A bus traveled on the highway and on local roads to complete a trip of 160 miles. The trip took...

1. TRANSLATE the problem information

• Given information:
  • Total distance: 160 miles
  • Total time: 4 hours
  • Highway speed: 55 mph
  • Local road speed: 25 mph
  • \(\mathrm{x}\) = time on highway (hours)
  • \(\mathrm{y}\) = time on local roads (hours)
• We need to find which system represents this situation

2. INFER what relationships we need

• We have two key constraints that must both be satisfied:
  • The total distance traveled must equal 160 miles
  • The total time must equal 4 hours
• This means we need two separate equations

3. TRANSLATE the distance relationship

• Distance on highway = speed × time = \(\mathrm{55x}\) miles
• Distance on local roads = speed × time = \(\mathrm{25y}\) miles
• Total distance equation: \(\mathrm{55x + 25y = 160}\)

4. TRANSLATE the time relationship

• Total time = time on highway + time on local roads
• Time equation: \(\mathrm{x + y = 4}\)

5. INFER the final system

• Our system of equations is:
  • \(\mathrm{55x + 25y = 160}\) (distance equation)
  • \(\mathrm{x + y = 4}\) (time equation)
• This matches choice B

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which numbers represent distance versus time, swapping 160 and 4 in their equations.

They might think: "The speeds are 55 and 25, so one equation should equal 4 (because there are 4 hours), and the other should equal 160." This leads them to set up \(\mathrm{55x + 25y = 4}\) and \(\mathrm{x + y = 160}\), which describes an impossible scenario where the trip was only 4 miles but took 160 hours.

This may lead them to select Choice A (\(\mathrm{55x + 25y = 4}\), \(\mathrm{x + y = 160}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which speed goes with which road type, incorrectly assigning 25 mph to highways and 55 mph to local roads.

They correctly identify that distance should equal 160 and time should equal 4, but write \(\mathrm{25x + 55y = 160}\) instead of \(\mathrm{55x + 25y = 160}\). This represents a scenario where highways are slower than local roads.

This may lead them to select Choice D (\(\mathrm{25x + 55y = 160}\), \(\mathrm{x + y = 4}\)).

The Bottom Line:

Success requires carefully tracking what each number in the problem represents (distance vs. time) and which speed corresponds to which type of road. The key insight is that distance = speed × time creates the first equation, while the constraint on total time creates the second equation.