If A can complete work in 6 days, B can complete the work in 8 days and C can complete the same work in 12 days. In how much time A, B, and C can complete the work.

Options

A. 3/4 days
B. 8/3 days
C. 5/2 days
D. 3/8 days

Answer

To solve the problem of how long A, B, and C can complete the work together, we can follow these steps:

### Step 1: Determine the work rates of A, B, and C.
- A can complete the work in 6 days, so A's work rate is \( \frac{1}{6} \) of the work per day.
- B can complete the work in 8 days, so B's work rate is \( \frac{1}{8} \) of the work per day.
- C can complete the work in 12 days, so C's work rate is \( \frac{1}{12} \) of the work per day.

### Step 2: Calculate the combined work rate of A, B, and C.
To find the combined work rate, we add the individual work rates together:
\[ \text{Combined work rate} = \frac{1}{6} + \frac{1}{8} + \frac{1}{12} \]

### Step 3: Find a common denominator.
The least common multiple (LCM) of 6, 8, and 12 is 24. We will convert each fraction to have a denominator of 24:
- \( \frac{1}{6} = \frac{4}{24} \)
- \( \frac{1}{8} = \frac{3}{24} \)
- \( \frac{1}{12} = \frac{2}{24} \)

### Step 4: Add the fractions.
Now, we can add the fractions:
\[ \frac{4}{24} + \frac{3}{24} + \frac{2}{24} = \frac{4 + 3 + 2}{24} = \frac{9}{24} \]

### Step 5: Simplify the combined work rate.
The combined work rate \( \frac{9}{24} \) can be simplified:
\[ \frac{9}{24} = \frac{3}{8} \]

### Step 6: Calculate the time taken to complete the work together.
To find the time taken to complete the work together, we take the reciprocal of the combined work rate:
\[ \text{Time} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \text{ days} \]

### Final Answer:
A, B, and C together can complete the work in \( \frac{8}{3} \) days, which is approximately 2.67 days or 2 days and 8 hours.

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