Stephanie, Regine, and Brian ran a 20 mile race. Stephanie and Regine's combined times exceeded Brian's time by exactly 2 hours. If nobody ran faster than 8 miles per hour, who could have won the race?
I. Stephanie II. Regine III. Brian
(A) I only (B) II only (C) III only (D) I or II only (E) I, II, or III
Whenever a problem gives you an upper or lower limit, plug in the limit in order to see how the problem is restricted.
In this problem, our upper limit is 8mph. No one is allowed to have a faster rate.
Let's start with Brian. Let's say that he wins by running at the fastest allowed speed of 8 mph.
Time = Distance/Rate
Brian's time would be 20/8 = 2.5 hours.
Stephanie and Regine's combined times exceeded Brian's time by exactly 2 hours: This means Stephanie and Regine's combined time would be 2.5 + 2 = 4.5 hours.
In this case, the upper limit for Stephanie and Regine also is 8mph. Neither can run faster because we want Brian to win. Let's see what happens when Stephanie and Regine each run at 8mph.
Stephanie's time would be 20/8 = 2.5 hours.
Regine's time would be 20/8 = 2.5 hours.
Their combined time would be 2.5 + 2.5 = 5 hours.
Too much, because we need their combined time to be 4.5 hours.
But the only way for their combined time to be 4.5 hours is if they run
faster. But they can't run faster because we want Brian to win.
So Brian can't win by going at the maximum rate of 8mph.
If Brian goes slower, the situation gets worse:
Let's say Brian runs at 5 mph.
Brian's time would be 20/5 = 4 hours.
This means Stephanie and Regine's combined time would be 4 + 2 = 6 hours.
In this case, the upper limit for Stephanie and Regine is 5mph. Neither can run faster because we want Brian to win. Let's see what happens when Stephanie and Regine each run at 5mph.
Stephanie's time would be 20/5 = 4 hours.
Regine's time would be 20/5 = 4 hours.
Their combined time would be 4 + 4 = 8 hours.
Too much, because we need their combined time to be 6 hours.
But the only way for their combined time to be 6 hours is if they run
faster. But they can't run faster because we want Brian to win.
So we're stuck. Brian can't win, poor guy.
Eliminate any answer choice that includes Brian: C and E. We're left with A, B, and D.
"None" is not included in the answer choices, so we know that someone has to be able to win. The problem makes no distinction between Stephanie and Regine; we know information only about their
combined time. If Stephanie can win, why couldn't Regine? If Regine can win, why couldn't Stephanie? So either must be able to win.
The correct answer is
D.
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