To find out after how many tickets both membership options cost the same, we can set up an equation based on the total cost for each option.
Let \( x \) represent the number of tickets purchased.
For the first membership option:
- Initial fee: \( 15 \)
- Cost per ticket: \( 10 \)
- Total cost for \( x \) tickets: \[ C_1 = 15 + 10x \]
For the second membership option:
- Initial fee: \( 45 \)
- Cost per ticket: \( 7.50 \)
- Total cost for \( x \) tickets: \[ C_2 = 45 + 7.50x \]
We want to find the value of \( x \) where both costs are equal: \[ 15 + 10x = 45 + 7.50x \]
Now, we'll solve this equation for \( x \).
-
Subtract \( 7.50x \) from both sides: \[ 15 + 10x - 7.50x = 45 \] \[ 15 + 2.5x = 45 \]
-
Subtract \( 15 \) from both sides: \[ 2.5x = 30 \]
-
Divide both sides by \( 2.5 \): \[ x = \frac{30}{2.5} = 12 \]
Thus, the two membership options will cost the same after purchasing 12 tickets.