Question

A movie theater offers two options for membership. The first option costs $15 per ticket but an initial fee of $20. The second option costs $12.50 per ticket but an initial fee of $50. After how many tickets are both options the same price?

Answer 1

To find the number of tickets after which both membership options cost the same, we can set up equations for both options.

Let \( x \) represent the number of tickets.

For the first membership option, the cost can be expressed as:
\[
\text{Cost}_1 = 20 + 15x
\]
where \( 20 \) is the initial fee and \( 15x \) is the cost per ticket.

For the second membership option, the cost can be expressed as:
\[
\text{Cost}_2 = 50 + 12.5x
\]
where \( 50 \) is the initial fee and \( 12.5x \) is the cost per ticket.

We need to find the value of \( x \) where these two costs are equal:
\[
20 + 15x = 50 + 12.5x
\]

Next, we can solve this equation for \( x \):

1. Start by isolating \( x \):
\[
20 + 15x - 12.5x = 50
\]
\[
20 + 2.5x = 50
\]

2. Subtract 20 from both sides:
\[
2.5x = 30
\]

3. Divide by 2.5 to solve for \( x \):
\[
x = \frac{30}{2.5} = 12
\]

Thus, the two options will cost the same after purchasing **12 tickets.**