An industrial facility uses two automated assembly units, Unit X and Unit Y, to fabricate … — GMAT

An industrial facility uses two automated assembly units, Unit $X$ and Unit $Y$ , to fabricate custom components. Working alone at its constant rate, Unit $X$ can complete a batch of components in $6$ hours, while Unit $Y$ can complete an identical batch working alone at its constant rate in $12$ hours. Unit $X$ begins working on a batch alone. After $2$ hours, Unit $Y$ joins Unit $X$ , and both units work together at their respective constant rates until the batch is complete. What is the total time, in hours, required to complete the batch from start to finish?

A
$2\frac{2}{3}$
B
$3\frac{1}{2}$
$4\frac{2}{3}$ Answer
D
$6$
E
$8$

Answer

4\frac{2}{3}

hours

The correct answer is

4\frac{2}{3}

hours. Unit

X

works alone for

2

hours, completing

2 \times \frac{1}{6} = \frac{1}{3}

of the job. This leaves

\frac{2}{3}

of the job remaining. Working together, Unit

X

and Unit

Y

have a combined rate of

\frac{1}{6} + \frac{1}{12} = \frac{1}{4}

of the job per hour. The time spent working together is

\frac{2/3}{1/4} = \frac{8}{3}

hours. Adding the initial

2

hours gives a total duration of

2 + \frac{8}{3} = 4\frac{2}{3}

hours.

Step-by-Step Solution

Calculate the individual work rates of Unit

X

and Unit

Y

Unit

X

's rate is

\frac{1}{6}

batch per hour, and Unit

Y

's rate is

\frac{1}{12}

batch per hour.

Work rate is defined as

\text{Rate} = \frac{1}{\text{Time}}

for completing one full job.

Determine the fraction of the batch completed by Unit

X

during its

2

hours of solo work.

Unit

X

completes

2 \times \frac{1}{6} = \frac{1}{3}

of the batch.

Work done equals

\text{Rate} \times \text{Time}

Find the remaining fraction of work to be completed.

The remaining work is

1 - \frac{1}{3} = \frac{2}{3}

of the batch.

Subtract the completed fraction from the total job (

1

Calculate the combined work rate when both units operate together.

Combined rate =

\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

batch per hour.

Simultaneous work rates add linearly.

Calculate the time needed for both units to finish the remaining work.

Time together =

\frac{2/3}{1/4} = \frac{2}{3} \times 4 = \frac{8}{3} = 2\frac{2}{3}

hours.

Time equals

\frac{\text{Remaining Work}}{\text{Combined Rate}}

Add the solo phase duration to the combined phase duration to get total elapsed time.

Total time =

2 + \frac{8}{3} = \frac{14}{3} = 4\frac{2}{3}

hours.

Total time is the sum of the times from each sequential stage.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question

Answer

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Key Concept

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