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?

Question

____ 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: \[ \boxed{10} \]