A commercial bakery operates two automated production lines, Line A and Line B, to produce… — GMAT

A commercial bakery operates two automated production lines, Line A and Line B, to produce specialized pastry boxes. Line A requires $4$ minutes of mixing and $2$ minutes of baking per box. Line B requires $3$ minutes of mixing and $5$ minutes of baking per box. On a given shift, the bakery utilizes a total of $180$ minutes of mixing time and $160$ minutes of baking time, with both lines running continuously without downtime.

If $x$ represents the number of pastry boxes produced by Line A and $y$ represents the number of pastry boxes produced by Line B during the shift, which of the following statements regarding the production parameters must be true? Select all that apply.

Line A produced $10$ more pastry boxes than Line B during the shift.Answer
The total number of pastry boxes produced by both lines combined is equal to $50$ .Answer
C
The linear equation modeling total mixing time across both lines is $4x + 2y = 180$ .
D
Line B accounted for $60\%$ of the total pastry boxes produced during the shift.
The system of linear equations modeling the resource constraints yields a unique solution of $x = 30$ and $y = 20$ .Answer

Answer

The statements that must be true are: Line A produced 10 more pastry boxes than Line B during the shift; the total number of pastry boxes produced by both lines combined is equal to 50; and the system of linear equations modeling the resource constraints yields a unique solution of

x = 30

and

y = 20

Solving the system of equations representing mixing time (

4x + 3y = 180

) and baking time (

2x + 5y = 160

) gives

x = 30

boxes for Line A and

y = 20

boxes for Line B. Consequently, Line A produced

30 - 20 = 10

more boxes than Line B, the total production is

30 + 20 = 50

boxes, and the system indeed yields the unique solution

(30, 20)

Step-by-Step Solution

Set up the linear equations based on mixing time and baking time constraints.

Mixing constraint:

4x + 3y = 180

; Baking constraint:

2x + 5y = 160

Each unit of

x

requires

4

min mixing and

2

min baking; each unit of

y

requires

3

min mixing and

5

min baking.

Solve the system of equations using elimination.

Multiply the baking equation by

2

4x + 10y = 320

. Subtracting the mixing equation (

4x + 3y = 180

) yields

7y = 140 \implies y = 20

. Substituting

y = 20

into

4x + 3(20) = 180

gives

4x = 120 \implies x = 30

Eliminating

x

yields the exact value for

y

, which then gives

x

Evaluate each candidate statement against the solution

(x, y) = (30, 20)

Difference:

x - y = 30 - 20 = 10

(True). Total:

x + y = 30 + 20 = 50

(True). Mixing equation:

4x + 3y = 180

, not

4x + 2y = 180

(False). Line B share:

\frac{20}{50} = 40\%

, not

60\%

(False). Unique solution

(30, 20)

(True).

Checking each statement determines which options are valid.

Key Concept

Algebraic modeling of multi-variable resource constraint word problems using linear systems of equations

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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