Two friends, Tanaya and Stephen were standing together. Tanaya begins

Question

Two friends, Tanaya and Stephen were standing together. Tanaya begins to walk in a straight line away from Stephen at a constant rate of 3 miles per hour. One hour later, Stephen begins to run in a straight line in the exact opposite direction at a constant rate of 9 miles per hour. If both Tanaya and Stephen continue to travel, what is the positive difference between the amount of time it takes Stephen to cover the exact distance that Tanaya has covered and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?

(A) 60 mins
(B) 72 mins
(C) 90 mins
(D) 100 mins
(E) 120 mins

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I did it like this

Let's assume that by the time Stephen catches up with tanaya she has travelled x miles. In addition she covered 3 miles in 1st hour. Time taken by pooja to travel x miles = time taken by stephen to cover 3+x miles.
so
(3+x)/9=x/3 =>9x=9+3x => 6x=9 => x=3/2 = 1.5 miles. so David travels 4.5 miles to meet tanya. time taken = 4.5/9 = 30 minutes.

time taken to cover twice the distance tanya covered = 9/9= 1 hours. so difference = 60-30 = 30 minutes. But answer is 90 minutes.

PS- I Picked the question from here https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/06 ... se-in-tsd/

gmatgrl · Accepted Answer

Assume Tanya travels for t hours.
So, Stephen travels for (t-1) hours

Since distance is the same
3t = 9(t-1)
t = 1.5 hours
So, Stephen travels for 0.5 hours to travel the same distance as Tanya.

Now, Stephen is supposed to travel twice the distance.
Again, suppose Tanya travels for t hours
So, Stephen travels for (t-1) hours
Since Stephen travels twice the distance as Tanya does, 
9(t-1) = 2*3t
t = 3 hours
So, Stephen travels for 2 hours to travel twice the distance as Tanya.

So, difference = 2 - 0.5 = 1.5 hours = 90 minutes.

Lucky2783 · Answer

Its not that hard as it seems by wording.

Let say tanya walked x miles after stephan started.
 \frac{X+3}{9}  = \frac{x}{3} 
X=1.5 so time taken by stephen = 4.5/9=30mins

Next equation is 2x+6/9= x/3 
x=6 so stephen would take 2hr

2hr-30min =90min

Zhenek · Answer

t1 - time it takes to cover the same distance tanya covered.
 3 + t1*3 = 9*t1, t1 = 1/2 
t2 - time it takes to cover twice of the distance Tanya covered
 2*(3+t2*3) = 9*t2, t2 = 2 
 t2 - t1 = 1.5 = 90  minutes, answer C.

manishbhusal · Answer

Would you mind telling how you got the second equation. I understood the L.H.S part i.e (2x+6) is twice the distance of what tanya had covered.so time taken = (2x+6)/9, but i couldn't understand the R.H.S part of second equation . i.e how we got x/3 and how (2x+6)/9 is equal to x/3.

Thanks
Manish

Lucky2783 · Answer

Hi Manish,

Lets assume that Tanya walked Y Miles when Stephen ran twice the distance she walked.

stephen had to run 2*(Y+3) .

2*(Y+3) /9 = Y/3 (tanya walked only Y Miles when stephen started to run )

hope it is clear .

mvictor · Answer

oh nice..that's a good brain killer.

after 1 hour, tanya traveled for 3 miles. Stephen 0.
after 2 hours, tanya - 6 miles, stephen -9 miles
after 3 hours, tanya 9 miles, stephen - 18 miles.
ok, so stephen needs 2 hours to cover twice the distance tanya has walked.

now, time needed for stephan to cover the same distance tanya has walked.
1 hour -> tanya 3, stephen -0.
we know the rates of each.
in 20 minutes, tanya walks 1 miles, stephen 3.
so after 1h 20 mins, tanya walked 4 miles. stephen in 20 mins - 3 miles.
so if 20 mins for tanya 1 mile and for stephen 3, then in 10 mins, tanya will walk 0.5 miles and stephen 1.5
so...after another 10 mins, tanya 4.5 miles, stephen 4.5 miles.
we can see that stephen needs 30 mins to cover the same distance that tanya has covered.

now, 30-120=-90. since we are asked for the POSITIVE difference, -90*-1 = 90 mins.

gracie · Answer

let t=Stephen's time
t+1=Tanaya's time
equation 1: 9t=3(t+1)
t=1/2 hour=30 minutes
equation 2: 9t=(2)(3)(t+1)
t=2 hours=120 minutes
120-30=90 minutes

Plunkster82 · Answer

I'll be honest this is a really poorly written question and I have to wonder if this was written by a native English speaker because what the question is asking for isn't an answer. The positive difference between the distance she's covered and twice the distance she's covered is going to change based on how long she's walking. We don't know how long she's been walking. This question needs to be rewritten.

homersimpsons · Answer

Hi  VeritasKarishma 
How do we solve this question using relative time?

Fdambro294 · Answer

Getting through the Concept of the Problem was difficult at first because it is a little different. Once you figure that out, the posters above are right, the Problem becomes easier.

1st) You want to find the Time at which Stephen will have covered the EXACT SAME DISTANCE that Tonya Covers (including her +1 hour head-start)

Setting up the R*T = D Chart:

Tonya ----- 3 m.p.h * (Ta + 1) = D
Stephen ----- 9 m.p.h. * (Ta) = D

we want D = D ---- so that they cover the Same Distance

3 * (Ta + 1) = 9 * (Ta)

Ta = 1/2 hour will pass (after Stephen leaves) in which they will have covered the same Distance

2nd) We want: Distance Stephen Travels = 2 x (Distance Tonya Travels)

Let this Time = Tb

9 * (Tb) = 2 *

9Tb = 6Tb + 6

3Tb = 6

Tb = 2 hours when Stephen will cover exactly TWICE THE DISTANCE that Tonya has covered (from the Time Stephen departs 1 hour after Tonya)

(+)Positive Difference between Ta - Tb = 1/2 hours - 2 hours = 1.5 hours or 90 minutes

-C-

KarishmaB · Answer

Following is what goes on in my mind when I read the question. I will quote sentences from the question to explain you the thought process.

