Stephanie, Regine, and Brian ran a 20 mile race. Stephanie

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.

Answer: D.

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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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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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 cannot hold true. Hence Brian could not have won the race.
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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

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.
So (D) is the answer.

Thanks
Prashant

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.

Answer: I and II only

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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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.

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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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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We see that since nobody ran faster than 8 miles per hour, then each of them took more than 20/8 = 2.5 hours to complete the 20-mile race.

If Stephanie took 2.6 hours and Regine took 2.7 hours to complete the race, then Brian took 2.6 + 2.7 - 2 = 3.3 hours to complete the race. In this case Stephanie is the winner of the race.

If we switch Stephanie’s time with Regine’s, then Regine is the winner of the race.

Finally, if we let Stepanie’s time = 2.5 + s and Regine’s time = 2.5 + r for some positive numbers s and r, then Brian’s time = 2.5 + s + 2.5 + r - 2 = 3 + s + r. We see that Brian took more time than either Stepanie or Regine to complete the race, which means he could never be the winner of the race. So only Stephanie or Regine could be the winner of the race.

Answer: D
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Whenever a problem gives you an upper or lower limit, TEST THE LMIIT to see how the problem is constrained.

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.


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