A specialty coffee roasting facility operates two roasting machines, Roaster X and Roaster… — GMAT

A specialty coffee roasting facility operates two roasting machines, Roaster X and Roaster Y, each producing roasted coffee beans at a constant hourly rate. On Monday, running Roaster X for 5 hours and Roaster Y for 4 hours yielded a total of $400\text{ kg}$ of roasted beans. On Tuesday, running Roaster X for 3 hours and Roaster Y for 7 hours yielded a total of $470\text{ kg}$ of roasted beans. Based on this information, match each quantity listed on the left with its corresponding value on the right.

Hourly production rate of Roaster X $40\text{ kg}$
Hourly production rate of Roaster Y $50\text{ kg}$
Combined production output of Roaster X and Roaster Y operating together for 1 hour $90\text{ kg}$
Net difference between the output of Roaster Y running for 2 hours and Roaster X running for 1 hour $60\text{ kg}$

Answer

Hourly production rate of Roaster X corresponds to

40\text{ kg}

; Hourly production rate of Roaster Y corresponds to

50\text{ kg}

; Combined production output of both roasters for 1 hour corresponds to

90\text{ kg}

; Net difference between 2 hours of Roaster Y and 1 hour of Roaster X corresponds to

60\text{ kg}

Formulating the system

5x + 4y = 400

and

3x + 7y = 470

and solving via elimination gives

x = 40\text{ kg/hr}

for Roaster X and

y = 50\text{ kg/hr}

for Roaster Y. Consequently, the 1-hour combined output

x + y = 90\text{ kg}

, and the net difference

2y - x = 2(50) - 40 = 60\text{ kg}

Step-by-Step Solution

Set up a system of linear equations using variables

x

and

y

for the hourly rates of Roaster X and Roaster Y.

Monday:

5x + 4y = 400

; Tuesday:

3x + 7y = 470

Total production equals the rate multiplied by time for each machine.

Eliminate variable

x

by multiplying the first equation by 3 and the second equation by 5.

First equation becomes

15x + 12y = 1200

; second equation becomes

15x + 35y = 2350

Align coefficients of

x

to eliminate

x

by subtraction.

Subtract the modified first equation from the modified second equation to solve for

y

23y = 1150 \implies y = 50\text{ kg/hr}$.

Simplifies the two-variable system to a single linear equation in

y

Substitute

y = 50

back into

5x + 4y = 400

to solve for

x

5x + 4(50) = 400 \implies 5x + 200 = 400 \implies 5x = 200 \implies x = 40\text{ kg/hr}$.

Determines the rate of Roaster X.

Calculate the composite values for the remaining matching targets.

Combined output:

x + y = 40 + 50 = 90\text{ kg}

. Difference:

2y - x = 2(50) - 40 = 60\text{ kg}

Evaluates expressions using solved individual variable values.

Key Concept

Solving systems of simultaneous linear equations in two variables.

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