A delivery truck's fuel efficiency varies with road conditions. On highway drives, the truck achieves between 8 and 12 miles...

1. TRANSLATE the problem information

• Given information:
  • Truck efficiency varies between 8 and 12 miles per gallon
  • Trip distance: 240 miles
  • Need to find: range of gallons (g) required
• Key relationship: \(\mathrm{gallons\ needed = distance ÷ efficiency}\)

2. INFER the approach needed

• To find the range of fuel needed, we must calculate fuel consumption at both efficiency extremes
• Higher efficiency (12 mpg) = less fuel needed
• Lower efficiency (8 mpg) = more fuel needed

3. Calculate fuel needed at maximum efficiency

• At 12 mpg: \(\mathrm{gallons = 240 ÷ 12 = 20\ gallons}\)
• This gives us the minimum fuel needed

4. CONSIDER ALL CASES by calculating the other extreme

• At 8 mpg: \(\mathrm{gallons = 240 ÷ 8 = 30\ gallons}\)
• This gives us the maximum fuel needed

5. TRANSLATE the range into inequality notation

• The truck needs between 20 and 30 gallons (inclusive)
• In inequality form: \(\mathrm{20 \leq g \leq 30}\)

Answer: D \(\mathrm{(20 \leq g \leq 30)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Students calculate fuel needed at only one efficiency level, typically the average or just one extreme.

For example, they might only calculate at 12 mpg (getting 20 gallons) and select Choice B \(\mathrm{(g \leq 20)}\), missing that efficiency can be as low as 8 mpg which would require more fuel.

Second Most Common Error:

Poor INFER execution: Students don't recognize the inverse relationship between efficiency and fuel consumption, potentially thinking higher efficiency means more fuel needed.

This conceptual confusion leads them to set up incorrect calculations or misinterpret which extreme represents minimum vs. maximum fuel needs, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that a range of input values (efficiency) produces a range of output values (fuel needed), and that they must calculate both extremes to capture the complete range.