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A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?

A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes

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A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?
A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes

Let's say the distance between the buses is $$d$$. We want to determine $$Interval=\frac{d}{b}$$, where $$b$$ is the speed of bus.

Let the speed of cyclist be $$c$$.

Every 12 minutes a bus overtakes cyclist: $$\frac{d}{b-c}=12$$, $$d=12b-12c$$;

Every 4 minutes cyclist meets an oncoming bus: $$\frac{d}{b+c}=4$$, $$d=4b+4c$$;

$$d=12b-12c=4b+4c$$, --> $$b=2c$$, --> $$d=12b-6b=6b$$.

$$Interval=\frac{d}{b}=\frac{6b}{b}=6$$

Answer: B (6 minutes).

Hope it helps.
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Originally posted by Bunuel on 04 Jan 2010, 04:44.
Last edited by Bunuel on 20 Sep 2019, 03:17, edited 3 times in total.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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lucalelli88 wrote:
A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?

A. 5 minutes
B. 6 minutes
C. 8 minutes
D. 9 minutes
E. 10 minutes

I didn't get how to solve this problem. Can someone explain me more detailed than solution provided by the test?
Thank you in advance!!

When solving motion problems, I can't do without some drawings.
So, here is my version:

Denote by B the speed of the bus and by C the speed of the bicycle. Both are assumed to be constant.
Let T be the constant time interval between consecutive buses. It means, the distance between two consecutive buses is BT.

First scenario: buses and bicycle moving in the same direction and buses overtake the bicycle. Refer to the first attached drawing.

When bus $$B$$ and bicycle $$C$$ are at point $$n,$$ next bus $$B^*$$ is at point $$m.$$
Bus $$B^*$$ will overtake bicycle $$C$$ at point $$p.$$
In 12 minutes, bus $$B^*$$ travels the distance $$mp$$ and bicycle $$C$$ travels the distance $$np.$$
We know that $$mp$$ is the distance between consecutive buses, therefore $$mp=BT.$$
Translated into an equation mp=mn+np, so:
$$12B=BT+12C$$ (1)

Second scenario: buses and bicycle moving in opposite directions and buses meet the bicycle. Refer to the second attached drawing.

When bus $$B$$ and bicycle $$C$$ are at point $$m$$, next bus $$B^*$$ is at point $$p.$$
Bus $$B^*$$ will meet bicycle C at point $$n.$$
In 4 minutes, bus $$B^*$$ travels the distance $$np$$ and bicycle $$C$$ travels the distance $$mn.$$
We know that $$mp$$ is the distance between consecutive buses, therefore $$mp=BT.$$
Translated into an equation mp=mn+np, so:
$$BT=4C+4B$$ (2)

Expressing $$BT$$ from both equations, we get $$12(B-C)=4(B+C)$$ from which $$B=2C$$ (3)
Substituting (3) in (2) for example, we get, $$2CT=8C+4C$$ from which $$T=6$$ (minutes).

Attachments BusBicycle1.jpg [ 7.78 KiB | Viewed 11431 times ] BusBicycle2.jpg [ 7.75 KiB | Viewed 11429 times ]

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GMAT 1: 750 Q50 V40 Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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5
B

$$\frac{L}{V_{bus}+V_{cyclist}}=4$$

$$\frac{L}{V_{bus}-V_{cyclist}}=12$$

$$4*(V_{bus}+V_{cyclist})=12*(V_{bus}-V_{cyclist})$$

$$V_{cyclist}=\frac12*V_{bus}$$

$$\frac{L}{V_{bus}+\frac12*V_{bus}}=4$$

$$\frac{L}{V_{bus}}=4*(1+\frac12)=6$$
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Walker,

If possible can can you dumb it down a little for us lowly earthlings . Thanks in advance.
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GMAT 1: 750 Q50 V40 Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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prasannar wrote:
Walker,

How do we know that the distance L traveled by the oncoming and overtaking buses is the same, Do we need to assume that it is same and calculate or am I missing anything.

Yes. we have to assume. If the distance is not the same, the buses will collect in the one of the end stations and we have many possible solutions.

neelesh wrote:
Walker, If possible can can you dumb it down a little for us lowly earthlings . Thanks in advance. 1. I wrote two equations for the time intervals of both the oncoming and overtaking buses using formula: t=L/V
2. I divided one equation by other one in order to exclude the distance between buses.
3. I found relationship between the speed of the buses and the speed of the cyclist
4. I used the finding and got L/Vbus (the time interval between consecutive buses) from first equation.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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topmbaseeker wrote:
A man cycling along the road noticed that every 12 minutes a bus overtakes him while every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at constant speed, what is the time interval between consecutive buses?

(C) 2008 GMAT Club - m08#18

* 5 minutes
* 6 minutes
* 8 minutes
* 9 minutes
* 10 minutes

d = distance the bus and cyclist togather run
c = cyclist's speed
b = bus's speed

d/(c+b) = 4
d/(b-c) = 12
d/(c+b) = d/3(b-c)
3c+3b = b-c
4c = 2b
b = 2c, or
c = c/2

we need bus's time to complet d distance. so replace c in terms of b:

d/(c+b) = 4
d/(b/2 + b) = 4
d/b = 4x3/2
d/b = 6

Thats the time taken to pass d for a bus. after every 6 minuets another bus comes.

