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Krishchamp
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Car B begins moving at 2 mph around a circular track with a radius of 18 Oct 2020, 09:49
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yangsta8
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. \(4\pi – 1.6\)

B. \(4\pi + 8.4\)

C. \(4\pi + 10.4\)

D. \(2\pi – 1.6\)

E. \(2\pi – 0.8\)


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The OA is pretty long and even solving it that way takes me +2 mins. Hopefully someone can offer a fast solution.
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Bunuel VeritasKarishma MathRevolution JeffTargetTestPrep BrentGMATPrepNow EMPOWERgmatRichC


Can you please tell how can we solve this one by equating the time they take to meet each other. We have 20pi - 20 as the distance they need to travel and speeds are also given. So if one travels X distance than the other would travel X- (20pi - 20) and thus we will find X (the distance at which they meet) But then how to proceed further? Please help me solve this one by this concept of equating time.
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Car B begins moving at 2 mph around a circular track with a radius of 20 Oct 2020, 23:53
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Krishchamp
yangsta8
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. \(4\pi – 1.6\)

B. \(4\pi + 8.4\)

C. \(4\pi + 10.4\)

D. \(2\pi – 1.6\)

E. \(2\pi – 0.8\)


Show Spoiler
The OA is pretty long and even solving it that way takes me +2 mins. Hopefully someone can offer a fast solution.

Bunuel VeritasKarishma MathRevolution JeffTargetTestPrep BrentGMATPrepNow EMPOWERgmatRichC


Can you please tell how can we solve this one by equating the time they take to meet each other. We have 20pi - 20 as the distance they need to travel and speeds are also given. So if one travels X distance than the other would travel X- (20pi - 20) and thus we will find X (the distance at which they meet) But then how to proceed further? Please help me solve this one by this concept of equating time.
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By taking distance X, you are using a longer more circutous method to arrive at the answer.

The distance they have to travel together is (20pi - 20 + 12) = 20pi - 8
The distance of X and [20pi - 8 - X] is travelled by them at speeds 2 mph and 3 mph in the same time.

X/2 = [20pi - 8 - X]/3

3X = 40pi - 16 - 2X

X = 8pi - 3.2

Car B travelled for 8pi - 3.2 miles. Time taken to cover this distance = (8pi - 3.2)/2 = 4pi - 1.6
Total time traveled by car B = 10 + 4pi - 1.6
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Car B begins moving at 2 mph around a circular track with a radius of 21 Oct 2020, 00:52
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distance travelled by B in 10 hours ; 20 miles
distance b/w A&B 20pi-20
relative speed of a&b ; 5mph
now for A 12 miles ahead of B ; 20pi-20+12 ; 20pi-8
time taken ; 20pi-8/5 ; 4pi-1.6
B has been travelling for 10 hrs so its time ; 4pi-1.6+10 ;
4pi+8.4 hours
option B

yangsta8
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. \(4\pi – 1.6\)

B. \(4\pi + 8.4\)

C. \(4\pi + 10.4\)

D. \(2\pi – 1.6\)

E. \(2\pi – 0.8\)




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The OA is pretty long and even solving it that way takes me +2 mins. Hopefully someone can offer a fast solution.
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Car B begins moving at 2 mph around a circular track with a radius of 18 Apr 2021, 09:43
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Radius of track is 10 miles so circumference is 20*pi i.e. the total length of the track.
B starts from X and travels for 10 hrs clockwise at 2 mph i.e. it travels 20 miles. Now car A starts from X counter clockwise. Distance between A and B is 20*pi - 20.
Now, to meet, they have to together cover this distance plus 12 miles more which they have to put between them.
Time taken to cover this distance by them = (20*pi - 20 + 12)/(3 + 2) = 4*pi - 1.6 hrs
Car B has been traveling for 10 + 4*pi - 1.6 = (4*pi + 8.4) hrs

Hi Karishma, i actually got this question wrong when i took the mgmat cat last week, i got confused on the explanation which is similar to yours (your diagram helps though), how did you derive this equation: (20*pi - 20 + 12)/(3 + 2), was this manipulated from Rate x Time = Distance? thanks.
Attachment:
Ques1.jpg

The red distance is what B has already covered at 2 mph in 10 hrs. This distance is 20 miles.
A and B are now moving towards each other (as shown by green arrows). To meet for the first time, they have to cover the remaining circumference of the track i.e. a distance of 20pi - 20. (20pi is the circumference of the circle out of which 20 has already been covered by B). They need to create a further 12 miles distance between them. So together they need to cover (20pi - 20 + 12) miles in all.
Since, A and B are moving towards each other, their relative speed (i.e. combined speed here) will be (3 + 2) mph.
So time taken for them to meet = D/S = (20pi - 20 + 12)/(3 + 2)

- Here, we are using the concept of Relative Speed. When two objects (speeds S1 and S2) move in opposite directions (towards each other or away from each other), they cover the distance between them (or create distance between them) at the rate of (S1 + S2). Here they are moving in opposite directions towards each other so their relative speed is sum of their speeds. After meeting, they are moving away from each other but their relative speed is still sum of their speeds.
When two objects move in same direction, their speeds get subtracted.
If this is unclear, I would suggest looking up the theory of relative speed for details.
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VeritasKarishma if circumference of circle is 20pi doesntt it mean that whole circumference is 20 miles ? And B covered a distance of a one whole circle instead of fraction i.e. 20p-20 ?
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Car B begins moving at 2 mph around a circular track with a radius of 18 Apr 2021, 23:09
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VeritasKarishma[/url] if circumference of circle is 20pi doesntt it mean that whole circumference is 20 miles ? And B covered a distance of a one whole circle instead of fraction i.e. 20p-20 ?
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Circumference of a circle is 2*pi*r which is the length of the track. Since radius is 10 miles, the length is 2*pi*10 = 20*pi which is something more than 60 miles because pi = 3.1416

