A movie theater offers a monthly membership that includes a fixed membership fee plus an additional charge for each movie...

1. TRANSLATE the problem information

• Given information:
  • Cost function: \(\mathrm{C(x) = 12x + 35}\)
  • Actual monthly cost: \(\$143\)
  • Need to find: number of movies watched (x)

• What this tells us: We need to solve \(\mathrm{C(x) = 143}\)

2. TRANSLATE this into an equation

• Set the function equal to the known cost:
  \(\mathrm{12x + 35 = 143}\)

3. SIMPLIFY to solve for x

• Subtract 35 from both sides:
  \(\mathrm{12x = 143 - 35}\)
  \(\mathrm{12x = 108}\)

• Divide both sides by 12:
  \(\mathrm{x = 108 ÷ 12 = 9}\)

4. Verify the answer

• Check: \(\mathrm{C(9) = 12(9) + 35 = 108 + 35 = 143}\) ✓

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors during the algebraic steps, particularly when calculating \(\mathrm{143 - 35}\) or \(\mathrm{108 ÷ 12}\).

For example, if a student calculates \(\mathrm{143 - 35 = 118}\) instead of 108, they get \(\mathrm{118 ÷ 12 ≈ 9.8}\), which they might round to 10. This may lead them to select Choice D (10).

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing the setup by trying to solve for the cost instead of the number of movies, or misunderstanding what the \(\$143\) represents in relation to the function.

This conceptual confusion about the function's input and output can cause students to set up incorrect equations, leading to random answer selection or guessing.

The Bottom Line:

This problem requires careful attention to arithmetic details within a straightforward algebraic framework. Students who understand the setup but rush through calculations often select incorrect answers that are close to the correct value.