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 $10 per ticket but an initial fee of $50. After how many tickets are both options the same price?

To determine after how many tickets both membership options cost the same, we can set up an equation.

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

For the first option, the total cost is given by:
\[
\text{Cost}_1 = 20 + 15x
\]

For the second option, the total cost is:
\[
\text{Cost}_2 = 50 + 10x
\]

We want to find the value of \( x \) where both costs are equal:
\[
20 + 15x = 50 + 10x
\]

Now, we'll solve for \( x \):
1. Subtract \( 10x \) from both sides:
\[
20 + 15x - 10x = 50
\]
Simplifying gives:
\[
20 + 5x = 50
\]

2. Subtract 20 from both sides:
\[
5x = 30
\]

3. Divide by 5:
\[
x = 6
\]

Thus, after purchasing **6 tickets**, both options will cost the same.

**Answer: 6 tickets**