `5 1/(17)`
D. `3 5/(17)`

Answer

To solve the problem of how many days A, B, and C can complete a piece of work together, we will follow these steps:

### Step 1: Determine the work done by each person in one day.
- A can complete the work in 4 days, so in one day, A completes \( \frac{1}{4} \) of the work.
- B can complete the work in 28 days, so in one day, B completes \( \frac{1}{28} \) of the work.
- C can complete the work in 56 days, so in one day, C completes \( \frac{1}{56} \) of the work.

### Step 2: Calculate the total work done by A, B, and C in one day.
To find the total work done by A, B, and C together in one day, we add their individual contributions:
\[ \text{Total work in one day} = \frac{1}{4} + \frac{1}{28} + \frac{1}{56} \]

### Step 3: Find a common denominator.
The least common multiple (LCM) of 4, 28, and 56 is 56. We will convert each fraction to have a denominator of 56:
- \( \frac{1}{4} = \frac{14}{56} \)
- \( \frac{1}{28} = \frac{2}{56} \)
- \( \frac{1}{56} = \frac{1}{56} \)

### Step 4: Add the fractions.
Now we can add the fractions:
\[ \text{Total work in one day} = \frac{14}{56} + \frac{2}{56} + \frac{1}{56} = \frac{14 + 2 + 1}{56} = \frac{17}{56} \]

### Step 5: Calculate the total time taken to complete the work.
If A, B, and C together complete \( \frac{17}{56} \) of the work in one day, we need to find out how many days it will take to complete the whole work (1 unit of work):
\[ \text{Time taken} = \frac{1 \text{ work}}{\frac{17}{56} \text{ work/day}} = \frac{56}{17} \text{ days} \]

### Conclusion:
Thus, A, B, and C together can complete the work in \( \frac{56}{17} \) days, which is approximately 3.29 days.

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