Manhattan Challenge Problem of the week! - 09/04/07

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

Answer: D

Let the race time of Stephanie = S
Let the race time of Regine = R
Let the race time of Brian = B

It is given that -- Stephanie and Regine's combined times exceed Brian's time by 2 hours.

Hence S + R = B + 2 ------- (1)

To win the race, an individual's time must be less than 1/3 of the the combined times of all the runners.

Therefore for Brian to win the race (=> that Brian would have the lowest time) his time would need to be less than 1/3 of the combined times for all the runners.

=> B is less than 1/3 of (S+R+B)

=>3B<> is less than (S+R+B)

=> 2B is less than (S+R)

Making use of (1) in above

2B<> is less than (S+R)

=> 2B<> is less than B+2

=> B<2


Thus to win the race Brian's time must be less than 2 hours. which is impossible as fastest Brian run is 8 miles/hr => that the least amount of time in which he can complete the 20 mile race is 2.5 hrs.

Hence Stephanie and Regine as possible winners. Since the question gives us same information about Stephanie and Regine, we cannot state either one as a possible winner.

Hence, the correct answer is D