B travels at 2 mph for 10 hrs so covers 20 miles. Hence in 10 hrs, of the 60+ miles of the track, B covers only 20 miles.
Now A and B need to cover the rest of 40+ miles together.
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Car B begins moving at 2 mph around a circular track with a radius of 14 May 2021, 16:28
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The distance left for A to complete one whole lap + twelve extra miles, driven at the cars' combined rate:

(20Pi - 20) + 12 = 5t

(20Pi - 8)/5 = t

Adding the first 10 hrs as fifths gives:

(20Pi + 42)/5 = 4Pi + 8,4
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Car B begins moving at 2 mph around a circular track with a radius of 23 May 2022, 18:14
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yangsta8
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. \(4\pi – 1.6\)

B. \(4\pi + 8.4\)

C. \(4\pi + 10.4\)

D. \(2\pi – 1.6\)

E. \(2\pi – 0.8\)


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The OA is pretty long and even solving it that way takes me +2 mins. Hopefully someone can offer a fast solution.
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It's fascinating to me that more than 50% of responses have been incorrect. If B has been driving for 10 hours before A even starts and we are asked how long B drives, it MUST be greater than 10.
A. Roughly 12.6-1.6 = 11. C'mon, that's obviously too low. Wrong.
B. Roughly 12.6+8.4 = 21. Okay, hold onto it.
C. Roughly 12.6+10.4 = 23. Okay, hold onto it.
D. Roughly 6.3-1.6 = 4.7. That's not even as long as B drove before A even started!! Wrong.
E. Roughly 6.3-0.8 = 5.5. That's not even as long as B drove before A even started!! Wrong.

At the worst, without doing anything other than using simple logic (and barely any math), we have a 50/50. And yet, more than 50% of people have gotten this question wrong. Missing the logic of the question is a HUGE problem. Please, on difficult problems, save yourself from picking a silly answer just by taking a few seconds to deploy some logic!!



Okay, so now you want to know how to solve this in a way that I haven't seen yet on this thread? I'm going to use two of my favorite techniques on geometry questions: ballparking and manipulating the figure.

First, there's really nothing about this problem that forces it to be a circular track rather than a straight road with two cars driving toward each other. My brain is much better at visualizing the straight road, so I'm going with that (I always look for opportunities to convert circles into straight lines when the circles are used as tracks)!!

The distance between the cars at the beginning is 20pi. Lol @ 20pi. Ballpark! 63 (blue arrow).

B starts driving for 10 hours, so it covers 20 miles (green arrow). That means B and A are now separated by 43 miles (red arrow). We need to close that gap to zero and then keep going until the cars are separated by 12 miles, now with B on the right and A on the left. So, they need to travel 55 miles combined. At a combined rate of 5mph, that's 11 hours on top of the 10 that B already traveled solo. That's a total of 21 hours. Look at the answer choices.

A. Roughly 12.6-1.6 = 11.
B. Roughly 12.6+8.4 = 21.
C. Roughly 12.6+10.4 = 23.
D. Roughly 6.3-1.6 = 4.7.
E. Roughly 6.3-0.8 = 5.5.

Answer choice B.


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Car B begins moving at 2 mph around a circular track with a radius of 13 Aug 2022, 01:03
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krishnanori
EASY EQUATION: I think the easy way to calculate is.

Distance travelled by B + Distance travelled by A = Circumference + 12
Let's sat the answer is T.

2T + 3(T-10) = (2 * Pi * 10 )+ 12
5t = 20 pi + 42
t= 4 pi + 8.4
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Hi, I was wondering what did i do wrong here.
I was able to setup a similar equation to this, which is

3T + 2(T+10) = 20pi + 12

Assuming that B travelled 10 hrs more than A.
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Car B begins moving at 2 mph around a circular track with a radius of 13 Aug 2022, 12:43
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krishnanori
EASY EQUATION: I think the easy way to calculate is.

Distance travelled by B + Distance travelled by A = Circumference + 12
Let's sat the answer is T.

2T + 3(T-10) = (2 * Pi * 10 )+ 12
5t = 20 pi + 42
t= 4 pi + 8.4


Hi, I was wondering what did i do wrong here.
I was able to setup a similar equation to this, which is

3T + 2(T+10) = 20pi + 12

Assuming that B travelled 10 hrs more than A.
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peaarrr

It appears that you have assigned T to be the amount of time that A travels, but the question asks for how long B travels. Did you remember to add 10 hours to account for "Ten hours later, Car A leaves from the same point in the opposite direction?"
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Car B begins moving at 2 mph around a circular track with a radius of 15 Aug 2025, 14:00
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Fun convergence problem. Remember to add on the original 10 hours at the end!:
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Car B begins moving at 2 mph around a circular track with a radius of 18 Sep 2025, 13:28
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