Jay walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour....

Step-by-Step Solution

1. TRANSLATE the problem information

• Given information:
  • Walking speed: 3 miles per hour
  • Walking time: \(\mathrm{w}\) hours
  • Running speed: 5 miles per hour
  • Running time: \(\mathrm{r}\) hours
  • Combined total distance: 14 miles

• What this tells us: We need to find how much distance is covered by each activity and sum them.

2. INFER the approach

• Since we know rate and time for each activity, we can find the distance for each using: \(\mathrm{distance = rate \times time}\)
• The total distance will be the sum of walking distance plus running distance
• This sum equals 14 miles, giving us our equation

3. TRANSLATE each activity into distance

• Walking distance = \(\mathrm{3\,mph \times w\,hours = 3w\,miles}\)
• Running distance = \(\mathrm{5\,mph \times r\,hours = 5r\,miles}\)

4. INFER the final equation

• Total distance = Walking distance + Running distance
• \(\mathrm{14 = 3w + 5r}\)
• Therefore: \(\mathrm{3w + 5r = 14}\)

Answer: A. \(\mathrm{3w + 5r = 14}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the \(\mathrm{distance = rate \times time}\) relationship and think they need to find time instead of distance. They might incorrectly think "\(\mathrm{time = distance/rate}\)" and set up something like \(\mathrm{14/3w + 14/5r}\), leading to expressions involving fractions.

This may lead them to select Choice B (\(\mathrm{1/3w + 1/5r = 14}\)) by incorrectly thinking they need reciprocals of the rates.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that they need to use \(\mathrm{distance = rate \times time}\) but misread the total distance as 112 instead of 14, possibly misinterpreting the problem statement.

This may lead them to select Choice D (\(\mathrm{3w + 5r = 112}\)) by using the correct formula structure but wrong total.

The Bottom Line:

This problem tests whether students can correctly apply the distance-rate-time relationship in a real-world context. The key insight is recognizing that when you know rate and time, you multiply to get distance, then add the distances from different activities to get the total.