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# Two cars run in opposite directions on a circular track. Car A travels

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Math Expert
Joined: 02 Sep 2009
Posts: 60480
Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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21 Jul 2017, 07:36
vp101 wrote:
I could not solve this due to my lack of understanding of nomenclature in the question. I could not understand what $$6\pi$$ is.

Is it simply 6<pi>? or is it like square of six <pi>?

Thanks.

It's $$6\pi$$, so 6 multiplied by pi.
Attachment:

2017-07-21_1835.png [ 2.51 KiB | Viewed 905 times ]

Does your screen show it in different way?
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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21 Jul 2017, 07:41
Sorry Bunuel, but I think this was a problem with my browser. I am sure there is a way in which pi can be depicted by using the forward slash keys. I pasted this URL in a different browser and my problem was solved! Thanks for the reply!
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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21 Jul 2017, 08:00
LCM of (L/a, ;L/b) is answer. L- Length of track; a- speed of car A, b- speed of car B

LCM (12pi/6pi, 12pi/8pi) = LCM (2, 1.5) = 6 (ANSWER)
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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15 Sep 2017, 19:33
Didn't think about the concept of relative speed. Didn't remember it, so I solved this in another way.

We know the length of the track is 12pi and we also know the average speed of both A and B.
So, in order for them to have ran a complete lap, the (average speed * time) / length of the track (as in, total distance) should be an integer, though naturally not the same for both of them.

The lowest number in the options provided is 6 hours, hence D is our answer.
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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21 Oct 2019, 15:02
Total distance for one lap = circumference of the circular track = 2 x pi x 6=12pi
Time required by A to complete 1 lap = 12pi/6pi=2hours
Time required by B to complete 1 lap = 12pi/8pi=1.5hours.
Time for them to meet again at point S is equal to the LCM (2 and 1.5).
But 1.5 is not an integer. So look for LCM of (4 and 3) and divided the result by 2 since we multiplied the numbers by 2
LCM (3 4) = 12
Therefore time required to meet again = 12/2=6hours.
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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21 Oct 2019, 17:38
Is the same concept applicable for linear motion?

Bunuel wrote:
Bunuel wrote:
Two cars run in opposite directions on a circular track. Car A travels at a rate of $$6\pi$$ miles per hour and Car B runs at a rate of $$8\pi$$ miles per hour. If the track has a radius of 6 miles and the cars both start from Point S at the same time, how long, in hours, after the cars depart will they again meet at Point S?

(A) 6/7 hrs
(B) 12/7 hrs
(C) 4 hrs
(D) 6 hrs
(E) 12 hrs

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

What would we usually do in such a question? Two cars start from the same point and run in opposite directions – their speeds are given. This would remind us of relative speed. When two objects move in opposite directions, their relative speed is the sum of their speeds. So we might be tempted to do something like this:

Perimeter of the circle = 2$$\pi$$r = 2$$\pi$$*6 = 12? miles

Time taken to meet = Distance/Relative Speed = 12$$\pi$$/(6? + 8?) = 6/7 hrs

But take a step back and think – what does 6/7 hrs give us? It gives us the time taken by the two of them to complete one circle together. In this much time, they will meet somewhere on the circle but not at the starting point. So this is definitely not our answer.

The actual time taken to meet at point S will be given by 12$$\pi$$/(8$$\pi$$ – 6$$\pi$$) = 6 hrs

This is what we mean by unexpected! The relative speed should be the sum of their speeds. Why did we divide the distance by the difference of their speeds? Here is why:

For the two objects to meet again at the starting point, obviously they both must be at the starting point. So the faster object must complete at least one full round more than the slower object. In every hour, car B – the one that runs at a speed of 8$$\pi$$ mph covers 2$$\pi$$ miles more compared with the distance covered by car A in that time (which runs at a speed of 6$$\pi$$ mph). We want car B to complete one full circle more than car A. In how much time will car B cover 12$$\pi$$ miles (a full circle) more than car A? In 12$$\pi$$/2$$\pi$$ hrs = 6 hrs.
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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21 Oct 2019, 19:08
Bunuel wrote:
Two cars run in opposite directions on a circular track. Car A travels at a rate of $$6\pi$$ miles per hour and Car B runs at a rate of $$8\pi$$ miles per hour. If the track has a radius of 6 miles and the cars both start from Point S at the same time, how long, in hours, after the cars depart will they again meet at Point S?

