A pharmaceutical facility uses two automated purification columns, Column X and Column Y, … — GMAT

A pharmaceutical facility uses two automated purification columns, Column $X$ and Column $Y$ , to process standard batches of a synthetic compound. Working alone at its constant rate, Column $X$ can purify a full batch in $15$ hours, while Column $Y$ can purify a full batch in $10$ hours working alone at its constant rate. Column $X$ begins purifying a batch alone. After $3$ hours, Column $Y$ is turned on to work alongside Column $X$ , and both columns work together at their respective constant rates until the batch is completely purified. What is the total time, in hours, from the moment Column $X$ started until the batch is completely purified?

A
4.8 hours
B
7.2 hours
7.8 hoursAnswer
D
9.6 hours
E
11.0 hours

Answer

7.8 hours

The correct answer of 7.8 hours is obtained by evaluating the work completed in stages. In the first 3 hours, Column X completes 3/15 or 1/5 of the batch, leaving 4/5 of the batch remaining. When Column Y joins Column X, their combined rate is 1/15 + 1/10 = 1/6 batch per hour. Completing the remaining 4/5 of the batch at a rate of 1/6 per hour requires (4/5) / (1/6) = 4.8 hours. Adding the initial 3 hours yields a total time of 7.8 hours.

Step-by-Step Solution

Determine the individual hourly work rates for each column

Rate of Column

X = \frac{1}{15}

batch per hour; Rate of Column

Y = \frac{1}{10}

batch per hour.

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

Calculate the fraction of work completed during the initial stage

Work done by

X = 3 \times \frac{1}{15} = \frac{3}{15} = \frac{1}{5}

of the batch.

Column

X

works alone for the first 3 hours.

Find the remaining fraction of work to be completed

Remaining work

= 1 - \frac{1}{5} = \frac{4}{5}

of the batch.

Subtract the completed fraction from the total job (1 full batch).

Determine the combined work rate of both columns working together

Combined rate

= \frac{1}{15} + \frac{1}{10} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6}

batch per hour.

Rates add when two entities work simultaneously.

Calculate the time required for both columns to finish the remaining work

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{4/5}{1/6} = \frac{4}{5} \times 6 = \frac{24}{5} = 4.8

hours.

Time equals work divided by rate.

Calculate the total elapsed time from the start

Total time

= 3 \text{ hours (alone)} + 4.8 \text{ hours (together)} = 7.8

hours.

Add the initial duration to the combined working duration.

Key Concept

Work Rate and Combined Work

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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