Question

Movie Theatre A has a popcorn machine that can hold 120 gallons and sells 7 gallons per hour. Movie Theatre B has a popcorn machine that can hold 150 gallons and sells 9 gallons per hour. When will both popcorn machines have the same amount of popcorn left?

Answer 1

To determine when both popcorn machines will have the same amount of popcorn left, we will first define the variables and set up equations for the amount of popcorn left in each machine over time.

Let's define \( t \) as the time in hours after the machines start popping popcorn.

For **Movie Theatre A**:
- Initial amount of popcorn = 120 gallons
- Sales rate = 7 gallons/hour

The amount of popcorn left in Theatre A after \( t \) hours:
\[
\text{Amount left in A} = 120 - 7t
\]

For **Movie Theatre B**:
- Initial amount of popcorn = 150 gallons
- Sales rate = 9 gallons/hour

The amount of popcorn left in Theatre B after \( t \) hours:
\[
\text{Amount left in B} = 150 - 9t
\]

We want to find the time \( t \) when both machines have the same amount of popcorn left. This means we need to solve the equation:

\[
120 - 7t = 150 - 9t
\]

Now, let's isolate \( t \):

1. Rearrange the equation:
\[
9t - 7t = 150 - 120
\]

2. Simplify:
\[
2t = 30
\]

3. Solve for \( t \):
\[
t = \frac{30}{2} = 15
\]

Now, we will double-check by substituting \( t = 15 \) back into the equations for the amounts of popcorn in both theatres.

For Theatre A:
\[
\text{Amount left in A} = 120 - 7(15) = 120 - 105 = 15 \text{ gallons}
\]

For Theatre B:
\[
\text{Amount left in B} = 150 - 9(15) = 150 - 135 = 15 \text{ gallons}
\]

Both machines will have 15 gallons left after 15 hours.

Thus, both popcorn machines will have the same amount of popcorn left after **15 hours**.