A delivery truck travels at 40;mph on highways and 24;mph in city traffic. On a particular route, the truck drove...

1. TRANSLATE the problem information

• Given information:
  • Highway speed: \(40\text{ mph}\)
  • City traffic speed: \(24\text{ mph}\)
  • Highway distance: \(x\text{ miles}\)
  • City traffic distance: \(y\text{ miles}\)
  • Total driving time: \(5.5\text{ hours}\)

• What we need: An equation representing this situation

2. INFER the approach

• Since we have distances and speeds, we need to find time for each segment
• Total time = highway time + city traffic time
• Use the relationship: \(\text{time} = \frac{\text{distance}}{\text{rate}}\)

3. TRANSLATE each time component

• Time on highways = distance ÷ rate = \(\frac{x\text{ miles}}{40\text{ mph}} = \frac{x}{40}\text{ hours}\)
• Time in city traffic = distance ÷ rate = \(\frac{y\text{ miles}}{24\text{ mph}} = \frac{y}{24}\text{ hours}\)

4. INFER the final equation

• Total time = highway time + city traffic time
• \(5.5 = \frac{x}{40} + \frac{y}{24}\)
• Therefore: \(\frac{x}{40} + \frac{y}{24} = 5.5\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which distance goes with which speed, writing the times as \(\frac{x}{24} + \frac{y}{40}\) instead of \(\frac{x}{40} + \frac{y}{24}\).

They might think "x goes with the first speed mentioned (24 mph)" rather than carefully matching highway distance (x) with highway speed (40 mph) and city distance (y) with city speed (24 mph).

This leads them to select Choice A (\(\frac{x}{24} + \frac{y}{40} = 5.5\)).

Second Most Common Error:

Conceptual confusion about the time formula: Students forget that \(\text{time} = \frac{\text{distance}}{\text{rate}}\) and instead multiply distance by rate, thinking this gives time.

This incorrect reasoning leads to distance × rate = time, giving them \(40x + 24y = 5.5\). When they see this doesn't match exactly, they might select Choice B (\(24x + 40y = 5.5\)) as the closest option.

The Bottom Line:

This problem requires carefully translating word descriptions into correct mathematical relationships while keeping track of which quantity corresponds to which variable. The key insight is recognizing that time calculations require dividing distance by rate, not multiplying.