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Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 59% (02:32) correct 41% (02:33) wrong based on 524 sessions

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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

Originally posted by jakolik on 06 Aug 2010, 01:42.
Last edited by Bunuel on 02 Dec 2012, 04:19, edited 1 time in total.
Renamed the topic and edited the question.
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Posts: 59266
Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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9
arps 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

Given that S+R=B+2, where S, R, and B are times in which Stephanie, Regine, and Brian completed the race.

Min time one could complete the race is 20/8=2.5 hours. Let's see if Brian could have won the race: if he ran at the 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 reason to distinguish Stephanie and Regine so if one could have won the race, another also could. So both could have won the race.

To elaborate more: the least time one could complete the race is 20/8=2.5 hours, hence $$S+R\geq{5}$$. Let's see if Brian could have won the race: best chances to win he would have if he ran at the fastest rate, so 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, but we know that $$S+B\geq{5}$$, so even if Brian ran at his fastest rate to win the race, given equation S+R=B+2 can not hold true. Hence Brian could not have won the race.

Hope it's clear.
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Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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15
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
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 59266
Re: Speed and Race Problem  [#permalink]

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7
6
jakolik 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 only
(D) I or II only
(E) I, II, or III

Given that S+R=B+2, where S, R, and B are times in which Stephanie, Regine, and Brian completed the race.

Min time one could complete the race is 20/8=2.5 hours. Let's see if Brian could have won the race: if he ran at the 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 reason to distinguish Stephanie and Regine so if one could have won the race, another also could. So both could have won the race.

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Re: Speed and Race Problem  [#permalink]

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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. - I still don't get this line , please explain
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Re: Speed and Race Problem  [#permalink]

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5
ichha148 wrote:
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. - I still don't get this line , please explain

The least time one could complete the race is 20/8=2.5 hours, hence $$S+R\geq{5}$$. Let's see if Brian could have won the race: best chances to win he would have if he ran at the fastest rate, so 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, but we know that $$S+B\geq{5}$$, so even if Brian ran at his fastest rate to win the race, given equation S+R=B+2 can not hold true. Hence Brian could not have won the race.
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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1
Ans is D
since s=d/t or t=d/s so lower the speed more the time
since combined time of Stephanie and Regine exceeds that of Brian so Brian was the slowest.
Thus someone between Stephanie and Regine might have won the race
Intern  Joined: 24 Jan 2011
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Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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We don’t know who has won the race and who has come last. All we know is the fastest time is 2.5 hrs and the fastest speed is 8mi/hr.
Let’s us assume that Brain won the race. He took 2.5 hrs to complete the race. In this case
S + R = 4.5
Now the minimum possible time S and R could have taken would be 2.25 hrs to complete the race, since if they took any time less than 2.25 hrs (which is already less than 2.5hrs), then it would mean that Brian wasn’t the first to win the race.

Note that if S (or R) took more than 2.5 hrs, it means R (or S) took less than 2.5 hrs, and hence Brian wouldn’t be winner in that case.
Now even if they took 2.25 hrs, still the case is not possible, because Brian needs to take the lowest time in order to be winner. So we come to conclusion that Brian cannot be a winner in any case.

Let’s assume that Stephanie won the race in record time of 2.5 hrs. In that case
B = R + 0.5
Note that even if R took the minimum possible time of (2.6 hrs), then Brian would be 3.1 hrs and Stephanie would be 2.5 hrs. This case is very much possible. So S came first, then R and B came last.
The same is true with
B = S + 0.5
R came first, S and then B.
So we see that B is a loser in any case, however R or S have equal chances to win.

Thanks
Prashant
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Re: Speed and Race Problem  [#permalink]

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5
1
A person with a speed of 8 miles per hour will finish the 20 mile race in:
$$t=20/8=5/2=2.5hrs$$

There is no person faster than that so the time of S, R and B must be at least 2.5 hours.

Let S, R and B be the time it took each person to finish the race.
S + R = B + 2

So the minimum possible value for S+R = 2.5 + 2.5 = 5 hrs
Therefore, the minimum possible value for B = 5-2 = 3 hrs.

Let us test:
B=3, S=2.5,R=2.5 ==> S and R could win.
B=4, S=3,R=3 ==> B still could not win. Even if we force either S or R to exceed B, the balance will still make B lose.

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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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Can someone clarify this.

1. Stephanie, Regine, and Brian ran a 20 mile race - [D = 20miles]
2. Stephanie and Regine's combined times exceeded Brian's time by exactly 2 hours - [S + R = B + 2]
3. nobody ran faster than 8 miles per hour - [Max Speed = 20/8 = 2.5m/h]

What if
A. S = 2, R = 1.8 then B = 1.8 so S would win the race.
B. S = 1.8, R = 2 then B = 1.8 so B would win the race.
C. S = 2.1, R = 2.1 then B = 2.2 so B would win the race.

Why not option E? Where am i going wrong?
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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Can someone clarify this.

1. Stephanie, Regine, and Brian ran a 20 mile race - [D = 20miles]
2. Stephanie and Regine's combined times exceeded Brian's time by exactly 2 hours - [S + R = B + 2]
3. nobody ran faster than 8 miles per hour - [Max Speed = 20/8 = 2.5m/h]

What if
A. S = 2, R = 1.8 then B = 1.8 so S would win the race.
B. S = 1.8, R = 2 then B = 1.8 so B would win the race.
C. S = 2.1, R = 2.1 then B = 2.2 so B would win the race.

Why not option E? Where am i going wrong?

