If I get 8 miles per gallon out of my car during the first leg of a journey but for some reason only get 4 miles per gallon during the second leg, what is my car’s performance over the entire trip?
Many students will add together the values given in the problem, 8 + 4 = 12, and conclude the rate is 12.
A more sophisticated response recognizes that two variables are involved. A student finds that the first leg has 8 miles for 1 gallon, the second leg has 4 miles for 1 gallon, so together you have 4 + 8 = 12 miles and 1 + 1 = 2 gallons, so the rate is 12 miles per 2 gallons, or 6 miles per gallon. However, this second method is also incorrect, for it assumes that the trip is exactly 8 miles for the first leg and 4 for the second leg. The number of miles traveled in each leg of the trip is not given in the problem, and so the problem is not solvable.
When working with rates problems, students frequently have trouble recognizing that a rate is a numerical description of a relationship between two varying quantities. In this example, it is common for students to confuse a rate of consumption with the actual quantity of consumption. Namely, if the car consumed gas at a rate of 8 miles per gallon, that does not mean that the car traveled only 8 miles.