“Two friends, Tanaya and Stephen were standing together.”

When I read the above, I think, “Ok, so two people, Tanaya and Stephen, are standing at the same point at a particular time, say 12 noon.”

“Tanaya begins to walk in a straight line away from Stephen at a constant rate of 3 miles/hour.”

Now I think, “Stephen is still standing where they were standing together but Tanaya starts walking away from Stephen at a speed of 3 miles/hour. Women!”

“One hour later, Stephen begins to run in a straight line in the exact opposite direction at a constant rate of 9 miles per hour.”

After a whole hour (i.e. at 1:00 pm), when Tanaya was actually 3 miles away from him, Stephen starts running in the opposite direction at 9 miles/hr, three times the speed at which Tanaya was walking away.

“what is the positive difference between the amount of time it takes Stephen to cover the exact distance that Tanaya has covered and …”

So, I need to find the time it takes Stephen to cover the exact distance that Tanaya has covered. To cover the same distance as Tanaya, Stephen needs to cover the distance that Tanaya is covering now plus he has to cover the extra 3 miles that Tanaya has already covered. The thing going for him is that he is running much faster than Tanaya. Out of his speed of 9 miles/hr, 3 miles/hr get used to cover what Tanaya is covering right now (since Tanaya’s speed is 3 miles/hr). So he uses the extra 6 miles/hr to cover the 3 miles that Tanaya has already covered! That means it will take him half an hour (3 miles/6 miles/hr) to cover the extra distance. In half an hour, i.e. at 1:30 pm, Stephen would have covered the same distance as Tanaya. At 1:30, Tanaya must be 4.5 miles (= 3 miles/hr*1.5 hrs) away from the starting point and Stephen would also be 4.5 miles away from the starting point (since he has covered the same distance as Tanaya)

“and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?”

I also need to find the time it takes Stephen to cover twice the distance that Tanaya has covered. So, out of his speed of 9 miles/hr, 6 miles/hr will be used to cover twice of what Tanaya will now cover at a speed of 3 miles/hr. The leftover 3 miles/hr (= 9 miles/hr – 6 miles/hr) of his speed will be used to cover an extra 4.5 miles. If you are wondering why he has to cover an extra 4.5 mile, recall that at 1:30 pm, both of them are at a distance of 4.5 miles from the starting point. Stephen should have been at a distance of 9 miles from the starting point if he wants to cover double the distance that Tanaya covers from the starting point. Hence, now he needs to cover an extra 4.5 miles. At a speed of 3 miles/hr (which we obtained above using 9 miles/hr – 6 miles/hr), it will take him 1.5 hrs to cover the extra 4.5 miles ( = 4.5 miles/3 miles per hour). Hence, at 3:00 pm he would have covered twice the distance that Tanaya would have covered since 12 noon.

The time difference between 1:30 pm and 3:00 pm is 1.5 hrs (90 mins). This is the required time difference.

Answer (C)

Archit3110 · Answer

for the 1st part of the question ; time it takes Stephen to match Tanaya
9t=3*(t+1)
t= 1/2 hrs ; 30 mins
now for twice the distance covered
9*t= 2*(3t+3)
9t= 6t+6
t= 2 hrs ; 120 mins 
+ve difference 120-30 ; 90 mins
option C

himgkp1989 · Answer

KarishmaB

“and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?”

I also need to find the time it takes Stephen to cover twice the distance that Tanaya has covered. So, out of his speed of 9 miles/hr,  6 miles/hr will be used to cover twice of what Tanaya will now cover at a speed of 3 miles/hr.  The leftover 3 miles/hr (= 9 miles/hr – 6 miles/hr) of his speed will be used to cover an extra 4.5 miles. If you are wondering why he has to cover an extra 4.5 mile, recall that at 1:30 pm, both of them are at a distance of 4.5 miles from the starting point. Stephen should have been at a distance of 9 miles from the starting point if he wants to cover double the distance that Tanaya covers from the starting point.  Hence, now he needs to cover an extra 4.5 miles.  At a speed of 3 miles/hr (which we obtained above using 9 miles/hr – 6 miles/hr), it will take him 1.5 hrs to cover the extra 4.5 miles ( = 4.5 miles/3 miles per hour). Hence, at 3:00 pm he would have covered twice the distance that Tanaya would have covered since 12 noon.

If Tanaya is fast that extra 4.5 miles can be covered from his relative speed. Why we are still considering that extra 4.5miles to be covered with speed 3MPH. I got so much confused now.

GmatPoint · Answer

One hour later, Tanaya will be 3 miles away from Stephen.
Let after Tanaya walks x miles, Stephen has covered the same distance as Tanya.
Also, since Stephen's speed is thrice that of Tanaya's, he would have covered 3x distance in the same time.

Thus, 3x = x + 3
x = 3/2 miles.
Time taken by Tanaya to cover the distance x will be (3/2)/3 = 1/2 hours = 30 minutes

Now, let's after Tanaya covers the distance y, Stephen has covered twice the distance Tanaya's. Also, since Stephen's speed is thrice that of Tanaya's, he would have covered a 3y distance simultaneously.

Thus, 3y = 2(y+3)
y = 6 miles.
Time taken by Tanaya to cover the distance y will be 6/3 = 2 hours = 120 minutes.

So, the difference between the time taken in the two cases will be 120 - 30 = 90 minutes.

Thus, the correct option is C.

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