We are told that nobody ran faster than 8 miles per hour, thus the maximum speed is 8 miles per hour not 2.5 miles per hour. Also, it would mean that minimum time one could complete the race is 20/8=2.5 hours, thus your examples are not valid.
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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Isn't this a much easier way to solve this?
I may be wrong:

There are 3 times, Ts Tr and (Ts+Tr-2). Now, we're told that the distance is 20 miles and the max they could have run is 8 m/h.
Let's say Brian beat Stephanie, and we'll start with Brian because Brian has the funkiest time formula. You would have
Ts+Tr-2<Ts, picking Stephanie for no reason, we can do this for Regine as well. You will get Tr<2hours. That means Regine is going 20miles/<2hours, which is >8m/h, which is not allowed. So he cannot beat Regine, and if you just look at the formula you can see he can't beat Stephanie for the same reason. So Brian cannot win. A quick comparison of 20/Ts < 20/Tr will show it is perfectly possible for S to beat R and vice versa.
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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2
I did it this way-

Let ts=time of steph
tr=time of reg
tb=time of brian

ts+tr=tb+2

From the problem its clear that Brian takes more time.
Since brian takes more time he could not have won the race. (speed and time are inverse proportional)
We have no info about the other 2.So, anyone could have won the race.
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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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

Time (s) + time (r) = time (b) +2

d=20

If no one ran faster than 8 miles/hour than no one completed the race in under 2.5 hours (time = distance/rate ===> time = 20/8 ===> time = 2.5 hours)

If S+R = B+2, Brian's time will always be greater than S or R. Because no one finishes in under 2.5 hours we can rule out maximum speeds. For example, Brian can not finish in 2.5 hours because S and R would have to have a combined time of 4.5 hours which we know is not possible because 4.5 combined would mean both of them ran it in under 2.5 hours. If Brian finished in 3 hours S and R's time could have been 2.5 each meaning their combined time is exactly two hours greater but their times separately are less than Brian's. The only way the equation can hold true is if Brian's time is equal to or greater than three in which case either S or R (or both) always run in less time.

ANSWER: (D) I or II only
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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Bunuel wrote:
arps 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

Given that S+R=B+2, where S, R, and B are times in which Stephanie, Regine, and Brian completed the race.

Min time one could complete the race is 20/8=2.5 hours. Let's see if Brian could have won the race: if he ran at the 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 reason to distinguish Stephanie and Regine so if one could have won the race, another also could. So both could have won the race.

To elaborate more: the least time one could complete the race is 20/8=2.5 hours, hence $$S+R\geq{5}$$. Let's see if Brian could have won the race: best chances to win he would have if he ran at the fastest rate, so 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, but we know that $$S+B\geq{5}$$, so even if Brian ran at his fastest rate to win the race, given equation S+R=B+2 can not hold true. Hence Brian could not have won the race.

Hope it's clear.

Hi Bunuel

Why so S+R>=5 ?????

Rgds
Prasannajeet
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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1
1
prasannajeet wrote:
Bunuel wrote:
arps 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

Given that S+R=B+2, where S, R, and B are times in which Stephanie, Regine, and Brian completed the race.

Min time one could complete the race is 20/8=2.5 hours. Let's see if Brian could have won the race: if he ran at the 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 reason to distinguish Stephanie and Regine so if one could have won the race, another also could. So both could have won the race.

To elaborate more: the least time one could complete the race is 20/8=2.5 hours, hence $$S+R\geq{5}$$. Let's see if Brian could have won the race: best chances to win he would have if he ran at the fastest rate, so 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, but we know that $$S+B\geq{5}$$, so even if Brian ran at his fastest rate to win the race, given equation S+R=B+2 can not hold true. Hence Brian could not have won the race.

Hope it's clear.

Hi Bunuel

Why so S+R>=5 ?????

Rgds
Prasannajeet

Nobody ran faster than 8 miles per hour --> the least time one could complete the race is 20/8=2.5 hours --> S+R>=2.5+2.5.

Hope it's clear.
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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3
arps 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

It's a good conceptual question. Here are my thoughts on it:

8 mph implies that each person took at least 20/8 = 2.5 hrs. They could have taken more time too.
So Stephanie and Regina's combined time is at least 2.5*2 = 5 hrs. So Brian's time taken is at least 3 hrs. Can Brian win? No. The difference between S and R's combined time and Brian's time is 2 hrs but each person takes more than 2.5 hrs.

If S and R together took 5 hrs, B took 3 hrs. Both S and R must have taken 2.5 hrs each.
If S and R together took 6 hrs, B took 4 hrs. Both S and R must have taken less than 4 hrs since each person takes at least 2.5 hrs.
If S and R together took 10 hrs, B took 8 hrs. Both S and R must have taken less than 8 hrs since each person takes at least 2.5 hrs.
Brian could never win.
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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I almost made a stupid mistake on this one, but quickly realized that it's obviously D.

Brian would need to complete it in 2.5 hours, meaning the other two would need to complete it in 4.5 combined, which would mean that they both would've been faster than the 8m/h limit given.

I almost stupidly confused mph with overall time!
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Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie  [#permalink]

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We know t1+t2=t3 +2 -- (1)

where t1, t2 and t3 are the time taken by Stephanie, Regine and Brian to complete the race.
Since the minimum value for the time taken is 2.5, t1 or t2 has to be lesser than t3 for equation 1 to hold true. So Brian could not have won the race. But we can find values for t1 and t2 such that stephanie and Regine could have won the race satisfying equation (1).
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Holistic and Systematic Approach Re: Stephanie, Regine, and Brian ran a 20 mile race. Stephanie   [#permalink] 23 Jun 2014, 22:19

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