During a month, Morgan ran r miles at 5 miles per hour and biked b miles at 10 miles per...

1. TRANSLATE the problem information

• Given information:
  • Morgan ran \(\mathrm{r}\) miles at \(5\) mph
  • Morgan biked \(\mathrm{b}\) miles at \(10\) mph
  • Total distance: \(\mathrm{r + b = 200}\) miles
  • She biked for twice as many hours as she ran

2. INFER the approach

• To work with "twice as many hours," we need to convert distances to time
• Use the relationship: \(\mathrm{Time = Distance \div Speed}\)
• This gives us: Time running = \(\mathrm{r/5}\) hours, Time biking = \(\mathrm{b/10}\) hours

3. TRANSLATE the time relationship into math

• "She biked for twice as many hours as she ran" becomes:
  \(\mathrm{b/10 = 2(r/5)}\)

4. SIMPLIFY the time equation

• \(\mathrm{b/10 = 2r/5}\)
• Multiply both sides by 10: \(\mathrm{b = 10 \times (2r/5) = 4r}\)

5. INFER the solution strategy

• Now we have two equations: \(\mathrm{r + b = 200}\) and \(\mathrm{b = 4r}\)
• Use substitution: replace \(\mathrm{b}\) with \(\mathrm{4r}\) in the distance equation

6. SIMPLIFY to solve for r

• \(\mathrm{r + 4r = 200}\)
• \(\mathrm{5r = 200}\)
• \(\mathrm{r = 40}\) miles

7. SIMPLIFY to find b

• \(\mathrm{b = 4r = 4(40) = 160}\) miles

Answer: D. 160

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "biked for twice as many hours as she ran" into a mathematical equation. They might set up \(\mathrm{b = 2r}\) (thinking twice as many miles instead of hours), missing that they need to work with time, not distance.

This leads them to solve \(\mathrm{r + 2r = 200}\), getting \(\mathrm{r = 66.67}\) and \(\mathrm{b = 133.33}\), which doesn't match any answer choice. This causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{b/10 = 2(r/5)}\) but make algebraic errors when simplifying. They might incorrectly get \(\mathrm{b = 2r}\) or \(\mathrm{b = 8r}\) instead of \(\mathrm{b = 4r}\).

With \(\mathrm{b = 2r}\): they get \(\mathrm{r = 66.67}\), \(\mathrm{b = 133.33}\) (no match)
With \(\mathrm{b = 8r}\): they get \(\mathrm{r = 22.22}\), \(\mathrm{b = 177.78}\) (no match)
This leads to confusion and random answer selection.

The Bottom Line:

This problem requires recognizing that "hours" and "miles" are different quantities, so you must convert between them using the speed relationship. Students who work directly with distances without considering time will struggle to set up the correct equation.