On a 210-mile trip, Cameron drove at an average speed of 60 miles per hour for the first x hours....

1. TRANSLATE the problem information

• Given information:
  • Total trip distance: 210 miles
  • First part: \(60\text{ mph}\) for x hours
  • Second part: \(50\text{ mph}\) for y hours
  • \(\mathrm{x = 1}\)

• What this tells us: We need to find how the two parts of the trip add up to the total distance.

2. TRANSLATE into mathematical equation

• Using \(\mathrm{Distance = rate \times time}\):
  • First part distance: \(60\mathrm{x}\text{ miles}\)
  • Second part distance: \(50\mathrm{y}\text{ miles}\)
  • Total: \(60\mathrm{x} + 50\mathrm{y} = 210\)

3. SIMPLIFY by substituting the known value

• Since \(\mathrm{x = 1}\):
  \(60(1) + 50\mathrm{y} = 210\)
  \(60 + 50\mathrm{y} = 210\)

4. SIMPLIFY to solve for y

• Subtract 60 from both sides:
  \(50\mathrm{y} = 210 - 60\)
  \(50\mathrm{y} = 150\)

• Divide both sides by 50:
  \(\mathrm{y} = 150 \div 50 = 3\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct distance equation from the word problem description. They might write separate equations like "\(60\mathrm{x} = \text{distance}_1\)" and "\(50\mathrm{y} = \text{distance}_2\)" without connecting them to the total distance of 210 miles.

This leads to confusion about how to proceed and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the solving process, such as incorrectly calculating \(210 - 60 = 140\) instead of 150, or dividing \(150 \div 50 = 4\) instead of 3.

This may lead them to select an incorrect answer choice if multiple choice options are available.

The Bottom Line:

This problem tests whether students can translate a real-world distance scenario into a mathematical equation and then execute basic algebraic solving. The key insight is recognizing that both parts of the trip contribute to the same total distance.