Question

A company has a job to prepare certain number of 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?

• A. 4 days
• B. \(\frac{4}{3}\) days
• C. 3 days
• D. 12 days

Hint:

When multiple entities work together, their combined rate of work is the sum of their individual rates. The total time taken is the reciprocal of the combined rate.

Answer 1

The Correct Option is B

Let the total work be represented by \( W \). The rates of work for each machine are:
- A can complete the work in 3 days, so its rate of work is \( \frac{W}{3} \).
- B can complete the work in 4 days, so its rate of work is \( \frac{W}{4} \).
- C can complete the work in 6 days, so its rate of work is \( \frac{W}{6} \).
When all three machines work together, their combined rate of work is the sum of their individual rates:
\[ \text{Combined rate of work} = \frac{W}{3} + \frac{W}{4} + \frac{W}{6}. \] To simplify this, we first find the least common denominator of 3, 4, and 6, which is 12:
\[ \frac{W}{3} = \frac{4W}{12}, \quad \frac{W}{4} = \frac{3W}{12}, \quad \frac{W}{6} = \frac{2W}{12}. \] Thus, the combined rate of work is:
\[ \frac{4W}{12} + \frac{3W}{12} + \frac{2W}{12} = \frac{9W}{12} = \frac{3W}{4}. \] This means the machines together complete \(\frac{3W}{4}\) of the work in one day.
Therefore, the time taken to complete the entire work is the reciprocal of the combined rate:
\[ \text{Time taken} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \text{ days}. \] Thus, the correct answer is \(\frac{4}{3}\) days.