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Word Problem

by [email protected] » Wed Dec 18, 2013 10:05 pm
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
D. only I or II only
E. I, II, or III

Ans D
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by theCodeToGMAT » Wed Dec 18, 2013 11:03 pm
Time of S = Ts
Time of R = Tr
Time of B = Tb

Ts + Tr = Tb + 2

20/S + 20/R = 20/B + 2
1/S + 1/R - 1/B = 1/10
1/S + 1/R - 1/B = 0.1

S,R,B <= 8

For any value of "S".. RHS will be negative because 1/8 = 0.125 ==>
(B-R)/(BR) = -ve ==> R > B

Similarly,

For any value of "R".. RHS will be negative because 1/8 = 0.125 ==>
(B-S)/(BS) = -ve ==> S > B

Hence, B will never win

Answer [spoiler]{D}[/spoiler]
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by vipulgoyal » Wed Dec 18, 2013 11:07 pm
wow, thanks for good qustion,This is more of a conceptual rather than math

Min time one could complete the race is 20/8=2.5 hours.
if Brian could have won the race with fastest rate, he would complete the race in 2.5 hours,
so combined time needed for Stephanie and Regine would be S+R=B+2=4.5 hours,
which is not possible as sum of two must be more than or equal the twice the least time: 2*2.5=5.
So Brian could not have won the race.
there is NO differance B/W, S and R hence any one could win the race
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by GMATGuruNY » Thu Dec 19, 2013 12:57 am
[email protected] wrote: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
D. only I or II only
E. I, II, or III

Ans D
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).
The correct answer must be A, B, or D.

"None" is not included among the answer choices, so we know that someone must 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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by [email protected] » Thu Dec 19, 2013 5:48 am
Nice explanation Mitch, thanks

GMATGuruNY wrote:
[email protected] wrote: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
D. only I or II only
E. I, II, or III

Ans D
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).
The correct answer must be A, B, or D.

"None" is not included among the answer choices, so we know that someone must 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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by [email protected] » Thu Dec 19, 2013 5:48 am
Nice explanation Mitch, thanks

GMATGuruNY wrote:
[email protected] wrote: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
D. only I or II only
E. I, II, or III

Ans D
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).
The correct answer must be A, B, or D.

"None" is not included among the answer choices, so we know that someone must 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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