Question

A movie theater offers two options for membership. The first option costs $15\$15$15 per ticket but an initial fee of $10\$10$10. The second option costs $10\$10$10 per ticket but an initial fee of $60\$60$60 . After how many tickets are both options the same price?

Answer 1

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

For the first membership option, the total cost (initial fee plus cost per ticket) can be expressed as:

\[
C_1 = 10 + 15x
\]

For the second membership option, the total cost can be expressed as:

\[
C_2 = 60 + 10x
\]

We want to find the value of \( x \) where both options have the same total cost, i.e., where \( C_1 = C_2 \).

Setting the costs equal to each other:

\[
10 + 15x = 60 + 10x
\]

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

First, we'll subtract \( 10x \) from both sides:

\[
10 + 15x - 10x = 60
\]

This simplifies to:

\[
10 + 5x = 60
\]

Next, we will isolate \( 5x \) by subtracting \( 10 \) from both sides:

\[
5x = 50
\]

Now we divide by \( 5 \):

\[
x = 10
\]

Thus, after purchasing \( \boxed{10} \) tickets, both membership options will have the same total cost.