Two volunteer teams, Team A and Team B, are packing food boxes for a local shelter. Team A… — ACT

Two volunteer teams, Team A and Team B, are packing food boxes for a local shelter. Team A packs at a constant rate of $15$ boxes per hour and begins packing at 8:00 a.m. Team B packs at a constant rate of $20$ boxes per hour and begins packing at 9:30 a.m. If both teams pack continuously at their respective rates, how many hours after Team A begins will both teams have packed the same total number of boxes?

Answer: 6 hours

Answer

Both teams will have packed the same total number of boxes

6

hours after Team A begins.

The correct answer is

6

. Let

x

be the number of hours Team A packs. Team B begins 1.5 hours later, so Team B packs for

x - 1.5

hours. Setting their total boxes packed equal gives

15x = 20(x - 1.5)

. Distributing yields

15x = 20x - 30

. Subtracting

20x

from both sides gives

-5x = -30

, which simplifies to

x = 6

Step-by-Step Solution

Define the variable for time and identify the time difference.

Let

x

be the number of hours Team A packs. Since Team B starts 1 hour and 30 minutes (which is

1.5

hours) later, Team B's time is represented as

x - 1.5

hours.

Establishing correct algebraic representations for time is necessary to set up the equation.

Set up an equation equating the total boxes packed by both teams.

The equation is

15x = 20(x - 1.5)

Since both teams pack a constant number of boxes per hour, multiplying their rate by their active time gives the total boxes packed.

Solve the linear equation for

x

Distribute the 20:

15x = 20x - 30

. Subtract

20x

from both sides:

-5x = -30

. Divide by

-5

x = 6

Solving the equation gives the number of hours after Team A starts when their packed boxes are equal.

Key Concept

Translating real-world rates and time shifts into linear equations and solving them.

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