Distance between A and B is 72 miles. Two men started walkin : GMAT Problem Solving (PS)
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# Distance between A and B is 72 miles. Two men started walkin

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Distance between A and B is 72 miles. Two men started walkin [#permalink]

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23 Dec 2009, 22:21
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Distance between A and B is 72 miles. Two men started walking from A and B at the same time towards each other. The person who started from A traveled uniformly with average speed 4 mph. While the other man traveled with varying speeds as follows: In first hour his speed was 2 mph, in the second hour it was 2.5 mph, in the third hour it was 3 mph, and so on. When will they meet each other?

(A) 35 miles away from A
(B) 32 miles away from B
(C) after 32 hours from start
(D) after 10 hours from start
(E) midway between A and B

i managed to solve it in my way which took 4 min , perhaps anyone can help me to solve it in 2 min
[Reveal] Spoiler: OA

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24 Dec 2009, 17:31
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xcusemeplz2009 wrote:
Distance between A and B is 72 miles. Two men started walking from A and B at the same time towards each other. The person who started from A travelled uniformly with average speed 4 mph. While the other man traveled with varying speeds as follows: In first hour his speed was 2 mph, in the second hour it was 2.5 mph, in the third hour it was 3 mph, and so on. When will they meet each other?

(A) 35 miles away from A
(B) 32 miles away from B
(C) after 32 hours from start
(D) after 10 hours from start
(E) midway between A and B

i managed to solve it in my way which took 4 min, perhaps anyone can help me to solve it in 2 min

They meet after n hours:
4n + [2 + 2.5 + 3 + ........ + {2 + (0.5(n-1)}] + x = 72, where x has to be 0.

Lets try for n = 8:
4(8) + [2 + 2.5 + 3 + 3.5 + 4 + 4.5+ 5 + 5.5] + x = 72
x = 10 ... not correct.

Lets try for n = 9:
4(9) + [2 + 2.5 + 3 + 3.5 + 4 + 4.5+ 5 + 5.5+6] + x = 72
x = 0 .... thats possible.

When they meet after 9 hours, that is mid point.

E.
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24 Dec 2009, 17:52
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Distance between A and B is 72 miles. Two men started walking from A and B at the same time towards each other. The person who started from A traveled uniformly with average speed 4 mph. While the other man traveled with varying speeds as follows: In first hour his speed was 2 mph, in the second hour it was 2.5 mph, in the third hour it was 3 mph, and so on. When will they meet each other?

(A) 35 miles away from A
(B) 32 miles away from B
(C) after 32 hours from start
(D) after 10 hours from start
(E) midway between A and B

Maybe there is an easier way...

Arithmetic progression for both:

$$a_1=4+2=6$$ and common difference $$d=0.5$$, $$n$$ number of terms (hours)

$$Sum=\frac{n}{2}(2a_1+d(n-1))=\frac{n}{2}(2*6+0.5(n-1))\geq72$$

$$0.5n^2+11.5n-144\geq0$$ --> $$n^2+23n-288\geq0$$ --> $$(n+32)(n-9)\geq0$$, $$-32\geq{n}\geq9$$

Hence for n=9(hours) the above equation is exactly 72.

In 9 hours men from A covers 9*4=36, which is half of the distance.

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25 Dec 2009, 17:58
thanks guys
its E
+1 kudos to bth
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28 Dec 2009, 03:02
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let journey hours be=n

when they meet distance travelled is =

hence
4n=n/2*(2a+(n-1)d)
where a=2 and d=.5
8=4+(n-1)*.5
n=9

now 4n=36(midway) ='E'
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Re: Distance between A and B is 72 miles. Two men started walkin [#permalink]

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26 Dec 2013, 07:48
Hello from the GMAT Club BumpBot!

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Re: Distance between A and B is 72 miles. Two men started walkin [#permalink]

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17 Oct 2015, 20:06
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Distance between A and B is 72 miles. Two men started walkin [#permalink]

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29 Oct 2015, 21:03
let t=A's and B's time to meeting
B's ratio of cumulative distance to time=(t+7)/4 mph
4t+(t)(t+7)/4=72
t^2+23t-288=0
(t+32)(t-9)=0
t=9 hours
4t=36 miles distance for A
72-36=36 miles distance for B
they meet midway between A and B

Last edited by gracie on 12 Nov 2015, 20:21, edited 2 times in total.
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Re: Distance between A and B is 72 miles. Two men started walkin [#permalink]

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29 Oct 2015, 21:40
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xcusemeplz2009 wrote:
Distance between A and B is 72 miles. Two men started walking from A and B at the same time towards each other. The person who started from A traveled uniformly with average speed 4 mph. While the other man traveled with varying speeds as follows: In first hour his speed was 2 mph, in the second hour it was 2.5 mph, in the third hour it was 3 mph, and so on. When will they meet each other?

(A) 35 miles away from A
(B) 32 miles away from B
(C) after 32 hours from start
(D) after 10 hours from start
(E) midway between A and B

i managed to solve it in my way which took 4 min , perhaps anyone can help me to solve it in 2 min

In such questions, I always look for the symmetrical situation to give me a ballpark figure or at least a range. For example, in many work-rate questions, it helps if you figure out the situation in which the rate of both people is the same.
Here the average speed of A (assuming guy who starts from A is A) is 4 mph and distance is 72 miles. So this means that A would take a total of 72/4 = 18 hours to cover the distance from A to B. If B's speed were also 4, both A and B would travel for 9 hours to meet in the middle.
B has uniform speed per hour so its average will be the simple average of all speeds: 2, 2.5, 3, 3.5, 4, 4.5, ...
4 is actually right in the middle of the 9 speeds and this implies that they both have the same average speed. So they will meet midway between A and B.

Had I obtained that B's average speed is more than 4 then that would tell me that they will meet closer to point A and will take less than 9 hours and so on...

You might feel that in this question it worked because the numbers were such but you will be surprised at the number of times such simple analysis will work because GMAT gives you numbers which fit in place.
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Re: Distance between A and B is 72 miles. Two men started walkin   [#permalink] 29 Oct 2015, 21:40
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