So it is B.

I know this is another hard question but when the concept is clear, it wont be difficult to get the result.. _________________
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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Why is the distance the same for both the cases?
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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topmbaseeker wrote:
Why is the distance the same for both the cases?

Hope the drawing helps understand how and in which direction the buses and cyclist are running.
Attachments Distance.doc [27 KiB]

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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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Thank you very much!!Great explanation!!+1 for you!

I think the question is pretty hard: do you think that is possible to find a question so hard in the real GMAT?
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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neoreaves wrote:
can you please explain how you got the answer ?

I might be VERY wrong..But when i don't get the answers..I go off track..So I'm not sure that this is the answer..
But marking the answer with some guess is better than NO guess.!!

Let say the road is a straight line AB.
The cyclist starts frm A and Bus starts frm B.

So in 4 min, let the distance traveled by Cyc is x, so distance traveled by Bus is (AB - x)
so 4 = x/c...........................1
and 4= (AB - x)/b..................................2
where c and b are respective speed of the cyclist and bus

Also in 12 mins, distance traveled by cyc is 3x.

so 12= 3x/(b-c) and x= 4c (frm 1)
so 1= c/(b-c)
and b=2c........................3

put the value of b frm 3 in equation 2.

We get AB=12c

So now the total distance is 12c and bus speed is 2c. The bus travels in 6 min...

It's A COMPLETE WILD GUESS..!! I was not able to get answer in 2 min..Let me knw the OA and OE.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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Bunuel wrote:

Let's say the distance between the buses is $$d$$. We want to determine $$Interval=\frac{d}{b}$$, where $$b$$ is the speed of bus.
Let the speed of cyclist be $$c$$.

$$Interval=\frac{d}{b}=\frac{6b}{b}=6$$

Hope it helps.

Thanks for the explanation. Please clarify the following doubts.
Aren't we calculating the interval between 2 buses that move towards each other?
If yes, then would the interval not be 6b/b+b ?
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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Fiver wrote:
Bunuel wrote:

Let's say the distance between the buses is $$d$$. We want to determine $$Interval=\frac{d}{b}$$, where $$b$$ is the speed of bus.
Let the speed of cyclist be $$c$$.

$$Interval=\frac{d}{b}=\frac{6b}{b}=6$$

Hope it helps.

Thanks for the explanation. Please clarify the following doubts.
Aren't we calculating the interval between 2 buses that move towards each other?
If yes, then would the interval not be 6b/b+b ?

Not sure I understood your question...

Anyway: question asks "what is the time interval between consecutive buses". Or time intervals between subsequent bus arrivals to a given bus stop (some static point). Which is: constant distance between two subsequent buses divided by the constant rate of these buses d/b. After some calculations we've gotten that d=6b, hence d/b=6b/b=6.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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noboru wrote:
A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?

5 minutes
6 minutes
8 minutes
9 minutes
10 minutes

There is only one thing you need to understand in this question - When buses are approaching him from both the sides at a constant speed, it doesn't matter whether the man is standing still or cycling, the number of buses that he will meet will be the same. Convince yourself by imagining the case where the man is standing still. He will meet a bus from each side after every few mins. When he starts cycling in a direction, he is cycling away from buses of one side but towards buses of the other side. Since in 12 mins he meets total 4 buses (1 + 3), in 6 mins he meets 2 buses, one from each side, if he were standing still. So buses ply at a frequency of 6 mins each.

Twist: Same scenario. If a man is sitting inside one bus, at what frequency will a bus from opposite side cross him?

Also try the same question by changing the time taken by buses to meet the man to 10 min and 8 min respectively (instead of 12 mins and 4 mins)
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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VeritasPrepKarishma wrote:
noboru wrote:
A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?

5 minutes
6 minutes
8 minutes
9 minutes
10 minutes

There is only one thing you need to understand in this question - When buses are approaching him from both the sides at a constant speed, it doesn't matter whether the man is standing still or cycling, the number of buses that he will meet will be the same. Convince yourself by imagining the case where the man is standing still. He will meet a bus from each side after every few mins. When he starts cycling in a direction, he is cycling away from buses of one side but towards buses of the other side. Since in 12 mins he meets total 4 buses (1 + 3), in 6 mins he meets 2 buses, one from each side, if he were standing still. So buses ply at a frequency of 6 mins each.

Twist: Same scenario. If a man is sitting inside one bus, at what frequency will a bus from opposite side cross him?

Also try the same question by changing the time taken by buses to meet the man to 10 min and 8 min respectively (instead of 12 mins and 4 mins)

Karishma,

I did not understand the TWIST qtn. Please do explain me the question and let me try to solve

2nd qtn:
same question by changing the time taken by buses to meet the man to 10 min and 8 min respectively (instead of 12 mins and 4 mins)

i take LCM of 10 and 8 that is 40
now, in 40 mins, i meet 9 buses (4+5)
==> 9buses --> 40 mins
==> 1 bus --> 40/9 mins
==> 2 buses --> 80/9 mins, that is the time interval b/w 2 buses

is this correct?