(A) 6/7 hrs
(B) 12/7 hrs
(C) 4 hrs
(D) 6 hrs
(E) 12 hrs

Kudos for a correct solution.

Total Distance=2*pi*6=12pi
Relative speed=8pi-6pi=2pi
So in 1 hour B will be 2 pi ahead of A or 2pi ahead from point S.
To cover 12pi or to go to point S, time taken will be=12pi/2pi
6 hours
D:)
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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22 Oct 2019, 00:20
As much I have studied in many official sources, relative speed of two vehicle moving in opposite direction is added, no subtracted as you did in your explanation.
Probably there might be another reason for subtracting speeds as clarified by Bunnel.

satya2029 wrote:
Bunuel wrote:
Two cars run in opposite directions on a circular track. Car A travels at a rate of $$6\pi$$ miles per hour and Car B runs at a rate of $$8\pi$$ miles per hour. If the track has a radius of 6 miles and the cars both start from Point S at the same time, how long, in hours, after the cars depart will they again meet at Point S?

(A) 6/7 hrs
(B) 12/7 hrs
(C) 4 hrs
(D) 6 hrs
(E) 12 hrs

Kudos for a correct solution.

Total Distance=2*pi*6=12pi
Relative speed=8pi-6pi=2pi
So in 1 hour B will be 2 pi ahead of A or 2pi ahead from point S.
To cover 12pi or to go to point S, time taken will be=12pi/2pi
6 hours
D:)
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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28 Oct 2019, 02:01
1
This is a very simple problem so need to make it unnecessarily complicated.

Let the speeds of Car A and Car B be Va and Vb respectively. The ratio of their speeds is:
Va/Vb=6pi/8pi=3/4 which means that B completes 4 revolutions in the time that A completes 3 revs after which they meet at their starting point S.
Time for A to complete 3 revs = 12pi*3/6pi = 6 hours.
Or, time for B to complete 4 revs = 12pi*4/8pi = (also) 6 hours. ANS: D
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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13 Nov 2019, 14:58
Bunuel wrote:
Bunuel wrote:
Two cars run in opposite directions on a circular track. Car A travels at a rate of $$6\pi$$ miles per hour and Car B runs at a rate of $$8\pi$$ miles per hour. If the track has a radius of 6 miles and the cars both start from Point S at the same time, how long, in hours, after the cars depart will they again meet at Point S?

(A) 6/7 hrs
(B) 12/7 hrs
(C) 4 hrs
(D) 6 hrs
(E) 12 hrs

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

What would we usually do in such a question? Two cars start from the same point and run in opposite directions – their speeds are given. This would remind us of relative speed. When two objects move in opposite directions, their relative speed is the sum of their speeds. So we might be tempted to do something like this:

Perimeter of the circle = 2$$\pi$$r = 2$$\pi$$*6 = 12? miles

Time taken to meet = Distance/Relative Speed = 12$$\pi$$/(6? + 8?) = 6/7 hrs

But take a step back and think – what does 6/7 hrs give us? It gives us the time taken by the two of them to complete one circle together. In this much time, they will meet somewhere on the circle but not at the starting point. So this is definitely not our answer.

The actual time taken to meet at point S will be given by 12$$\pi$$/(8$$\pi$$ – 6$$\pi$$) = 6 hrs

This is what we mean by unexpected! The relative speed should be the sum of their speeds. Why did we divide the distance by the difference of their speeds? Here is why:

For the two objects to meet again at the starting point, obviously they both must be at the starting point. So the faster object must complete at least one full round more than the slower object. In every hour, car B – the one that runs at a speed of 8$$\pi$$ mph covers 2$$\pi$$ miles more compared with the distance covered by car A in that time (which runs at a speed of 6$$\pi$$ mph). We want car B to complete one full circle more than car A. In how much time will car B cover 12$$\pi$$ miles (a full circle) more than car A? In 12$$\pi$$/2$$\pi$$ hrs = 6 hrs.

