Question

A movie theater offers two options for membership. The first option costs $15 per ticket but an initial fee of $5. 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 out after how many tickets the two membership options cost the same, we can set up an equation for each option.

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

**Option 1**: Costs \( 5 + 15x \) (initial fee of $5 plus $15 for each ticket)

**Option 2**: Costs \( 50 + 12.50x \) (initial fee of $50 plus $12.50 for each ticket)

We want to find \( x \) such that the costs for both options are equal:

\[
5 + 15x = 50 + 12.50x
\]

Now, let's solve for \( x \):

1. Subtract \( 12.50x \) from both sides:

\[
5 + 15x - 12.50x = 50
\]

This simplifies to:

\[
5 + 2.5x = 50
\]

2. Subtract 5 from both sides:

\[
2.5x = 45
\]

3. Divide both sides by 2.5:

\[
x = \frac{45}{2.5} = 18
\]

Thus, the two membership options cost the same after \( \boxed{18} \) tickets.