A manufacturing company produces two models of office chairs, Model P and Model Q. Produci… — GRE General Test

A manufacturing company produces two models of office chairs, Model P and Model Q. Producing each Model P chair requires $3$ hours of assembly and $1$ hour of finishing. Producing each Model Q chair requires $2$ hours of assembly and $2$ hours of finishing. On a given day, the assembly department was scheduled for $96$ total hours of work and the finishing department was scheduled for $48$ total hours of work. If both departments operated at full capacity and used all scheduled hours, how many Model P chairs were produced on that day?

Answer: 24

Answer

Letting

p

be the number of Model P chairs and

q

be the number of Model Q chairs, the hours used by the assembly department give

3p + 2q = 96

and the hours used by the finishing department give

p + 2q = 48

. Subtracting the finishing equation from the assembly equation eliminates

2q

, resulting in

2p = 48

, or

p = 24

Step-by-Step Solution

Define variables for the unknown quantities.

Let

p

equal the number of Model P chairs produced and

q

equal the number of Model Q chairs produced.

Assigning variables to the unknown quantities enables the construction of algebraic equations.

Formulate a system of linear equations representing total department hours.

Assembly department equation:

3p + 2q = 96

; Finishing department equation:

p + 2q = 48

The sum of hours required for both models in each department must equal that department's total scheduled hours.

Solve the system of equations for

p

using the elimination method.

Subtracting

p + 2q = 48

from

3p + 2q = 96

yields

2p = 48

, which gives

p = 24

Eliminating

q

directly provides the value for

p

, which corresponds to the target quantity requested in the problem.

Key Concept

Modeling real-world resource allocation using systems of linear equations

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