Question

A movie theater offers two options for membership. The first option costs $10\$10$10 per ticket but an initial fee of $15. The second option costs $7.50 per ticket but an initial fee of $40 . After how many tickets are both options the same price?



____ tickets

Answer 1

To find out after how many tickets both membership options are the same price, we can set up equations based on the cost of each option.

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

1. **Cost of the first option**:
The first option has an initial fee of $15 and a cost of $10 per ticket. Therefore, the total cost for the first option can be expressed as:
\[
C_1 = 15 + 10x
\]

2. **Cost of the second option**:
The second option has an initial fee of $40 and a cost of $7.50 per ticket. Thus, the total cost for the second option is:
\[
C_2 = 40 + 7.5x
\]

We need to find the value of \( x \) where both costs are equal:
\[
C_1 = C_2
\]

Setting the equations equal to each other:
\[
15 + 10x = 40 + 7.5x
\]

Next, we will solve for \( x \). First, subtract \( 7.5x \) from both sides:
\[
15 + 10x - 7.5x = 40
\]
\[
15 + 2.5x = 40
\]

Now, subtract 15 from both sides:
\[
2.5x = 40 - 15
\]
\[
2.5x = 25
\]

Next, divide both sides by 2.5:
\[
x = \frac{25}{2.5} = 10
\]

Thus, the number of tickets after which both options cost the same is:
\[
\boxed{10}
\]