MATH ITEM:

On a race track exactly one mile in circumference, a driver completes the first lap at a constant rate of 60 miles per hour and the second lap at a constant rate of 30 miles per hour. Find the driver’s average speed, in miles per hour, for the two laps.

(A) 30 mph
(B) 40 mph
(C) 45 mph
(D) 50 mph
(E) 90 mph

MATH EXPLANATION:

Correct Answer: (B)
A common test-taking error is choosing an "obvious" answer to a difficult problem. The answer here is not something as obvious as (60 + 30)/2 = 45. That’s too easy. Instead, recall that:

Distance = (Rate)(Time)

From this we can conclude the following:

Time = (Distance)/(Rate)
Average Rate = (Total Distance)/(Total Time)

How long does it take for the driver to turn two laps?

Time (first lap) = Distance/Rate = 1/60
Time (second lap) = Distance/Rate = 1/30
Total time = 1/60 + 1/30 = 1/20

Now we’re ready to solve the problem:

Total distance = 2 miles
Average Rate = (Total Distance)/(Total Time) = 2/(1/20) = 40

Too much arithmetic? Just use a little common sense. Since the driver spent less time going fast and more time going slow, the average of the two has to be slower rather than faster-slower, that is, than the average of 60 and 30, which is 45 mph. So, you can eliminate (C), (D), and (E). Then, the driver went 60 mph for part of the time, so the average couldn’t be 30 mph. Eliminate (A). The correct answer has to be (B), 45 mph.

Another method is to use the total time and see how many laps could be driven at an average rate of 45 mph (the midpoint value of the answer choices, as well as the most common error). Distance = (Rate)(Time), so Distance = (45)(1/20) = 2.25 miles. This is more than the 2 miles around the track, so the rate must be less than 45 mph. Eliminate (C), (D), and (E). We can also eliminate (A) for the reason mentioned above—since the driver went more than 30 mph (60 mph) for part of the trip, the average must be above 30 mph. Thus, (B) is the correct answer.