Three automated server clusters—Cluster X, Cluster Y, and Cluster Z—process large-scale da… — GMAT

Three automated server clusters—Cluster $X$ , Cluster $Y$ , and Cluster $Z$ —process large-scale data analytics tasks. Working alone at their respective constant rates, Cluster $X$ can complete a standard workload in $12$ hours, Cluster $Y$ can complete it in $20$ hours, and Cluster $Z$ can complete it in $30$ hours. Cluster $X$ and Cluster $Y$ begin processing a standard workload together. After $4$ hours, Cluster $X$ goes offline due to maintenance while Cluster $Y$ continues working alone. Exactly $2$ hours after Cluster $X$ goes offline, Cluster $Z$ is brought online to assist Cluster $Y$ . How many additional hours will it take Cluster $Y$ and Cluster $Z$ working together to complete the remaining portion of the workload?

$4.4$ hoursAnswer
B
$5.6$ hours
C
$2.8$ hours
D
$18.33$ hours
E
$6.2$ hours

Answer

The additional time required for Cluster Y and Cluster Z to finish the remaining workload is

4.4

hours.

The combined rate of Cluster X and Cluster Y is

\frac{1}{12} + \frac{1}{20} = \frac{2}{15}

job/hr. In

4

hours, they complete

4 \times \frac{2}{15} = \frac{8}{15}

of the job, leaving

\frac{7}{15}

. Cluster Y then works alone for

2

hours, completing

2 \times \frac{1}{20} = \frac{1}{10}

of the job. The remaining work is

\frac{7}{15} - \frac{1}{10} = \frac{11}{30}

. Finally, Cluster Y and Cluster Z work together at a combined rate of

\frac{1}{20} + \frac{1}{30} = \frac{1}{12}

job/hr. The time needed to finish is

\frac{11/30}{1/12} = 4.4

hours.

Step-by-Step Solution

Calculate individual work rates for each cluster per hour.

r_X = \frac{1}{12}

r_Y = \frac{1}{20}

, and

r_Z = \frac{1}{30}

of the total workload per hour.

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

Calculate the work completed in Stage 1 when Cluster X and Cluster Y work together for 4 hours.

Combined rate

r_{X+Y} = \frac{1}{12} + \frac{1}{20} = \frac{5 + 3}{60} = \frac{8}{60} = \frac{2}{15}

. Work done in 4 hours =

4 \times \frac{2}{15} = \frac{8}{15}

. Remaining work =

1 - \frac{8}{15} = \frac{7}{15}

Multiply the combined rate of Clusters X and Y by 4 hours to find the fraction of work completed.

Calculate the work completed in Stage 2 when Cluster Y works alone for 2 hours.

Work done by Y in 2 hours =

2 \times \frac{1}{20} = \frac{1}{10} = \frac{3}{30}

. Remaining work =

\frac{7}{15} - \frac{1}{10} = \frac{14}{30} - \frac{3}{30} = \frac{11}{30}

Subtract the work done by Cluster Y during its solo 2-hour interval from the remaining work.

Calculate the time required for Cluster Y and Cluster Z working together to complete the remaining

11/30

of the work.

Combined rate

r_{Y+Z} = \frac{1}{20} + \frac{1}{30} = \frac{3 + 2}{60} = \frac{5}{60} = \frac{1}{12}

. Time

t = \frac{11/30}{1/12} = \frac{11}{30} \times 12 = \frac{132}{30} = 4.4

hours.

Divide the remaining workload fraction by the combined hourly work rate of Clusters Y and Z.

Key Concept

Combined Work Rate in Multi-Stage Problems

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic