A community theater has 450 seats. For a certain performance, 60 of these seats were reserved. Of the seats that...

1. TRANSLATE the problem information

• Given information:
  • Total theater seats: \(\mathrm{450}\)
  • Reserved seats: \(\mathrm{60}\)
  • Student seats are \(\mathrm{20\%}\) of NON-RESERVED seats (not total seats)
• What we need to find: Number of seats sold to students

2. INFER the solution strategy

• Key insight: We can't directly take \(\mathrm{20\%}\) of all \(\mathrm{450}\) seats
• We must first find how many seats were available (not reserved)
• Then calculate \(\mathrm{20\%}\) of those available seats

3. SIMPLIFY to find non-reserved seats

• \(\mathrm{Available\ seats = Total\ seats - Reserved\ seats}\)
• \(\mathrm{Available\ seats = 450 - 60 = 390\ seats}\)

4. SIMPLIFY to find student seats

• Student seats = \(\mathrm{20\%}\) of available seats
• Convert percentage: \(\mathrm{20\% = 0.20}\)
• Student seats = \(\mathrm{0.20 \times 390 = 78\ seats}\)

Answer: B) 78

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread the problem and calculate \(\mathrm{20\%}\) of ALL \(\mathrm{450}\) seats instead of just the non-reserved seats.

They think: "\(\mathrm{20\%}\) of \(\mathrm{450 = 0.20 \times 450 = 90\ seats}\)"

This leads them to select Choice C (90) without realizing they skipped the crucial first step of subtracting reserved seats.

Second Most Common Error:

Poor INFER reasoning: Students correctly find the \(\mathrm{390}\) non-reserved seats but then get confused about what to do next and mistakenly calculate something other than \(\mathrm{20\%}\) of \(\mathrm{390}\).

Some might try: "\(\mathrm{390 - 20\% = 390 - 78 = 312}\)" or make other logical errors about what the \(\mathrm{20\%}\) represents.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can carefully parse multi-step percentage problems. The key challenge is recognizing that the percentage applies to a subset (non-reserved seats) rather than the original total. Students who rush through the TRANSLATE step often miss this crucial distinction.