A can do a piece of work in 8 days, B can do it in 16 days, while C can do it in 80 days. In how many days they can complete the whole work, working together?

Options

A. 5
B. 6
C. `8(2)/(3)`
D. `20(2)/(5)`

Answer

To solve the problem step by step, we follow these steps:

### Step 1: Determine the work done by each person in one day.
- **A can complete the work in 8 days**, so the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{8} \]

- **B can complete the work in 16 days**, so the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{16} \]

- **C can complete the work in 80 days**, so the work done by C in one day is:
\[ \text{Work done by C in one day} = \frac{1}{80} \]

### Step 2: Calculate the total work done by A, B, and C in one day.
- Now, we add the work done by A, B, and C together:
\[ \text{Total work done in one day} = \frac{1}{8} + \frac{1}{16} + \frac{1}{80} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple (LCM) of 8, 16, and 80 is 80. We convert each fraction:
\[ \frac{1}{8} = \frac{10}{80}, \quad \frac{1}{16} = \frac{5}{80}, \quad \frac{1}{80} = \frac{1}{80} \]
- Now, we can add them:
\[ \text{Total work done in one day} = \frac{10}{80} + \frac{5}{80} + \frac{1}{80} = \frac{16}{80} \]

### Step 4: Simplify the total work done in one day.
- Simplifying \(\frac{16}{80}\):
\[ \frac{16}{80} = \frac{1}{5} \]

### Step 5: Calculate the total time taken to complete the work together.
- Since they complete \(\frac{1}{5}\) of the work in one day, the total time taken to complete the whole work (1 unit of work) is the reciprocal of their combined work rate:
\[ \text{Total time} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer:
- Therefore, A, B, and C together can complete the work in **5 days**.

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