A movie theater offers two options for membership. The first option costs $10 per ticket but an initial fee of $10. The second option cost $5 per ticket but an initial fee of $60. After how many tickets are both options the same?

Answer 1

To determine after how many tickets the total costs of the two membership options are the same, we can set up equations for both membership options and then find the number of tickets at which these equations are equal.

1. First Option:

   • Initial fee: $10
   • Cost per ticket: $10
   • Total cost after \( x \) tickets: \[ C_1 = 10 + 10x \]

2. Second Option:

   • Initial fee: $60
   • Cost per ticket: $5
   • Total cost after \( x \) tickets: \[ C_2 = 60 + 5x \]

Next, we need to find \( x \) such that \( C_1 = C_2 \): \[ 10 + 10x = 60 + 5x \]

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

1. Subtract \( 5x \) from both sides: \[ 10 + 10x - 5x = 60 \] This simplifies to: \[ 10 + 5x = 60 \]

2. Subtract 10 from both sides: \[ 5x = 60 - 10 \] This simplifies to: \[ 5x = 50 \]

3. Divide both sides by 5: \[ x = \frac{50}{5} = 10 \]

Thus, the two options will cost the same after 10 tickets.

We can verify by calculating the total cost for 10 tickets with each option:

• First Option: \[ C_1 = 10 + 10 \times 10 = 10 + 100 = 110 \]

• Second Option: \[ C_2 = 60 + 5 \times 10 = 60 + 50 = 110 \]

Both options have a total cost of $110 after 10 tickets, confirming our solution is correct. The answer is:

\[ \boxed{10} \]