Regards,
Murali.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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muralimba wrote:
VeritasPrepKarishma wrote:
noboru wrote:
A man cycling along the road noticed that every 12 minutes a bus overtakes him and every 4 minutes he meets an oncoming bus. If all buses and the cyclist move at a constant speed, what is the time interval between consecutive buses?

5 minutes
6 minutes
8 minutes
9 minutes
10 minutes

There is only one thing you need to understand in this question - When buses are approaching him from both the sides at a constant speed, it doesn't matter whether the man is standing still or cycling, the number of buses that he will meet will be the same. Convince yourself by imagining the case where the man is standing still. He will meet a bus from each side after every few mins. When he starts cycling in a direction, he is cycling away from buses of one side but towards buses of the other side. Since in 12 mins he meets total 4 buses (1 + 3), in 6 mins he meets 2 buses, one from each side, if he were standing still. So buses ply at a frequency of 6 mins each.

Twist: Same scenario. If a man is sitting inside one bus, at what frequency will a bus from opposite side cross him?

Also try the same question by changing the time taken by buses to meet the man to 10 min and 8 min respectively (instead of 12 mins and 4 mins)

Karishma,

I did not understand the TWIST qtn. Please do explain me the question and let me try to solve

2nd qtn:
same question by changing the time taken by buses to meet the man to 10 min and 8 min respectively (instead of 12 mins and 4 mins)

i take LCM of 10 and 8 that is 40
now, in 40 mins, i meet 9 buses (4+5)
==> 9buses --> 40 mins
==> 1 bus --> 40/9 mins
==> 2 buses --> 80/9 mins, that is the time interval b/w 2 buses

is this correct?

Regards,
Murali.

The twist question - now imagine that there is a man sitting in one of the buses. My question is, at what frequency (i.e. after what time interval) will buses from opposite direction cross him (or his bus).
(e.g. a bus from opposite direction crosses his bus every t mins - so I want the value of t). This information is in addition to the given original question.

and yes, your answer to the next question is correct.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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Same scenario. If a man is sitting inside one bus, at what frequency will a bus from opposite side cross him?

Ok let me answer the twist question. Let's say the bus in which the man is sitting is not moving. Then a bus from opposite direction crosses him every 6 mins because that is the frequency of the buses. Now, since his bus is also actually moving with the same speed towards the buses coming from opposite direction, he will meet those buses in 3 mins (half the distance will be covered by his bus and half by the bus coming from opposite direction). So he will meet a bus every 3 minutes.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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Bunuel wrote:
Fiver wrote:
Bunuel wrote:

Let's say the distance between the buses is $$d$$. We want to determine $$Interval=\frac{d}{b}$$, where $$b$$ is the speed of bus.
Let the speed of cyclist be $$c$$.

$$Interval=\frac{d}{b}=\frac{6b}{b}=6$$

Hope it helps.

Thanks for the explanation. Please clarify the following doubts.
Aren't we calculating the interval between 2 buses that move towards each other?
If yes, then would the interval not be 6b/b+b ?

Not sure I understood your question...

Anyway: question asks "what is the time interval between consecutive buses". Or time intervals between subsequent bus arrivals to a given bus stop (some static point). Which is: constant distance between two subsequent buses divided by the constant rate of these buses d/b. After some calculations we've gotten that d=6b, hence d/b=6b/b=6.

Thanks bunnel for the explanation.
But I have a doubt here. what is the time interval between consecutive buses?
Isn't it the time interval between consecutive buses going in one direction. If that is the case then ans should be 12 min.
Thanks.
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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2
Thanks bunnel for the explanation.
But I have a doubt here. what is the time interval between consecutive buses?
Isn't it the time interval between consecutive buses going in one direction. If that is the case then ans should be 12 min.
Thanks.

yeah if you standing at a busstop then whats the time interval of arrival of bus.--> you may take the question this way.

use relative velocity:

vel of bus=a
vel of cyclist=b
distance between them=d

now,
d=12(a-b)=4(a+b)

hence d=6a
so time interval is d/a=6 min

hope this clarifies...!!
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Re: A man cycling along the road noticed that every 12 minutes a bus overt  [#permalink]

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You see in ONE DIRECTION, the distance between two buses is "d", the speed of the buses is "b"(lets just say its km/minute).
So to find this interval, we take this distance divided by speed of bus(all in ONE DIRECTION). so d/b.

Now through algebra 12(a-b)=4(a+b) , we find that distance "d" is equal to "6b". remember d is distance between two buses in ONE DIRECTION. And so we do the division and get the answer 6 minutes.

Just because we do this algebra equation d=12(a-b)=4(a+b), doesn't mean that the nature of b,c or d has changed. It stays true to its original meaning.

No idea what else can clear up your confusion :S
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