Problem Solving

Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Quickest approach is to ballpark. Pi ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20pi ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only answer choice B works: 4pi + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B

Many thanks man, similar explanation but quicker

why are we using combined rate why can't we just divide 52 by 2. since we just have to find car B's hours. please
clear my doubt

GMATGuruNY wrote:

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Quickest approach is to ballpark. Pi ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20pi ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only answer choice B works: 4pi + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B

gmatapril wrote:why are we using combined rate why can't we just divide 52 by 2. since we just have to find car B's hours. please
clear my doubt

GMATGuruNY wrote:

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Quickest approach is to ballpark. Pi ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20pi ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only answer choice B works: 4pi + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B

After B has traveled for 10 hours alone, A starts to travel as well.
At this point we have to combine A's rate with B's rate because the two cars are traveling toward each other.
Since B's rate is 2 mph, each hour B travels 2 miles toward A.
Since A's rate is 3 mph, each hour A travels 3 miles toward B.
Thus, every hour A and B travel 2+3 = 5 miles toward each other.

thanks a lot for clearing my doubt. just one more question if both the cars were moving in same direction then we would had subtracted their Rates.
thank you.

GMATGuruNY wrote:

gmatapril wrote:why are we using combined rate why can't we just divide 52 by 2. since we just have to find car B's hours. please
clear my doubt

GMATGuruNY wrote:

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Quickest approach is to ballpark. Pi ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20pi ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only answer choice B works: 4pi + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B

After B has traveled for 10 hours alone, A starts to travel as well.
At this point we have to combine A's rate with B's rate because the two cars are traveling toward each other.
Since B's rate is 2 mph, each hour B travels 2 miles toward A.
Since A's rate is 3 mph, each hour A travels 3 miles toward B.
Thus, every hour A and B travel 2+3 = 5 miles toward each other.

gmatapril wrote:thanks a lot for clearing my doubt. just one more question if both the cars were moving in same direction then we would had subtracted their Rates.
thank you.

GMATGuruNY wrote:

gmatapril wrote:why are we using combined rate why can't we just divide 52 by 2. since we just have to find car B's hours. please
clear my doubt

GMATGuruNY wrote:

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Quickest approach is to ballpark. Pi ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20pi ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only answer choice B works: 4pi + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B

After B has traveled for 10 hours alone, A starts to travel as well.
At this point we have to combine A's rate with B's rate because the two cars are traveling toward each other.
Since B's rate is 2 mph, each hour B travels 2 miles toward A.
Since A's rate is 3 mph, each hour A travels 3 miles toward B.
Thus, every hour A and B travel 2+3 = 5 miles toward each other.

Yes, if A and B were moving in the same direction, we would subtract the slower rate from the faster rate.
Since A = 3 mph and B = 2 mph, in one hour A will travel 3-2 = 1 more mile than B.
Thus, if A and B leave from the same location at the same time, the rate at which A will move ahead of B = 3-2 = 1 mph.

It is a pretty good question. The lesson that I learned here were couple:
1) look for what the question is asking. I stopped solving when I found the total time it takes for them to travel the asked distance, when the question was asking how much time B kept travelling.
2) This is a collision problem, we ADD the rates, not subtract.

It is helpful to draw a diagram on this one:

Draw a circle with an arrow going clock-wise, indicating that B is traveling in that direction.

Now, RTD chart

R T D
2mph x 10 h = 20 miles

so, B has been traveling for 20 miles when A starts.

The entire track is 2 pi r = 20 pi (This means that B is somewhere on the track, I drew another picture to just keep that in mind)

Now, the question is asking how long will it take for them to get past each other + add 12 miles.

So, we have total distance = 20pi - 20 + 12 = 20 pi - 8

Therefore,

20 pi 8 = (5) T
T = 4 pi - 1.6

Now, the fun has not stopped yet, we need to find the total time B has been traveling.

B had already traveled 10 hours so we add this in:

(4 pi - 1.6) + 10 = 4pi - 8.4 Answer.

Hi,

I guess we can't include 10 hrs which B already covered as the ques says for how many hours will car B have been travelling WHEN the cars have passed each other for the first time and put another 12 miles between them.
It is not asking the total time B traveled.

i.e. (20pie- 20) + 12 = (20pie - 8) miles / 5

Please clear my doubt.

GMATGuruNY wrote:

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Quickest approach is to ballpark. Pi ≈ 3.

Car B is traveling for more than 10 hours, so answer choices D and E are too small, and A is unlikely. The correct answer is either B or C.

Circumference of track = 20pi ≈ 60 miles.
In 10 hours, distance for B = 2*10 = 20 miles.
60-20 = 40 miles between A and B.
A and B now have to travel the 40 miles between them and then -- after they meet -- keep traveling in opposite directions until there is another 12 miles between them. So the total distance that they need to travel is 40+12 = 52 miles.
When things work together, we can add their rates. Combined rate for A+B = 3+2 = 5 miles/hour.
Time for A+B = Distance/Rate = 52/5 = 10.4 hours.
Since B traveled for 10 hours before A joined in, the total time for B = 10 + 10.4 = 20.4 hours.
Only answer choice B works: 4pi + 8.4 ≈ 12 + 8.4 = 20.4.

The correct answer is B

Ankur87 wrote:Hi,

I guess we can't include 10 hrs which B already covered as the ques says for how many hours will car B have been travelling WHEN the cars have passed each other for the first time and put another 12 miles between them.
It is not asking the total time B traveled.

i.e. (20pie- 20) + 12 = (20pie - 8) miles / 5

Please clear my doubt.

Hi Ankur,

If you carefully review what that question is asking is then you will come to a conclusion that 10 hours have to be indeed added into the final answer for B.
"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 "

Have been traveling = Continuous tense, the action started in the past and still continues... B started long back and the poor chap is still driving!
and = +
So essentially its asking "How much distance traveled UNTIL NOW" + "How much for another 12 miles"

Hope that clears the doubt.

Regards,
Vivek

waltz2salsa wrote:Ques: 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)?
a) 4pi- 1.6
b) 4pi+ 8.4
c) 4pi+ 10.4
d) 2pi- 1.6
e) 2pi- 0.8

OA:B

I am not satisfied with the explanation provided. Can someone please help!

Regards,
Waltz2Salsa

Check the solution...

Answer: Option B