The distance a train travels between 2 cities is 540 miles. The train travels at a speed of 80 miles per hour. Matt set up these rates to find the time it took the train to travel between the 2 cities.

Question

80 miles
1 hour
= ? hours
540 miles

What was Matt’s error?
The first rate should be written as 80 hours to 1 mile.
The second rate should be written as a rate of distance to time.
The first rate should compare miles to miles.
Matt did not make an error.

Answer 1

Matt's setup actually does contain an error. The correct interpretation of the train's speed should express the relationship between distance and time in a clear way.

The correct rate to represent how distance relates to time is typically expressed in the format of distance to time (miles per hour), which is what Matt did. However, his framing in the question seems misleading in that it suggests a direct correlation of hours to miles that isn't quite appropriate.

The error in logic can be based on the structure of his representation:

The second rate should be written as a rate of distance to time.

This is because rates should equate the distance traveled to the time taken. Hence, the train’s speed of 80 miles per hour can properly be set against the total distance (540 miles) to find the time taken.

To calculate the time it took, Matt could find the time using the formula:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{540 \text{ miles}}{80 \text{ miles/hour}} = 6.75 \text{ hours} \]

So while he didn't make a mathematical error, the framing of how the rates are compared could have been clearer. Thus, it's mostly about how he expressed the relationship rather than a computational error.