Hi,

If the speed of car B had been 18$$\pi$$, then Speed Car B - Speed Car A = 18$$\pi$$ - 6 $$\pi$$ = 12 $$\pi$$
Following the approach above, 12$$\pi$$ / 12$$\pi$$ = 1 hour. But this is not possible since in one hour, Car A will have completed only half a lap. In this case what we get is that Car A has completed 1/2 lap and Car B has completed 1.5 laps, so they meet but not at the starting point. Am I doing something wrong?

Thanks!
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Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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15 Dec 2019, 15:49
lucisgm4 wrote:
Bunuel wrote:
Bunuel wrote:
Two cars run in opposite directions on a circular track. Car A travels at a rate of $$6\pi$$ miles per hour and Car B runs at a rate of $$8\pi$$ miles per hour. If the track has a radius of 6 miles and the cars both start from Point S at the same time, how long, in hours, after the cars depart will they again meet at Point S?

(A) 6/7 hrs
(B) 12/7 hrs
(C) 4 hrs
(D) 6 hrs
(E) 12 hrs

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

What would we usually do in such a question? Two cars start from the same point and run in opposite directions – their speeds are given. This would remind us of relative speed. When two objects move in opposite directions, their relative speed is the sum of their speeds. So we might be tempted to do something like this:

Perimeter of the circle = 2$$\pi$$r = 2$$\pi$$*6 = 12? miles

Time taken to meet = Distance/Relative Speed = 12$$\pi$$/(6? + 8?) = 6/7 hrs

But take a step back and think – what does 6/7 hrs give us? It gives us the time taken by the two of them to complete one circle together. In this much time, they will meet somewhere on the circle but not at the starting point. So this is definitely not our answer.

The actual time taken to meet at point S will be given by 12$$\pi$$/(8$$\pi$$ – 6$$\pi$$) = 6 hrs

This is what we mean by unexpected! The relative speed should be the sum of their speeds. Why did we divide the distance by the difference of their speeds? Here is why:

For the two objects to meet again at the starting point, obviously they both must be at the starting point. So the faster object must complete at least one full round more than the slower object. In every hour, car B – the one that runs at a speed of 8$$\pi$$ mph covers 2$$\pi$$ miles more compared with the distance covered by car A in that time (which runs at a speed of 6$$\pi$$ mph). We want car B to complete one full circle more than car A. In how much time will car B cover 12$$\pi$$ miles (a full circle) more than car A? In 12$$\pi$$/2$$\pi$$ hrs = 6 hrs.

Hi,

If the speed of car B had been 18$$\pi$$, then Speed Car B - Speed Car A = 18$$\pi$$ - 6 $$\pi$$ = 12 $$\pi$$
Following the approach above, 12$$\pi$$ / 12$$\pi$$ = 1 hour. But this is not possible since in one hour, Car A will have completed only half a lap. In this case what we get is that Car A has completed 1/2 lap and Car B has completed 1.5 laps, so they meet but not at the starting point. Am I doing something wrong?

Thanks!

Bunuel could you help with this question I have? Thank you!
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Joined: 04 Feb 2018
Posts: 45
Re: Two cars run in opposite directions on a circular track. Car A travels  [#permalink]

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22 Dec 2019, 22:10
Note that because A is slower than B, they will only meet at the starting point again when the difference in the distances travelled is equal to the circumference of the circle. The circumference of the circle is equal to 2x6xpi = 12pi.
Time required for B to travel 12pi more than A = 12pi/relative speed = 12pi/2pi=6hours.
Re: Two cars run in opposite directions on a circular track. Car A travels   [#permalink] 22 Dec 2019, 22:10

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