Maria bought 6 movie tickets that were each the same price. She paid an additional $12 service fee for the...

1. TRANSLATE the problem information

• Given information:
  • 6 movie tickets, each costing the same unknown amount
  • Additional $12 service fee for the entire purchase
  • Total amount paid = $78
  • Need to find: price of 1 movie ticket
• Let x = the original price of 1 movie ticket in dollars

2. TRANSLATE the problem into an equation

• Cost of 6 tickets = \(\mathrm{6x}\) dollars
• Total cost = cost of tickets + service fee
• Total cost = \(\mathrm{6x + 12 = 78}\)

3. SIMPLIFY to solve for x

• Starting equation: \(\mathrm{6x + 12 = 78}\)
• Subtract 12 from both sides: \(\mathrm{6x = 66}\)
• Divide both sides by 6: \(\mathrm{x = 11}\)

Answer: $11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may set up the wrong equation by misunderstanding the problem structure. They might think the service fee applies to each ticket rather than the entire purchase, leading to an equation like \(\mathrm{6(x + 12) = 78}\).

Solving this incorrect equation:

\(\mathrm{6(x + 12) = 78}\)

\(\mathrm{x + 12 = 13}\)

\(\mathrm{x = 1}\)

which isn't even among the answer choices. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE execution: Students might correctly identify the components but make an error in the equation setup, such as writing \(\mathrm{6x - 12 = 78}\) (subtracting instead of adding the service fee).

This gives:

\(\mathrm{6x = 90}\)

\(\mathrm{x = 15}\)

This may lead them to select Choice D ($15).

The Bottom Line:

Success depends on carefully reading the problem to understand that the $12 service fee is added once to the entire purchase, not to each individual ticket. The key insight is recognizing the problem structure: (number of items × price per item) + one-time fee = total cost.