A data processing center uses two server clusters, Cluster X and Cluster Y, operating at c… — GRE General Test

A data processing center uses two server clusters, Cluster $X$ and Cluster $Y$ , operating at constant individual processing rates. The ratio of the rate of Cluster $X$ to the rate of Cluster $Y$ is $3 : 5$ . Cluster $X$ alone can process a standard dataset of size $D$ gigabytes in 20 hours.

If Cluster $X$ and Cluster $Y$ work together for 4 hours at their initial rates, and then Cluster $X$ 's rate is increased by $33\frac{1}{3}\%$ while Cluster $Y$ 's rate is increased by $20\%$ , how many additional hours will it take for the two clusters working together at their new rates to complete the remaining portion of dataset $D$ ?

Cevap: 2.8 hours

Cevap

It will take 2.8 additional hours for the two clusters working together at their new rates to complete the remaining portion of dataset

D

Representing Cluster

X

's rate as

3k

and Cluster

Y

's rate as

5k

establishes the dataset size

D = 20 \times 3k = 60k

. During the first 4 hours, both clusters process

4 \times (3k + 5k) = 32k

GB, leaving

28k

GB remaining. After rate increases, Cluster

X

's rate becomes

4k

and Cluster

Y

's rate becomes

6k

, resulting in a new combined rate of

10k

. Dividing the remaining

28k

GB by

10k

GB/hr yields

2.8

hours.

Adım Adım Çözüm

Define variables for the initial rates and dataset size based on the given ratio.

Let the processing rate of Cluster

X

r_X = 3k

GB/hr and Cluster

Y

r_Y = 5k

GB/hr for some constant

k > 0

. Since Cluster

X

alone completes dataset

D

in 20 hours,

D = 3k \times 20 = 60k

GB.

Relating the ratio of individual rates to total work defines all quantities in terms of a single parameter

k

Calculate the amount of work finished during the initial joint operation.

The combined initial rate is

r_X + r_Y = 3k + 5k = 8k

GB/hr. Working together for 4 hours completes

8k \times 4 = 32k

GB.

Working simultaneously means their processing rates add together.

Find the remaining work and the updated processing rates after adjustments.

Remaining dataset volume =

60k - 32k = 28k

GB. Cluster

X

's new rate =

3k \times \left(1 + \frac{1}{3}\right) = 4k

GB/hr. Cluster

Y

's new rate =

5k \times 1.20 = 6k

GB/hr. New combined rate =

4k + 6k = 10k

GB/hr.

Modifying the individual rates changes the combined throughput for the remaining task.

Compute the additional time required to process the remaining dataset.

Additional time =

\frac{28k \text{ GB}}{10k \text{ GB/hr}} = 2.8

hours.

Dividing the remaining work volume by the new combined rate gives the exact required time.

Anahtar Kavram

Combined work rates, ratio proportionality, and percentage rate adjustments.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru