A movie theater sells two types of tickets, adult tickets for $12 and child tickets for $8. If the theater...

1. TRANSLATE the problem information

• Given information:
  • Adult tickets cost $12 each
  • Child tickets cost $8 each
  • Total tickets sold = 30
  • Total revenue = $300
  • Need to find: Amount spent on adult tickets

• What this tells us: We need two equations to solve for two unknowns

2. INFER the approach

• Since we have two unknowns (adult tickets and child tickets), we need a system of equations
• Let \(\mathrm{a}\) = number of adult tickets, \(\mathrm{c}\) = number of child tickets
• Set up equations from the constraints:
  • Total tickets: \(\mathrm{a + c = 30}\)
  • Total revenue: \(\mathrm{12a + 8c = 300}\)

3. SIMPLIFY by solving the system

• From equation 1: \(\mathrm{c = 30 - a}\)
• Substitute into equation 2:
  \(\mathrm{12a + 8(30 - a) = 300}\)
  \(\mathrm{12a + 240 - 8a = 300}\)
  \(\mathrm{4a = 60}\)
  \(\mathrm{a = 15}\)

4. INFER the final answer

• We found \(\mathrm{a = 15}\) (number of adult tickets)
• The question asks for amount spent on adult tickets
• Amount = \(\mathrm{15 × \$12 = \$180}\)

Answer: 180

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students solve for the number of adult tickets (\(\mathrm{a = 15}\)) but forget that the question asks for the total amount spent on adult tickets, not the number of tickets.

They correctly find \(\mathrm{a = 15}\) but then answer 15 instead of calculating \(\mathrm{15 × \$12 = \$180}\). This happens because they get focused on solving the algebra and lose track of what the original question was asking.

Second Most Common Error:

Poor TRANSLATE execution: Students set up incorrect equations, often confusing the price per ticket with total amounts, or mixing up which variable represents which ticket type.

For example, they might write \(\mathrm{8a + 12c = 300}\) (reversing the coefficients) or misread the constraint equations entirely. This leads to wrong values for the number of tickets and consequently the wrong final answer.

The Bottom Line:

This problem tests both systematic equation setup and careful attention to what the question is actually asking. The algebra itself is straightforward once properly set up, but students often stumble on translating word constraints into equations or remembering to answer the right question.