A car's onboard computer models the remaining driving distance, in miles, with the function \(\mathrm{d(g) = 460 - 22g}\), where...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{d(g) = 460 - 22g}\)
  • \(\mathrm{d}\) = remaining driving distance in miles
  • \(\mathrm{g}\) = gallons of gasoline already used
• Question asks: How many gallons used per mile driven?

2. INFER what the function coefficients mean

• This is a linear function with \(\mathrm{slope = -22}\)
• The slope tells us: for each 1 gallon used, remaining distance decreases by 22 miles
• This means the car travels 22 miles for every 1 gallon used
• In other words: 22 miles per gallon

3. INFER the relationship needed

• We found: 22 miles per gallon
• Question asks for: gallons per mile
• These are reciprocal relationships: gallons per mile = \(\mathrm{1/(miles\text{ }per\text{ }gallon)}\)

4. SIMPLIFY to find the answer

• Gallons per mile = \(\mathrm{1/22}\)
• Convert to decimal (use calculator): \(\mathrm{1 ÷ 22 \approx 0.045}\)

Answer: (B) 0.045

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret what the slope coefficient represents. They might think the -22 directly gives gallons per mile, leading them to select Choice (C) (0.22) by dropping the negative sign, or Choice (D) (22) by ignoring the negative entirely.

Second Most Common Error:

Missing reciprocal relationship: Students correctly identify that the car gets 22 miles per gallon but fail to recognize that the question asks for gallons per mile (the reciprocal). This leads them to think the answer should be related to 22, causing them to select Choice (D) (22).

The Bottom Line:

This problem tests whether students can extract rate information from linear function slopes and understand reciprocal rate relationships. The key insight is that function coefficients often represent rates in real-world contexts, and students must be flexible in converting between different rate expressions.