man travelled with varying speeds

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 am; Location: India; Thanked: 267 times; Followed by:80 members; GMAT Score:760

man travelled with varying speeds

by sanju09 » Wed Dec 23, 2009 4:17 am

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 travelled 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

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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Source: Beat The GMAT — Problem Solving | Wed Dec 23, 2009 4:17 am

rohan_vus: Master | Next Rank: 500 Posts; Posts: 189; Joined: Thu Apr 03, 2008 2:03 pm; Location: USA; Thanked: 21 times

by rohan_vus » Wed Dec 23, 2009 5:09 am

IMO E...

Did backsolving, one choice had 10 hours , so started with that , but found A + B exceeding the total distance , then went to cross check with 9 hours , which turns out to satisfy the conditions.

In 9 hrs , A travels 36 miles
in 9 hrs , B travels - 2 + 2.5 + 3.5...... ( till the 9th term)..this is AP with common diff of .5 and is equal to 36

Last edited by rohan_vus on Wed Dec 23, 2009 5:42 am, edited 1 time in total.

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 am; Location: India; Thanked: 267 times; Followed by:80 members; GMAT Score:760

by sanju09 » Wed Dec 23, 2009 5:27 am

Great going rohan, you got it; in fact backsolving is a great strategy on GMAT Quantitative section. Another classical approach, anyone?

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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linkinpark: Master | Next Rank: 500 Posts; Posts: 124; Joined: Tue Dec 22, 2009 8:32 am; Location: Karjat; Thanked: 3 times

by linkinpark » Wed Dec 23, 2009 8:28 am

interesting question

here time will be same for both A and B say its T

so for A it will be 72/4 and for B it will be 72/(sum of speeds)/n, where n = number of different speeds

after solving I get 4n = (sum of diff speeds). I'm stuck here

how do I find n and sum

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Wed Dec 23, 2009 6:31 pm

linkinpark wrote:interesting question

here time will be same for both A and B say its T

so for A it will be 72/4 and for B it will be 72/(sum of speeds)/n, where n = number of different speeds

after solving I get 4n = (sum of diff speeds). I'm stuck here how do I find n and sum

The algebra is incredibly complicated - we need to use arithmetic series to find the distance travelled by B; backsolving is definitely the way to go.

As an aside, the total distance travelled is 72; that's not how far each of them travels alone. So, our final equation would be:

72 = 4n + (sum of arithmetic series for B)

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 am; Location: India; Thanked: 267 times; Followed by:80 members; GMAT Score:760

by sanju09 » Wed Dec 23, 2009 11:58 pm

Stuart Kovinsky wrote:
linkinpark wrote:interesting question

here time will be same for both A and B say its T

so for A it will be 72/4 and for B it will be 72/(sum of speeds)/n, where n = number of different speeds

after solving I get 4n = (sum of diff speeds). I'm stuck here how do I find n and sum
The algebra is incredibly complicated - we need to use arithmetic series to find the distance travelled by B; backsolving is definitely the way to go.

As an aside, the total distance travelled is 72; that's not how far each of them travels alone. So, our final equation would be:

72 = 4n + (sum of arithmetic series for B)

Let me finish Stuart's take please...

Actually, he took n to be the number of hours after what they met, ?if? it's an integer, then we can talk about forming an AP for (person from B)'s run, as under:

sum of arithmetic series for B = n/2 {2*6 + (n - 1) (0.5)}

I think that to make sure that n is a positive integer, we should ideally write it as an inequality here, I mean...

4 n + n/2 {2*6 + (n - 1) (0.5)} â‰¤ 72

this ends up in...

(n - 9) (n + 32) â‰¤ 0

therefore n â‰¤ 9 permits us to go with n = 9; and in 9 hours person from A will travel 36 miles and will meet person from B too. This point must be the midway between A and B. Go with E.

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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pink_08: Senior | Next Rank: 100 Posts; Posts: 87; Joined: Wed Dec 31, 2008 7:01 pm

by pink_08 » Thu Dec 24, 2009 2:53 pm

where did u get that 6 in

sum of arithmetic series for B = n/2 {2*6 + (n - 1) (0.5)}

Is the sum of AP = (a1+an)/2 ??

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Thu Dec 24, 2009 6:11 pm

pink_08 wrote:where did u get that 6 in

sum of arithmetic series for B = n/2 {2*6 + (n - 1) (0.5)}

Is the sum of AP = (a1+an)/2 ??

It's actually n(a1 + an)/2.

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maihuna: Legendary Member; Posts: 1578; Joined: Sun Dec 28, 2008 1:49 am; Thanked: 82 times; Followed by:9 members; GMAT Score:720

by maihuna » Fri Dec 25, 2009 5:35 am

sanju09 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 travelled 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

Assuming that other man moved in AP: 2, 2.5, 3, 3.5, 5 etc.

On an average one is @4mph, another is minimum @2mph, so a min of @6mph a max of 12 hrs are needed.

A mid of 4 hrs
==Option C is out. Here is a table covering hr and distance covered:

Hour One's speed Two's speed Distance covered
==========================================
1 4 2 6
2 4 2.5 6.5
3 4 3 7
4 4 3.5 7.5
4 4 4 8
4 4 4.5 8.5
======================================

Basically it turns out to be an AP which S = 72 with terms 6 6.5 7 7.5
72 = n/2(2*6 + (n-1)/2) = n/4(24+n-1) = n/4(23+n)
72*4 = n(23+n)
9*(9+23) = n(23+n)

So they will meet in 9th hours. In 9 hrs they might have gone from A by 9*4 = 36 miles. which is the mid point, I will choose E.

Charged up again to beat the beast