A solar-powered desalination facility uses two purification units, Unit X and Unit Y, to p… — GMAT

A solar-powered desalination facility uses two purification units, Unit $X$ and Unit $Y$ , to process standard batches of seawater. Working alone at its constant rate, Unit $X$ can desalinate a full batch of seawater in $12$ hours. Working alone at its constant rate, Unit $Y$ can desalinate an identical batch of seawater in $18$ hours. Unit $X$ begins processing a batch alone and operates for $3$ hours. Then, Unit $Y$ is also turned on, and both units work together at their respective constant rates to complete the remaining portion of the batch. What is the total number of hours required to process the entire batch from the time Unit $X$ was first turned on?

A
$5.4$
B
$7.2$
$8.4$ Answer
D
$14.25$
E
$25.5$

Answer

8.4 hours

Unit X completes 1/12 of the batch per hour, so in 3 hours it processes 3/12 = 1/4 of the batch. This leaves 3/4 of the batch unfinished. Working together, Unit X and Unit Y have a combined rate of 1/12 + 1/18 = 5/36 batch per hour. Dividing the remaining 3/4 batch by 5/36 gives 27/5 = 5.4 hours for the joint operation phase. Summing the initial 3 hours and the 5.4 hours yields a total processing time of 8.4 hours.

Step-by-Step Solution

Calculate individual hourly work rates for each unit

Unit X rate =

\frac{1}{12}

batch/hour, Unit Y rate =

\frac{1}{18}

batch/hour

Work rate is the reciprocal of the time required to complete one full job.

Determine the fraction of work completed during the first 3 hours

Work completed by Unit X =

3 \times \frac{1}{12} = \frac{1}{4}

of the batch

Unit X operates alone for 3 hours at a rate of

\frac{1}{12}

batch per hour.

Calculate the remaining fraction of work to be done

Remaining work =

1 - \frac{1}{4} = \frac{3}{4}

of the batch

Subtract the completed fraction from the total job equal to 1 whole batch.

Find the combined hourly work rate of both units

Combined rate =

\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36}

batch/hour

When entities work together, their individual rates add up.

Calculate the time required for both units to complete the remaining work

Time together =

\frac{\frac{3}{4}}{\frac{5}{36}} = \frac{3}{4} \times \frac{36}{5} = \frac{27}{5} = 5.4

hours

Time equals remaining work divided by combined rate.

Add the initial single-unit operating time to the combined operating time

Total time =

3 + 5.4 = 8.4

hours

The question asks for the total elapsed time from when Unit X first started.

Key Concept

Work Rate and Combined Work

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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