A discount airline sells a certain number of tickets, x, for $90 each. It sells the number of remaining tickets,...

1. TRANSLATE the problem information

• Given information:
  • x tickets cost \($90\) each
  • y tickets cost \($250\) each
  • Total tickets sold: 120
  • Total money collected: \($27,600\)

• What this tells us: We have two different constraints that must both be satisfied simultaneously.

2. INFER the approach

• We need two separate equations because we have two separate constraints
• First constraint: total number of tickets
• Second constraint: total amount of money collected

3. TRANSLATE each constraint into an equation

For total tickets:

• x tickets + y tickets = 120 total tickets
• Equation 1: \(\mathrm{x + y = 120}\)

For total revenue:

• Money from x tickets: \($90 \times \mathrm{x} = \mathrm{90x}\) dollars
• Money from y tickets: \($250 \times \mathrm{y} = \mathrm{250y}\) dollars
• Total money collected: \(\mathrm{90x + 250y = $27,600}\)
• Equation 2: \(\mathrm{90x + 250y = 27,600}\)

4. INFER the final system

Our system of equations is:

• \(\mathrm{x + y = 120}\)
• \(\mathrm{90x + 250y = 27,600}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students correctly identify that they need \(\mathrm{x + y = 120}\), but then mistakenly think the total revenue should be \(\mathrm{120(27,600)}\) instead of just \(\mathrm{27,600}\).

Their reasoning: "Since there are 120 tickets and the total is \($27,600\), maybe I multiply them together?" This shows confusion about what each number represents - they're mixing up the constraint (total tickets = 120) with a calculation involving that constraint.

This may lead them to select Choice B (\(\mathrm{120(27,600)}\)).

Second Most Common Error:

Weak TRANSLATE execution: Students mix up which totals go with which equations, putting \(\mathrm{27,600}\) as the total number of tickets instead of \(\mathrm{120}\).

Their reasoning: "Both 120 and 27,600 are totals, so maybe the bigger number goes with the first equation?" This shows they haven't carefully tracked what each constraint actually represents.

This may lead them to select Choice C (\(\mathrm{x + y = 27,600}\)).

The Bottom Line:

This problem tests whether students can systematically translate multiple real-world constraints into separate mathematical equations. The key insight is recognizing that each constraint (total tickets, total money) requires its own equation with the correct total on the right side.