Two server clusters, Alpha and Beta, process data jobs of Type 1 and Type 2. Cluster Alpha… — GMAT

Two server clusters, Alpha and Beta, process data jobs of Type 1 and Type 2. Cluster Alpha processes $15$ Type 1 jobs and $25$ Type 2 jobs in a total time of $210$ minutes. Cluster Beta processes $25$ Type 1 jobs and $15$ Type 2 jobs in a total time of $190$ minutes. Assuming constant processing rates per job type across both clusters, how many total minutes will it take to process a workload consisting of $30$ Type 1 jobs and $10$ Type 2 jobs?

Answer: 180 minutes

Answer

The total processing time required is 180 minutes.

Let

x

be the processing time in minutes for a Type 1 job, and

y

be the processing time in minutes for a Type 2 job. Translating the given data gives the simultaneous equations

15x + 25y = 210

and

25x + 15y = 190

. Dividing both equations by

5

yields

3x + 5y = 42

and

5x + 3y = 38

. Adding these simplified equations yields

8x + 8y = 80

, so

x + y = 10

. Subtracting the first from the second gives

2x - 2y = -4

, so

x - y = -2

. Solving

x + y = 10

and

x - y = -2

gives

x = 4

and

y = 6

. The required workload duration is

30(4) + 10(6) = 120 + 60 = 180

minutes.

Step-by-Step Solution

Set up a linear system of simultaneous equations based on the rates given.

15x + 25y = 210

and

25x + 15y = 190

, where

x

and

y

are the processing times per Type 1 and Type 2 job.

Establishing accurate algebraic expressions translates the real-world scenario into a solvable model.

Simplify the system by dividing through by common factors.

3x + 5y = 42

and

5x + 3y = 38

Simplification reduces arithmetic complexity and minimizes computational errors.

Solve for variables x and y using symmetric combinations (adding/subtracting equations).

x = 4

and

y = 6

Symmetric linear systems can be solved efficiently by taking the sum and difference of the equations.

Evaluate the target expression

30x + 10y

30(4) + 10(6) = 180

Multiplying individual job times by requested quantities yields the total workload time.

Key Concept

Simultaneous Linear Systems and Symmetric Reduction

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