A company has a job to prepare certain number cans and there are three machines A, B and C for this job. A can complete the job in 3 days, B can complete the job in 4 days, and C can complete the job in 6 days. How many days will the company take to complete the job if all the machines are used simultaneously?

Options

A. 4/3 days
B. 3 days
C. 3 days
D. 12 days

Answer

To solve the problem of how many days the company will take to complete the job if all machines A, B, and C are used simultaneously, we can follow these steps:

### Step 1: Determine the work done by each machine in one day.
- Machine A can complete the job in 3 days. Therefore, the work done by A in one day is:
\[ \text{Work by A in one day} = \frac{1}{3} \text{ of the job} \]

- Machine B can complete the job in 4 days. Therefore, the work done by B in one day is:
\[ \text{Work by B in one day} = \frac{1}{4} \text{ of the job} \]

- Machine C can complete the job in 6 days. Therefore, the work done by C in one day is:
\[ \text{Work by C in one day} = \frac{1}{6} \text{ of the job} \]

### Step 2: Calculate the total work done by all machines in one day.
To find the total work done by all machines in one day, we add the work done by each machine:
\[ \text{Total work in one day} = \text{Work by A} + \text{Work by B} + \text{Work by C} \]
\[ = \frac{1}{3} + \frac{1}{4} + \frac{1}{6} \]

### Step 3: Find a common denominator and add the fractions.
The least common multiple (LCM) of 3, 4, and 6 is 12. We convert each fraction:
\[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12} \]
Now, add them together:
\[ \text{Total work in one day} = \frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{4 + 3 + 2}{12} = \frac{9}{12} = \frac{3}{4} \text{ of the job} \]

### Step 4: Calculate the time taken to complete the job.
If the machines together complete \(\frac{3}{4}\) of the job in one day, then the time taken to complete the entire job (1 job) is:
\[ \text{Time} = \frac{1 \text{ job}}{\frac{3}{4} \text{ job/day}} = \frac{4}{3} \text{ days} \]

### Final Answer:
The company will take \(\frac{4}{3}\) days to complete the job if all machines are used simultaneously.

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