Three automated data-processing pipelines, A, B, and C, process incoming dataset files at … — GMAT

Three automated data-processing pipelines, $A$ , $B$ , and $C$ , process incoming dataset files at constant individual rates. Pipeline $A$ operating alone can complete a standard processing job in $10$ hours, while Pipeline $B$ operating alone can complete the same job in $15$ hours.

The processing begins with Pipelines $A$ and $B$ working together at their standard rates for $2$ hours. At that point, Pipeline $A$ encounters a network slowdown that reduces its processing rate by $50\%$ for the remainder of the job. Pipelines $A$ and $B$ continue working at these modified rates for another $2$ hours, after which Pipeline $C$ is brought online to assist them. If Pipeline $C$ operating alone at its constant rate could finish the entire job in $10$ hours, how many total hours from the start of the processing will it take for the job to be completed?

Answer: 6 hours

Answer

The total time required to complete the job is 6 hours.

To solve multi-stage work rate problems, determine the rate per unit of time for each participant, multiply by the duration of each operational phase, and track the remaining fraction of the job. In the first 2 hours, Pipelines

A

and

B

complete

\frac{1}{3}

of the job. In the next 2 hours with Pipeline

A

at half speed (rate

\frac{1}{20}

), they complete an additional

\frac{7}{30}

of the job, leaving

\frac{13}{30}

unfinished. When Pipeline

C

(rate

\frac{1}{10}

) joins, the combined rate becomes

\frac{13}{60}

per hour, taking exactly 2 more hours to complete the remaining

\frac{13}{30}

of the job. Adding all three intervals (

2 + 2 + 2

) yields a total time of 6 hours.

Step-by-Step Solution

Determine the individual hourly work rates for each pipeline

Rate of Pipeline

A = \frac{1}{10}

job/hr, Rate of Pipeline

B = \frac{1}{15}

job/hr, Rate of Pipeline

C = \frac{1}{10}

job/hr.

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

Calculate work completed during Stage 1 (first 2 hours)

Combined rate =

\frac{1}{10} + \frac{1}{15} = \frac{1}{6}

job/hr. Work done =

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

of the job.

Both Pipeline

A

and Pipeline

B

operate together at their standard rates for 2 hours.

Calculate work completed during Stage 2 (next 2 hours)

Modified rate of Pipeline

A = 0.50 \times \frac{1}{10} = \frac{1}{20}

job/hr. Combined rate =

\frac{1}{20} + \frac{1}{15} = \frac{7}{60}

job/hr. Work done =

2 \times \frac{7}{60} = \frac{7}{30}

of the job.

Pipeline

A

's efficiency drops by 50%, while Pipeline

B

remains unchanged.

Calculate the remaining work after 4 hours

Total work completed =

\frac{1}{3} + \frac{7}{30} = \frac{17}{30}

. Remaining work =

1 - \frac{17}{30} = \frac{13}{30}

Subtracting cumulative work done from the total job (1) leaves the remaining portion.

Calculate the duration of Stage 3 when Pipeline

C

joins

Combined rate =

\frac{1}{20} + \frac{1}{15} + \frac{1}{10} = \frac{13}{60}

job/hr. Time required =

\frac{13/30}{13/60} = 2

hours.

Dividing remaining work by the combined rate of all three active pipelines gives the remaining time.

Sum the durations of all stages to find total elapsed time

Total time =

2\text{ hours} + 2\text{ hours} + 2\text{ hours} = 6\text{ hours}

The question asks for total elapsed time from the start of processing.

Key Concept

Multi-stage combined work rate problems require tracking accumulated work completed across distinct intervals with changing worker rates.

Question

Answer

Step-by-Step Solution

Key Concept

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