A production facility uses two automated machines, Alpha and Beta, which operate at consta… — GMAT

A production facility uses two automated machines, Alpha and Beta, which operate at constant hourly production rates. Machine Alpha produces $x$ units per hour and Machine Beta produces $y$ units per hour. When both machines operate together for 4 hours, followed by Machine Alpha operating alone for 3 hours, the facility produces a total of 310 units. When Machine Beta operates alone for 2 hours, followed by both machines operating together for 5 hours, the facility produces a total of 325 units. If Machine Alpha's hourly rate is increased by $20\%$ and Machine Beta's hourly rate is decreased by $20\%$ , how many total units will both machines produce if they work together for 8 hours at their new rates?

Cevap: 448 units

Cevap

The total number of units produced by both machines working together for 8 hours under the modified rates is 448.

Setting up the system of equations based on individual hours worked yields

7x + 4y = 310

and

5x + 7y = 325

. Solving this system gives original rates

x = 30

units/hr and

y = 25

units/hr. Applying the

20\%

increase to Alpha (

36

units/hr) and

20\%

decrease to Beta (

20

units/hr) gives a combined rate of

56

units/hr. Multiplying by

8

hours gives the final answer of

448

units.

Adım Adım Çözüm

Translate the given operational scenarios into a system of two linear equations.

Equation 1:

7x + 4y = 310

; Equation 2:

5x + 7y = 325

Operating both machines for 4 hours and Alpha alone for 3 hours means Alpha works

4 + 3 = 7

hours while Beta works

4

hours, giving

7x + 4y = 310

. Operating Beta alone for 2 hours and both for 5 hours means Alpha works

5

hours while Beta works

2 + 5 = 7

hours, giving

5x + 7y = 325

Solve the system of linear equations for variables

x

and

y

x = 30

and

y = 25

Eliminating

y

by multiplying the first equation by 7 and the second by 4 yields

49x - 20x = 2170 - 1300 \implies 29x = 870 \implies x = 30

. Substituting

x = 30

back into

7x + 4y = 310

gives

210 + 4y = 310 \implies 4y = 100 \implies y = 25

Calculate the modified production rates after percentage adjustments.

New rate for Alpha is

36

units/hr; new rate for Beta is

20

units/hr.

20\%

increase on

x = 30

yields

30 \times 1.20 = 36

. A

20\%

decrease on

y = 25

yields

25 \times 0.80 = 20

Compute total combined output over 8 hours.

Total units produced =

448

Combined modified rate is

36 + 20 = 56

units per hour. Total production over 8 hours is

56 \times 8 = 448

Anahtar Kavram

Linear Equations in One and Two Variables

Tahmini Süre:2m 30s

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