That logic (3-2) applies to calculate the time required to keep 12 miles between car A and car B after they meet, but the 1st part is different since the distance between car A and car B when car A start is only 20 miles and not 20pi - 20 miles

The 1st equation will be 20 + 2t = 3t ==> t = 20 hours

Hope this helps.

Yes, the speed of car A relative to car B does change to 3 - 2 = 1 mph.
Also, the question becomes simpler since after 10 hrs, car B is 20 miles ahead of car A. Car A is faster and has to catch up to B and go 12 miles ahead. So relative to B, car A has to cover 32 miles which it will do in 32 hrs (since relative speed of A relative to B is 1 mph)
Add another 10 to it to account for the 10 hrs B spent initially and you get 42 hrs.
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Car B begins moving at 2 mph around a circular track with a radius of 10 miles. Ten hours later, Car A leaves from the same point in the opposite direction, traveling at 3 mph. For how many hours will Car B have been traveling when car A has passed and moved 12 miles beyond Car B?

R=10
c=2(pi)r
Track circumference =20(pi)
In 10 hours car B will have traveled 10*2=20 miles
So when car A starts, car B will have a 20 mile head start on it.
When A leaves, it leaves in the opposite direction. Therefore, it's not simply 20 miles behind B. For example, look at a clock. Pretend B left from where 12 is on the clock and is currently sitting on where 4 is. If A left and followed B it would be 1/3rd of the clocks circumference behind B. However, if it leaves in the opposite direction it has all the numbers between 12 and 4 between it and B, or 2/3rds of the clocks circumference between it and B. Therefore, the distance between A and B is:

20(pi)-20

The time it takes for them to pass one another is the distance they must travel to do so [20(pi)-20] divided by their two rates of travel (2 and 3 miles/hour)

[20(pi)-20] / (2+3)
[20(pi)-20] / (5)
Time = 4(pi)-4

The time it takes for A to move 12 miles AWAY from B is their combined rate of speed:
T = 12/(2+3)
This caused me much confusion at first. I treated it as if A and B were moving in the same direction and I was looking for how fast A was pulling ahead of B. They are moving in opposite directions at 2 and 3 miles per hour respectively. It would be no different than if one car was moving away from point x at a speed of (2+3) The distance it would put between itself and X would be the same distance A and B put between them at 3 and 2 Miles/hour respectively!

The time it takes for A and B to move 12 miles away from one another is 12/5 = 2.4 hours.

Therefore, it takes 4(pi)-4 hours for them to reach one another + another 2.4 hours for them to move another 12 miles away from one another. Keep in mind, we also need to add in the 10 hours car B traveled before car A left because the question is looking for the total number of hours car B has been on the road when car A is ten miles past it in the opposite direction.

Therefore, Car B has been traveling for 10+4(pi)-4+2.4 hours

Answer: (B) 4(pi)+8.4 hours

It can also be solved by Tabular form as is suggested in the GMAT Club Math Book.

Ugh. That's so sleazy to call the first car "Car B" and the second car "Car A". That's what tripped me up.

This is actually not that hard if you have your basics right!! I learnt this tabular format in the Math GMAT Club book. Might help you out with such questions. It has helped me for sure.

Yes, actual GMAT questions will not try to trick you in such an uncool manner. If you get tricked by something in GMAT, it will be conceptual such that when you see the explanation you will go 'oh wow!'
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the length of the circular track is ~63 miles(2*pi*r).
B and A are travelling in opp directions
B started earlier at 2 mph, travelling for 10 hrs=dist. covered 20 miles.
now A starts from opp direction at 3 mph from same point(the key clue) and both A and B will cover ~43 miles at the combined speed of 5 mph which give time as 8.6 hrs for each A and B.
question also involves additional travel of 12 miles in opp direction which results in additional 2.4 hrs for each A and B.
so car B has been travelling for 10 hrs+8.6 hrs +2.4 hr=21 hrs.

option B) 4pi + 8.4 = 20.97 hrs = ~21 hrs

the length of the circular track is ~63 miles(2*pi*r).
B and A are travelling in opp directions
B started earlier at 2 mph, travelling for 10 hrs=dist. covered 20 miles.
now A starts from opp direction at 3 mph from same point(the key clue) and both A and B will cover ~43 miles at the combined speed of 5 mph which give time as 8.6 hrs for each A and B.
question also involves additional travel of 12 miles in opp direction which results in additional 2.4 hrs for each A and B.
so car B has been travelling for 10 hrs+8.6 hrs +2.4 hr=21 hrs.

option B) 4pi + 8.4 = 20.97 hrs = ~21 hrs

Check the attached solution...

