Movie Theatre A has a popcorn machine with a 100-gallon capacity, and sells 3 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity, and sells 5 gallons of popcorn per hour. Write an equation to show when the two popcorn machines will have the same amount of popcorn left. Use x

Question

as your variable.(1 point)
Responses

A 150+3x=100+5x
150 plus 3 x equals 100 plus 5 x

B 3x−100=5x−150
3 x minus 100 equals 5 x minus 150

C 100+3x=150+5x
100 plus 3 x equals 150 plus 5 x

D 100−3x=150−5x

Answer 1

To determine when the two popcorn machines will have the same amount of popcorn left, we need to consider how much popcorn each machine starts with and how much popcorn is being sold over time.

Let \( x \) represent the number of hours that have passed.

• Movie Theatre A starts with 100 gallons and sells 3 gallons per hour. Therefore, the amount of popcorn left in Theatre A after \( x \) hours is: \[ 100 - 3x \]

• Movie Theatre B starts with 150 gallons and sells 5 gallons per hour. Therefore, the amount of popcorn left in Theatre B after \( x \) hours is: \[ 150 - 5x \]

To find the point when both theaters have the same amount of popcorn left, we can set these two expressions equal to each other: \[ 100 - 3x = 150 - 5x \]

Now, simplifying this equation:

1. Add \( 5x \) to both sides: \[ 100 - 3x + 5x = 150 \] Which simplifies to: \[ 100 + 2x = 150 \]

2. Subtract 100 from both sides: \[ 2x = 50 \]

3. Divide by 2: \[ x = 25 \]

Now, let's evaluate the answer choices given:

• A: \( 150 + 3x = 100 + 5x \) — This is incorrect because it is not correctly representing the initial amounts and sales.
• B: \( 3x - 100 = 5x - 150 \) — This can be rearranged but is not in the proper form.
• C: \( 100 + 3x = 150 + 5x \) — This is incorrect for the same reason as A.
• D: \( 100 - 3x = 150 - 5x \) — This correctly represents the amount of popcorn left in both theaters.

The correct answer is: D: \( 100 - 3x = 150 - 5x \).