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why is the distance between buses a constant value?

Assume that starting from a bus station, all buses run at the same speed of 50 mph. Say a bus starts at 12:00 noon. Another starts at 1:00 pm i.e. exactly one hr later on the same route. Can we say that the previous bus is 50 miles away at 1:00 pm? Yes, so the distance between the two buses initially will be 50 miles. The 1 o clock bus also runs at 50 mph. Will the distance between these two buses always stay the same i.e. the initial 50 miles? Since both buses are moving at the same speed of 50 mph, relative to each other, they are not moving at all and the distance between them remains constant.

The exact same concept is used in this question.
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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?

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!!

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: \(\frac{d}{b+c}=4\), \(d=4b+4c\);

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

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24 Aug 2012, 22:52

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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).

Answer B.

Attachments

BusBicycle1.jpg [ 7.78 KiB | Viewed 1978 times ]

BusBicycle2.jpg [ 7.75 KiB | Viewed 1976 times ]

_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

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?

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!!

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: \(\frac{d}{b+c}=4\), \(d=4b+4c\);

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?

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!!

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: \(\frac{d}{b+c}=4\), \(d=4b+4c\);

what does " every 4 minutes he meets an oncoming bus. " refers? please help

FTGNGU, it means every 4 minutes, the cyclist meets a bus moving in opposite direction. So, the relative speed between them is the absolute value of [speed of cyclist - speed of the bus]

Hope it helps,
_________________

Please +1 KUDO if my post helps. Thank you.

"Designing cars consumes you; it has a hold on your spirit which is incredibly powerful. It's not something you can do part time, you have do it with all your heart and soul or you're going to get it wrong."

what does " every 4 minutes he meets an oncoming bus. " refers? please help

The cyclist is moving in one direction, say due east. Every 4 mins, he meets a bus coming towards him (going due West)

Cyc ----> <-------- Bus (Bus coming towards him)

Cyc-><-Bus (Meets the bus)

<------Bus Cycl -------> (They cross each other)

This happens with a different bus every 4 mins.

Every 4 mins, he meets an oncoming bus. So he meets a bus coming towards him at 12:00 noon. Then, at 12:04 pm, he meets another bus coming towards him. Then again at 12:08 pm and so on.

Their relative speed = Speed of cyclist + Speed of Bus
_________________

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

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23 Jun 2013, 22:41

1. In the case of the overtaking bus, the speed of the cyclist needs to be deducted from that of the bus and in the case of the oncoming bus their speed needs to be added. 2. These two speeds are in the ratio 1:3 because the time is in the ratio 3:1. 3. s1+s2 =3x -----(1) and s2-s1 = x --- (2) Therefore s2 = 2x --- (3) where s1 is the speed of the cyclist and s2 is the speed of the bus. 4. To find the time interval between consecutive buses, we need to assume the case when the buses alone are running and the cyclist is stationary. That is we need to consider s2 only and assume s1=0. 5. From (1) and (3) we know that the speed of (3) is 1.5 times less than the speed of (1.) Therefore the time taken will be 1.5 times more than in the case of (1) 6. In the case of (1), the time taken is 4 min. Therefore in the case of (3) it is 4*1.5=6 min. That is when the cyclist is stationary, the buses cross him every 6 min.

Therefore the time interval between 2 consecutive buses is 6 min
_________________

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

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23 Jun 2013, 23:24

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Another approach:

1. Assume the combined speed of the cyclist and the bus is s1+ s2. s1 is the speed of the cyclist and s2 is the speed of the bus. 2. From the time a bus starts till the time it meets the cyclist, assume the distance traveled is d and the time taken is m 3. In the case of overtaking the difference in the speeds needs to be taken since both are traveling in the same direction. The relative speed is s2- s1. Assume to travel the same distance d the time taken is n 5. From (2) and (3), we have, m(s1+s2) = n(s2-s1)=d. we know m is 4 min and n is 12 min. 6. d= 4(s1+s2) = 12(s2-s1) 7. We want to find d/ s2 = 12(s2-s1)/s2 = 12(s2/s2) - 12(s1/s2) = 12-6= 6 min
_________________

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

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08 Aug 2013, 16:13

Hello.

While I understand the algebra behind the problem, I'm not sure I understand the logic. Consecutive buses are passed every four minutes and the cyclist is passed every 12. I'm unable to rationalize how the time between consecutive buses is 6 minutes? I'm not even sure I understand the context of "consecutive" in this problem.

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

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09 Aug 2013, 01:22

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!!

Time interval between consecutive buses!!!!! Already given !!!!

this question is FLAWED....... We can calculate when they will overtake each other..... 100% FLAWED question
_________________

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

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09 Aug 2013, 01:25

Asifpirlo wrote:

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!!

Time interval between consecutive buses!!!!! Already given !!!!

this question is FLAWED....... We can calculate when they will overtake each other..... 100% FLAWED question

Hello, It will be interesting to understand your logic behind branding this question as '100 % flawed"? Can you kindly elucidate?
_________________

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

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09 Aug 2013, 08:04

1

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While I wouldn't say the question is flawed, the wording is quite ambiguous.

mau5 wrote:

Asifpirlo wrote:

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!!

Time interval between consecutive buses!!!!! Already given !!!!

this question is FLAWED....... We can calculate when they will overtake each other..... 100% FLAWED question

Hello, It will be interesting to understand your logic behind branding this question as '100 % flawed"? Can you kindly elucidate?

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

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09 Aug 2013, 12:35

mau5 wrote:

Asifpirlo wrote:

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!!

Time interval between consecutive buses!!!!! Already given !!!!

this question is FLAWED....... We can calculate when they will overtake each other..... 100% FLAWED question

Hello, It will be interesting to understand your logic behind branding this question as '100 % flawed"? Can you kindly elucidate?

i already mentioned but may be you didn't see.
_________________

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

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13 Aug 2013, 12:03

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?

This problem took me a while to figure out but I think I got it!

The interval refers to the time (i.e. time = distance/speed)

The interval for the bus is t=d/b where b represents the constant speed of both buses.

When the bus overtakes the cyclist, it represents the bus gaining on the cyclist (i.e. the bus speed is faster than the cyclist) so every time the bus overtakes the cyclist, which is every 12 minutes, we have:

12 = d/(b-c) where b-c is the rate that the bus gains on the cyclist. 12b-12c = d

When the oncoming bus meets the cyclist, it represents their combined forward speeds towards one another. Every time the bus meets the cyclist, which is every 4 minutes, we have:

4 = d/(b+c) 4b+4c = d Note, we use c for both bus speeds because the question tells us that they move at the same constant speed.

12b-12c = d 4b+4c = d

12b-12c = 4b+4c 8b = 16c b = 2c

12b-12c = d 12(2c) - 12c = d

My problem is this: how do I know to solve for d? In other words, why would I know: b = 2c, 12b-12c = d 12*(2c) - 12c = d 24c - 12c = d 12c = d isn't correct? How would I know to convert b=2c to c=b/2 to plug in so I can get a value for d thus getting me the right answer?

c=b/2 12b-12c = d 12b - 12(b/2) = d 12b - 6b = d 6b = d

t=d/b (the interval for both buses) t=(6b/b) t = 6

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?

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!!

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: \(\frac{d}{b+c}=4\), \(d=4b+4c\);

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?

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!!

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/b-c---Why it is so if objects are moving in same direction. d/b+c---Why it is so if objects are moving in opposite direction.

Rgds Prasannajeet

When 2 objects move in the same direction their relative speed is the difference of their individual speeds. When 2 objects move in the opposite direction their relative speed is the sum of their individual speeds.
_________________

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!!

Time interval between consecutive buses!!!!! Already given !!!!

this question is FLAWED....... We can calculate when they will overtake each other..... 100% FLAWED question

There is nothing wrong with the question.
_________________

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