Answer: option B

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The problem gets easier when we get rid of unnecessary abstraction caused by presence of pie. Just take the length of the lap as 60 miles (2*pie*10). Second, understand that the cars are moving towards each other/from each other hence you need to add up the individual speeds (3+2).

So in first 10 hours B covered 20 miles (2 *10) and when A started off only 40 miles separated them on the lap. This will be covered in 8 hours (40/5). Finally after they meet and go in opposite directions again 12 miles will be covered in 2.4 hours (12/5).

So in total this sums up to 10+8+2.4 = 20.4 hours for B. Answer B only fits if we take pie for 3.

there are 3 legs to B's trip:
leg 1=10 hours before A starts
leg 2=(20⫪-20)/(2+3)=4⫪-4 hours before meeting A
leg 3=12/(2+3)=2.4 hours before A moves 12 miles beyond B
total time for B's trip=4⫪+8.4 hours

d = 2*pi*r = 20pi
let pi = 3 (roughly), hence d would be 66

after 10 hrs B would have travelled 20 miles, so remaining d = 46

A travels in opposite direction, so inorder to meet time taken would be 46/2+3 = 9.2
time for further 12 miles would be 12/2+3 = 2.4

B has already been travelling for 10 hours, so total time = 9.2+2.4+10 = 21 (approx)

from answer choices B and C are closest
B is 4pi +8.4 => 12+8.4 = 20.4
C is 4pi + 10 => 12+10.4 = 22.4

Since B is closer, I went with B

This is still a very challenging question for me, particularly because the distance of the track can loop

For example, what if B was going at a rate of 8 miles per hour for 20 hours before A started - how then would you calculate the distance between them?

If the track is only 20*pi miles (roughly 60.2 miles) and car B has driven 160 miles, how would that change the set up to find the distance between A and B when A is starting?

circumference is 2pie R => 20pie

after 10hr => B's distance from X => 20pie - 20
Now A start towards B => rel speed = 5 M/h
T1 = time to meet or rendezvous => 4pie -4

Now relative speed = 5 but distance more to cover is 12M => 2.4 hr
T2 => 10 + 4pie - 4 + 2.4
T total => 4 pie + 8.4

B it is

In my opinion, this one is a little bit misleading, as we are told that is a circular track and we are asked "when car A has passed and moved 12 miles beyond Car B?", one could understand that the linear distance between cars is the one that has to be found. Therefore, that unlucky exam taker would start searching for a chord 12 miles long.

Car B begins moving at 2 mph around a circular track with a radius of 10 miles. Ten hours later, Car A leaves from the same point in the opposite direction, traveling at 3 mph. For how many hours will Car B have been traveling when car A has passed and moved 12 miles beyond Car B?

A. 4π–1.6

B. 4π+8.4

C. 4π+10.4

D. 2π–1.6

E. 2π–0.8

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the length of the circular track = 20π miles
every 1 hour car A and car B move together = 5 miles
let t be "hours"

car A and B first met + 12 miles = the length of the circular track = the distance of car A + the distance of car B
20π + 12 miles = 20miles(distance A) + 5miles(A and B together every hour) * t(hours)
--> 20π - 8 = 5t
--> 4(5π - 2) = 5t
--> 0.8(5π - 2) = t
--> 4π - 1.6 = t

the question is asking hours of car B,
therefore t hours (4π - 1.6) + 10 hours (the hours that B moved first) = 4π + 8.4

the answer is B

Given: 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.
Asked: 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)?

Car B travels = 2*10 = 20 miles before Car A starts
Circumference of the track =2π∗10=20π miles

Distance travelled by cars the cars have passed each other for the first time and put another 12 miles between them (measured around the curve of the track) = 20\pi - 20 + 12 = 20\pi - 8 miles
Relative speed of the cars = 2 + 3 = 5 mph

Time taken by both cars=(20π−8)/5=4π−1.6

Hours Car B have been traveling =4π−1.6+10=4π+8.4hours

IMO B
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Hello Bunuel
Is there any formula to directly get the answer here or reduce the steps? If yes, please share the same