A municipal water treatment facility utilizes three primary purification units—Unit P, Uni… — GMAT

A municipal water treatment facility utilizes three primary purification units—Unit P, Unit Q, and Unit R—to process large water batches. Operating alone at its constant rate, Unit P can process a full batch in $12$ hours, Unit Q can process a full batch in $18$ hours, and Unit R can process a full batch in $36$ hours. All three units begin processing a batch together. After $3$ hours, Unit P stops operating due to scheduled maintenance. Unit Q and Unit R continue operating together for an additional $2$ hours, after which Unit Q is shut down. Unit R then completes the remainder of the batch working alone. How many total hours does it take from start to finish to process the complete batch?

Answer: 17 hours

Answer

The total time required to process the complete batch from start to finish is 17 hours.

To find the total time from start to finish, analyze the work completed in each of the three stages:

1. In Stage 1 (3 hours), all three units operate together. Their combined rate is

\frac{1}{12} + \frac{1}{18} + \frac{1}{36} = \frac{3 + 2 + 1}{36} = \frac{6}{36} = \frac{1}{6}

batch per hour. In

3

hours, they complete

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

of the total batch.

2. In Stage 2 (2 hours), Unit P stops and Units Q and R continue together. Their combined rate is

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

batch per hour. In

2

hours, they complete

2 \times \frac{1}{12} = \frac{1}{6}

of the total batch.

3. After Stage 2, the total fraction of work completed is

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

of the batch. The remaining fraction of work is

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

of the batch.

4. In Stage 3, Unit R completes the remaining

\frac{1}{3}

batch alone at its rate of

\frac{1}{36}

batch per hour. The time taken by Unit R is

\frac{1/3}{1/36} = 12

hours.

Adding the durations of all three stages gives

3 + 2 + 12 = 17

total hours.

Step-by-Step Solution

Calculate individual work rates

Rate of P =

\frac{1}{12}

batch/hr, Rate of Q =

\frac{1}{18}

batch/hr, Rate of R =

\frac{1}{36}

batch/hr

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

Calculate work done during Stage 1 (first 3 hours)

Combined rate of P, Q, R =

\frac{1}{12} + \frac{1}{18} + \frac{1}{36} = \frac{6}{36} = \frac{1}{6}

batch/hr. Work done =

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

batch

All three units operate simultaneously for 3 hours.

Calculate work done during Stage 2 (next 2 hours)

Combined rate of Q and R =

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

batch/hr. Work done =

2 \times \frac{1}{12} = \frac{1}{6}

batch

Unit P stops working, leaving Q and R to operate together for 2 hours.

Determine remaining work and time required for Stage 3

Remaining work =

1 - \left(\frac{1}{2} + \frac{1}{6}\right) = \frac{1}{3}

batch. Time for R alone =

\frac{1/3}{1/36} = 12

hours

Unit R must complete the remaining fraction of the batch operating alone at its constant rate.

Calculate total elapsed time

3\text{ hours} + 2\text{ hours} + 12\text{ hours} = 17\text{ hours}

Sum the durations of all three distinct operational stages.

Key Concept

Combined Work Rate in Multi-Stage Work Scenarios

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