Car B starts at point X and moves clockwise around

Step by step analyzes:

B speed: \(2\) mph;
A speed: \(3\) mph (travelling in the opposite direction);
Track distance: \(2*\pi*r=20*\pi\);

What distance will cover B in 10h: \(10*2=20\) miles
Distance between B and A by the time, A starts to travel: \(20*\pi-20\)

Time needed for A and B to meet distance between them divided by the relative speed: \(\frac{20*\pi-20}{2+3}= \frac{20*\pi-20}{5}=4*\pi-4\), as they are travelling in opposite directions relative speed would be the sum of their rates;

Time needed for A to be 12 miles ahead of B: \(\frac{12}{2+3}=2.4\);

So we have three period of times:
Time before A started travelling: \(10\) hours;
Time for A and B to meet: \(4*\pi-4\) hours;
Time needed for A to be 12 miles ahead of B: \(2.4\) hours;

Total time: \(10+4*\pi-4+2.4=4*\pi+8.4\) hours.

Answer: B.

Also discussed here: rates-on-a-circular-track-86675.html#p849908

Hope it helps.
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It is a long question stem so make sure you analyze each statement as you read it otherwise you will waste a lot of time re-reading the question again and again. Let me tell you how:

Car B starts at point X and moves clockwise around a circular track at a constant rate of 2 mph.

Ok, car B covers 2 miles every hour moving clockwise. We don't know the track length yet.

Ten hours later, Car A leaves from point X and travels counter-clockwise around the same circular track at a constant rate of 3 mph.
Car A is faster and covers 3 miles every hour counter clockwise. Since their directions are opposite, their relative speed = 2+3 = 5 mph. We still don't know the track length so we cannot say where car B was when car A started. All we know is that in 10 hrs, car B traveled 20 miles

If the radius of the track is 10 miles,

Now we know the track length. It is \(2{\pi}r = 2{\pi}*10 = 20{\pi}.\) It is greater than 20 miles that car B covered in 10 hrs so car B had not finished one round. Of the \(20{\pi}\), it had covered 20 miles.

for how many hours will Car B have been traveling when the cars have passed each other for the first time and put another 12 miles between them (measured around the curve of the track)?


To pass each other for the first time, cars A and B together need to cover the remaining distance on the circle i.e. \(20{\pi} - 20\). To create a distance of another 12 miles, they together need to travel 12 miles more away from each other

Time taken = Total distance to be traveled/ Relative Speed
Time taken = \((20{\pi} - 20 + 12)/5 = (20{\pi} - 8)/5 hrs = 4{\pi} - 1.6 hrs\)
Car B must have been traveling for \(4{\pi} - 1.6 + 10 = 4{\pi} + 8.4 hrs\)

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i don't know where i am making the mistake ...

Car B has already traveled (2 mph)(10 hours) = 20 miles in a clockwise direction by the time Car A starts moving 10 hours later in a counterclockwise direction (and from the same starting point).

let x distance from Car A starting point the two cars meet

then distance covered by car B=2pi*10-20-x
time taken by car B=(2pi*10-20-x)/2
distance covered by car A=x
time taken by car A=x/3

equating the two,
x/3=(2pi*10-20-x)/2
2x=60*pi-60-3x
x=(60*pi-60)/5
x=12(pi-1)
car B has to travel another 12 miles,therefore time taken by car B=(2pi*10-20-x+12)/2

or (20*pi-8-12*pi+12)/2=4*pi+2

What does the red part even mean? If it means the distance between B and A by the time, A starts to travel then we know exactly what it is. It equal to: \(20*\pi-20\) (car-b-starts-at-point-x-and-moves-clockwise-around-128215.html#p1050349).
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Everything is correct till here. Now, x is the distance traveled by A so time taken = x/3 = 12(pi-1)/3 = 4(pi-1)

Now notice that they have to together put 12 miles distance between them. They are moving in opposite directions so time taken to cover 12 miles together = 12/(2+3) = 2.4 hrs (relative speed concepts)

Total time taken = 4(pi-1) + 2.4 = 4*pi - 1.6 hrs
Car B has traveled 10 hrs extra so it must have been traveling for 4*pi - 1.6 + 10 = 4*pi + 8.4 hrs
car B has to travel another 12 miles,therefore time taken by car B=(2pi*10-20-x+12)/2
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Hi Karishma,

I have been fan of your logical approach, which I keep applying wherever possible. More so when I am not very comfortable with algebra part of word problems. I am attempting to solve this question based on logic.

I will consider Pi as 3. So the distance round the circle is 20pi or 60 mi approx.
Out of this Car B has already traveled 20 mi in 10 hours. Thus we are left with 40 mi to be covered by A and B together but driving in opposite direction @ 3km/hr and 2km/hr respectively. Thus to cover balance distance plus 12km i.e. 52miles, I need 52/5=10.4 hours.

