On a car trip, Rhett and Jessica each drove for part of the trip, and the total distance they drove...

1. TRANSLATE the problem information

• Given information:
  • Rhett drove at \(\mathrm{35\ mph}\) for \(\mathrm{r\ hours}\)
  • Jessica drove at \(\mathrm{40\ mph}\) for \(\mathrm{j\ hours}\)
  • Total distance was under \(\mathrm{220\ miles}\)

2. TRANSLATE distances using the distance formula

• Since \(\mathrm{Distance = speed \times time}\):
  • Rhett's distance = \(\mathrm{35 \times r = 35r\ miles}\)
  • Jessica's distance = \(\mathrm{40 \times j = 40j\ miles}\)

3. INFER the total distance expression

• Total distance = Rhett's distance + Jessica's distance
• Total distance = \(\mathrm{35r + 40j\ miles}\)

4. TRANSLATE the constraint into mathematical form

• "Under 220 miles" means "less than 220 miles"
• So: \(\mathrm{35r + 40j \lt 220}\)

Answer: B. \(\mathrm{35r + 40j \lt 220}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Mixing up which person drove at which speed

Students might accidentally assign \(\mathrm{40\ mph}\) to Rhett and \(\mathrm{35\ mph}\) to Jessica, creating the expression \(\mathrm{40r + 35j}\) instead of \(\mathrm{35r + 40j}\). This happens when students don't carefully track which variable corresponds to which person while reading through the problem.

This may lead them to select Choice D (\(\mathrm{40r + 35j \lt 220}\)) if they get the inequality direction right, or Choice C (\(\mathrm{40r + 35j \gt 220}\)) if they make both errors.

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing "under" with "over"

Some students misinterpret "under 220 miles" as meaning the distance should be greater than 220 miles, leading them to use \(\mathrm{\gt}\) instead of \(\mathrm{\lt}\). This is often a rushed reading error or confusion about the meaning of "under."

This may lead them to select Choice A (\(\mathrm{35r + 40j \gt 220}\)).

The Bottom Line:

Success on this problem requires careful attention to detail when translating words to math symbols. Students must accurately match speeds to people and correctly interpret comparison words like "under."