The time required to travel 80 miles at a speed of x miles per hour is 8 hours more than...

1. TRANSLATE the problem information

• Given information:
  • Distance: \(\mathrm{80}\) miles (same for both scenarios)
  • Speed 1: \(\mathrm{x}\) mph
  • Speed 2: \(\mathrm{(x + 5)}\) mph
  • Time at speed \(\mathrm{x}\) is \(\mathrm{8}\) hours MORE than time at speed \(\mathrm{(x + 5)}\)

2. TRANSLATE the relationship into equations

• Time = Distance ÷ Speed, so:
  • Time at speed \(\mathrm{x}\): \(\mathrm{\frac{80}{x}}\) hours
  • Time at speed \(\mathrm{(x + 5)}\): \(\mathrm{\frac{80}{x + 5}}\) hours
• The key relationship: \(\mathrm{\frac{80}{x} = \frac{80}{x + 5} + 8}\)

3. SIMPLIFY by rearranging to standard form

• Move terms: \(\mathrm{\frac{80}{x} - \frac{80}{x + 5} = 8}\)
• Find common denominator: \(\mathrm{\frac{80(x + 5) - 80x}{x(x + 5)} = 8}\)
• Simplify numerator: \(\mathrm{\frac{80x + 400 - 80x}{x(x + 5)} = 8}\)
• This gives us: \(\mathrm{\frac{400}{x(x + 5)} = 8}\)

4. SIMPLIFY by clearing the fraction

• Cross multiply: \(\mathrm{400 = 8x(x + 5)}\)
• Expand: \(\mathrm{400 = 8x^2 + 40x}\)
• Divide by \(\mathrm{8}\): \(\mathrm{50 = x^2 + 5x}\)
• Standard form: \(\mathrm{x^2 + 5x - 50 = 0}\)

5. SIMPLIFY using the quadratic formula

• For \(\mathrm{x^2 + 5x - 50 = 0}\): \(\mathrm{a = 1, b = 5, c = -50}\)
• \(\mathrm{x = \frac{-5 \pm \sqrt{25 + 200}}{2}}\)
• \(\mathrm{x = \frac{-5 \pm \sqrt{225}}{2}}\)
• \(\mathrm{x = \frac{-5 \pm 15}{2}}\)
• Two solutions: \(\mathrm{x = \frac{-5 + 15}{2} = 5}\) and \(\mathrm{x = \frac{-5 - 15}{2} = -10}\)

6. APPLY CONSTRAINTS to select final answer

• Since speed must be positive in real-world context: \(\mathrm{x = 5}\) mph

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to correctly set up the relationship "8 hours more than." They might write \(\mathrm{\frac{80}{x} = \frac{80}{x + 5} - 8}\) instead of \(\mathrm{\frac{80}{x} = \frac{80}{x + 5} + 8}\), reversing the logic of which time is greater.

This fundamental setup error leads to the quadratic equation \(\mathrm{x^2 + 5x + 50 = 0}\), which has no real solutions, causing confusion and forcing them to guess randomly among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the initial equation but make algebraic errors when clearing fractions or expanding terms. Common mistakes include sign errors when distributing or calculation errors when applying the quadratic formula.

These computational mistakes often lead to incorrect values that don't match any answer choice, again leading to guessing.

The Bottom Line:

This problem requires careful translation of the English comparison into mathematical relationships, followed by systematic algebraic manipulation. The key insight is recognizing that "8 hours more" means the slower speed \(\mathrm{(x)}\) takes longer, so its time equals the faster speed's time PLUS \(\mathrm{8}\) hours.