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.
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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: \[ C_1 = 15 + 10x \]
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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: \[ \boxed{10} \]