Marisol drove 3 hours from City A to City B. The equation below estimates the distance d, in miles, Marisol...

1. TRANSLATE the problem information

• Given information:
  • Marisol drove for 3 hours total
  • Equation: \(\mathrm{d = 45t}\)
  • \(\mathrm{d}\) = distance in miles after \(\mathrm{t}\) hours
  • We need to identify what 45 represents

2. INFER the connection to distance formulas

• The equation \(\mathrm{d = 45t}\) follows the pattern of the distance formula
• Standard distance formula: distance = rate × time, or \(\mathrm{d = rt}\)
• Comparing \(\mathrm{d = 45t}\) to \(\mathrm{d = rt}\), we can see that 45 takes the place of \(\mathrm{r}\) (rate)

3. TRANSLATE the units to confirm

• Since \(\mathrm{d}\) is in miles and \(\mathrm{t}\) is in hours
• The coefficient 45 must have units of miles/hour
• This confirms 45 represents speed (miles per hour)

Answer: C. Marisol drove at an average speed of about 45 miles per hour

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students see the number 45 and connect it to other numbers in the problem rather than understanding its role as a coefficient in the equation.

They might think "45 must be related to the total distance" and calculate 45 miles as the total distance, leading them to select Choice B (45 miles total distance). They fail to recognize that if the total distance were 45 miles and the trip took 3 hours, then \(\mathrm{d = 45}\) when \(\mathrm{t = 3}\), which would mean \(\mathrm{45 = 45(3) = 135}\), creating a contradiction.

Second Most Common Error:

Missing conceptual knowledge about the distance formula: Students don't connect \(\mathrm{d = 45t}\) to the standard \(\mathrm{d = rt}\) formula, so they can't identify what role 45 plays.

This leads to confusion about what coefficients mean in linear equations, causing them to guess among the remaining choices or select Choice D (45 hours) by mixing up the coefficient with the time variable.

The Bottom Line:

This problem tests whether students can interpret coefficients in linear equations within real-world contexts, specifically recognizing rate in the distance formula.