Thus answer is 10hours plus 10.4 hours i.e. 20.4 hours and answer choice B (4pi+8.4) is correct.

I think there is a flaw in this question...

moving on a circular track could not be CONSTANT RATE! it has variable velocity So you could not solve the problem with Rate= Distance/ time Formula. Because this formulation is just for straight distances not for circular ones...

Yes, that's absolutely fine. I followed the exact same approach above (but with pi). I hope you understand that the approximation of pi worked because the answer was also in terms of pi. If you had a pure numerical answer, you would not have assumed pi = 3. Here, pi acted like a variable. Say, if you put pi = 10, you will still get the answer (of course you will need to substitute pi = 10 in the options too)
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Velocity has two components - speed and direction. Velocity is variable here only because of changing direction.

Speed = Distance/Time is applicable for circular distances as well since speed is not a vector.

GMAT only deals with speed, not with velocity. Constant rate implies a constant speed.
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"for how many hours will Car B have been traveling when the cars have passed each other for the first time and put another 12 miles between them"
i have a little problem with the language of the question stem ..isint the question asking that we have to calculate the time of travel since they met for the first time and to the point when the distance between them is 12 miles .in that case the answer wud have been straight 12 /5 hrs . i feel the language is little faulty .karisma ,buneul plz claify

The language of the question isn't very good. This does lead to different interpretations by different people sometimes. But most people will interpret it as 'from the starting point to right now (which is the point when the cars have passed each other for the first time and put another 12 miles between them)'


In GMAT, the language is always very clear. e.g. you could be given "After some time, the cars had put a distance of 12 miles between them after meeting for the first time. For how many total hrs did car B travel till this time?" - In this case, your answer is the one given above.

or you could have "After meeting for the first time, the cars continued on their respective paths and put a distance of 12 miles between them (they have met only once). How long has car B been traveling since meeting car A for the first time?" - In this case, your answer is 12/5
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It is easier to understand the question using a drawing:

\(A_1\) and \(B_1\)
are the points A and B are respectively after their first time pass. If we denote by T the time B travels from X to \(B_1\), then the total distance the two cars travel until the situation depicted in the figure is one whole circumference of the circle plus the overlap of 12 miles. Car A travels \(T-10\) hours.
We can write the following equation:
\(2T+3(T-10)=2\pi10+12\)
or \(5T=20\pi+42\),
from which we get \(T=4\pi+8.4\).

Answer: B

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Car B starts at point X and moves clockwise around a circular track at a constant rate of 2 mph. Ten hours later, Car A leaves from point X and travels counter-clockwise around the same circular track at a constant rate of 3 mph. If the radius of the track is 10 miles, for how many hours will Car B have been traveling when the cars have passed each other for the first time and put another 12 miles between them (measured around the curve of the track)?

This problem deals with the distance the cars have moved over a certain period of time so we should find the distance (circumference) of the track:
r=10
c=(pi)d
c=(pi)(20)
c=20(pi)

Car B leaves 10 hours before car A. When car A starts traveling B has traveled: distance=rate*time: distance=2miles/hour*10 hours: D=20 miles.

We need to know how long it takes for them to reach one another then how long it takes for them to put another to miles between the two of them.

When car A leaves B has already traveled for 10 hours and 20 miles. Therefore, car A and B have [(pi)20 - 20] miles to travel before they reach one another.

Their combined rate is 2miles/hour and 3miles/hour = 5 miles/hour

Time = distance/rate
Time = [(pi)20 - 20]/5
Time = [(pi)4 - 4]

It takes [(pi)4 - 4] hours for them to reach one another.

We now have to figure out how long it takes for them to travel past one another and put 12 miles between them:

Distance = r*t
12 = (2+3)*t
12/5 = time

It takes another 2.4 hours for the two cars to travel a combined 12 hours away from one another.

So, Car B has been on the circuit for 10 hours + (pi)4 - 4 hours + 2.4 hours: (pi)4+8.4 hours total.

ANSWER: B. 4pi + 8.4

Hi ,

Can anyone help me with the level of question ?650+??

thanks

You can check difficulty level of a question along with the stats on it in the first post. For this question Difficulty = 700 Level. The difficulty level of a question is calculated automatically based on the timer stats from the users which attempted the question.
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The statistics at the top of the question provide a lot of information about difficulty level.


This question has a difficulty rating out of 95%, which is the highest rating.
Also notice that only 48% of students correct answered the question, and the average student took 3:01 to solve the question.

All this information makes this question a 750+ level GMAT question.

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