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

A commercial bakery operates two automated production lines, Line A and Line B, to fulfill a production order for $B$ boxes of goods:

- Line A operates at a constant rate of $r$ boxes per hour ( $r > 0$ ) for $t$ hours.
- Line B operates at a constant rate that is $25\%$ higher than Line A's rate and works for $2$ hours longer than Line A.
- Line B produces twice as many boxes as Line A during their respective operational periods.
- The total contract volume $B$ is the sum of the boxes produced by Line A and Line B.

Which of the following statements MUST be true regarding this algebraic model? (Select ALL that apply.)

Line A operates for $t = \frac{10}{3}$ hours (3 hours and 20 minutes).Answer
The total contract volume $B$ is equal to $10r$ boxes.Answer
C
The overall average production rate over the combined total operating time of both lines is $1.125r$ boxes per hour.
D
If both lines were to work together simultaneously to complete the total order of $B$ boxes, the required time would equal the sum of their individual operating times ( $\frac{26}{3}$ hours).
E
Dividing both sides of the equation $1.25r(t + 2) = 2rt$ by $r$ loses $r = 0$ as a physically valid non-zero rate solution.

Answer

The correct statements are the one establishing Line A's operating time as 10/3 hours and the one establishing the total volume as 10r boxes.

The model yields

t = \frac{10}{3}

hours by equating

1.25r(t + 2) = 2rt

and dividing by

r

. Substituting

t = \frac{10}{3}

into the total output expression

B = rt + 1.25r(t + 2)

gives

B = \frac{10}{3}r + \frac{20}{3}r = 10r

. Thus, both the operating time statement for Line A and the total contract volume statement are mathematically true.

Step-by-Step Solution

Formulate algebraic expressions for the output of each production line.

Line A output:

Q_A = r \cdot t

. Line B rate is

1.25r

and operating time is

t + 2

, so Line B output:

Q_B = 1.25r(t + 2)

Word problem translation requires modeling quantities produced as rate multiplied by time.

Set up the relational equation based on the condition that Line B produces twice as much as Line A.

1.25r(t + 2) = 2(r \cdot t) \implies 1.25r(t + 2) = 2rt

Translates the given relationship

Q_B = 2 Q_A

into a single variable equation for

t

Solve for

t

by dividing out non-zero

r

1.25(t + 2) = 2t \implies 1.25t + 2.5 = 2t \implies 0.75t = 2.5 \implies t = \frac{2.5}{0.75} = \frac{10}{3}

hours.

Since rate

r > 0

, dividing by

r

isolates

t

Calculate individual outputs and total contract volume

B

Q_A = r \cdot \frac{10}{3} = \frac{10}{3}r

Q_B = 2 Q_A = \frac{20}{3}r

. Total volume

B = Q_A + Q_B = \frac{10}{3}r + \frac{20}{3}r = 10r

Summing the output of both lines gives the exact total contract volume in terms of rate

r

Key Concept

Algebraic Modeling of Multi-Rate Work Systems

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