A cargo helicopter delivers only 100-pound packages and 120-pound packages. For each delivery trip, the helicopter must carry at least...

1. TRANSLATE the problem constraints into mathematical language

• Given information:
  • Helicopter carries only 100-pound and 120-pound packages
  • Must carry at least 10 packages total
  • Total weight can be at most 1,100 pounds
  • Want to maximize 120-pound packages

• Let \(\mathrm{a}\) = number of 120-pound packages, \(\mathrm{b}\) = number of 100-pound packages
• This gives us: \(\mathrm{120a + 100b \leq 1,100}\) (weight constraint)
• And: \(\mathrm{a + b \geq 10}\) (package count constraint)

2. INFER the optimization strategy

• To maximize \(\mathrm{a}\) (120-pound packages), we need to minimize \(\mathrm{b}\) (100-pound packages)
• From \(\mathrm{a + b \geq 10}\), we get \(\mathrm{b \geq 10 - a}\)
• So the minimum value of \(\mathrm{b}\) is exactly \(\mathrm{(10 - a)}\)

3. SIMPLIFY by substituting the minimum b value

• Substitute \(\mathrm{b = 10 - a}\) into the weight constraint:
• \(\mathrm{120a + 100(10 - a) \leq 1,100}\)
• \(\mathrm{120a + 1000 - 100a \leq 1,100}\)
• \(\mathrm{20a + 1000 \leq 1,100}\)
• \(\mathrm{20a \leq 100}\)
• \(\mathrm{a \leq 5}\)

4. APPLY CONSTRAINTS to find the maximum

• Since \(\mathrm{a}\) must be a whole number (can't have fractional packages)
• The maximum number of 120-pound packages is 5

Answer: C. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret "at most" and "at least," setting up inequalities with wrong direction signs (\(\mathrm{\lt}\) instead of \(\mathrm{\leq}\), or \(\mathrm{\geq}\) instead of \(\mathrm{\leq}\)). They might write \(\mathrm{120a + 100b \geq 1,100}\) thinking the helicopter needs to carry exactly 1,100 pounds or more.

This leads to completely wrong constraints and typically causes them to get stuck and guess randomly.

Second Most Common Error:

Poor INFER reasoning: Students correctly set up the inequalities but don't realize they need to minimize \(\mathrm{b}\) to maximize \(\mathrm{a}\). Instead, they might try to solve the system without this key insight, leading to multiple possible solutions without identifying the maximum value of \(\mathrm{a}\).

This may lead them to select Choice B (4) by testing values randomly rather than systematically finding the maximum.

The Bottom Line:

This problem requires students to translate optimization language into mathematical constraints and then use substitution strategically - two distinct skills that must work together for success.