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A bus from city M is traveling to city N at a constant speed

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A bus from city M is traveling to city N at a constant speed [#permalink] New post 10 Jul 2019, 11:26
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Question Stats:

20% (00:10) correct 80% (03:52) wrong based on 10 sessions
A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for two hours. The following day the buses do the return trip at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A.48

B.72

C.96

D.120

E. 192
[Reveal] Spoiler: OA

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Re: A bus from city M is traveling to city N at a constant speed [#permalink] New post 11 Jul 2019, 04:58
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Carcass wrote:
A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for two hours. The following day the buses do the return trip at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A.48
B.72
C.96
D.120
E. 192


Buses traveling at SAME speeds meet in point P after driving for 2 hours.
IMPORTANT: Since the 2 buses are traveling at the SAME speed, then point P must be HALF WAY between city M and city N

Also recognize that the TOTAL travel time = 4hrs (2 hrs for each bus)

TOTAL distance traveled = (distance one bus traveled) + (distance other bus traveled)
Distance = (travel time)(rate)
Let r = the rate that EACH bus is traveling.
So, we get: TOTAL distance traveled = 2r + 2r = 4r
So, from the above fact, the distance from point P to city M (and to city N) = 4r/2 = 2r

One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?
Another way to read this is: One bus leaves its city. ONE HOUR LATER, the other bus leaves its city.
Let's say Bus A leaves city M at noon, and Bus B leaves city N at 1pm.
Let t = Bus A's travel time until they meet
So, Let t-1 = Bus B's travel time until they meet

Since the TOTAL travel time = 4hrs, we can write: t + (t-1) = 4
Solve to get: t = 2.5
So, Bus A traveled for 2.5 hours, Bus B traveled for 1.5 hours,
This means Bus A traveled further. In fact, Bus A travels PAST point P (which is halfway between cities M and N) for an ADDITIONAL 24 miles
In other words, Bus A's travel distance = 2r + 24
Since Bus A traveled for 2.5 hours at a rate of r, we can write: 2.5r = 2r + 24
Multiply both sides by 2 to get: 5r = 4r + 48
Solve: r = 48

TOTAL distance traveled = 4r
= 4(48)
= 192

Answer: E

Cheers,
Brent
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Re: A bus from city M is traveling to city N at a constant speed [#permalink] New post 11 Jul 2019, 05:11
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This was good :)
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Re: A bus from city M is traveling to city N at a constant speed [#permalink] New post 11 Jul 2019, 07:14
GreenlightTestPrep wrote:
Carcass wrote:
A bus from city M is traveling to city N at a constant speed while another bus is making the same journey in the opposite direction at the same constant speed. They meet in point P after driving for two hours. The following day the buses do the return trip at the same constant speed. One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?

A.48
B.72
C.96
D.120
E. 192


Buses traveling at SAME speeds meet in point P after driving for 2 hours.
IMPORTANT: Since the 2 buses are traveling at the SAME speed, then point P must be HALF WAY between city M and city N

Also recognize that the TOTAL travel time = 4hrs (2 hrs for each bus)

TOTAL distance traveled = (distance one bus traveled) + (distance other bus traveled)
Distance = (travel time)(rate)
Let r = the rate that EACH bus is traveling.
So, we get: TOTAL distance traveled = 2r + 2r = 4r
So, from the above fact, the distance from point P to city M (and to city N) = 4r/2 = 2r

One bus is delayed 24 minutes and the other leaves 36 minutes earlier. If they meet 24 miles from point P, what is the distance between the two cities?
Another way to read this is: One bus leaves its city. ONE HOUR LATER, the other bus leaves its city.
Let's say Bus A leaves city M at noon, and Bus B leaves city N at 1pm.
Let t = Bus A's travel time until they meet
So, Let t-1 = Bus B's travel time until they meet

Since the TOTAL travel time = 4hrs, we can write: t + (t-1) = 4
Solve to get: t = 2.5
So, Bus A traveled for 2.5 hours, Bus B traveled for 1.5 hours,
This means Bus A traveled further. In fact, Bus A travels PAST point P (which is halfway between cities M and N) for an ADDITIONAL 24 miles
In other words, Bus A's travel distance = 2r + 24
Since Bus A traveled for 2.5 hours at a rate of r, we can write: 2.5r = 2r + 24
Multiply both sides by 2 to get: 5r = 4r + 48
Solve: r = 48

TOTAL distance traveled = 4r
= 4(48)
= 192

Answer: E

Cheers,
Brent



Very tough :cry:
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Re: A bus from city M is traveling to city N at a constant speed [#permalink] New post 11 Jul 2019, 08:09
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AlaminMolla wrote:

Very tough :cry:


Very tough indeed! A 165+ level question for sure.

Cheers,
Brent
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Re: A bus from city M is traveling to city N at a constant speed [#permalink] New post 21 Jul 2019, 23:06
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I usually teach problems like this with a picture, even if only to help guide my algebra. Because the trains are going at the same speed, the meeting point must be precisely halfway between the positions the trains are in at the moment they start moving toward one another. When they leave at the same time, it will look like this:

|------|------*|------|------|

It takes one hour to go from one tick to the next one. The trains meet at the distance covered by one train in two hours, and since that is the exact halfway point, the total distance will be four hours worth of travel.

|xxxxxx|------|---*---|------|

For the second trip, one train has been moving for a full hour before the other one moves, so it must be one hour's worth of distance past its starting point before the other train starts. From that point and the other train's starting point, we can see the middle is exactly a half-hour's distance from the old meeting point.

We are told that this distance, which we just figured out to be a half-hour's worth of travel, is 24 miles. We can then multiply that number by 8 (there are eight half-hours in four hours) to get the answer of 192.

Some students do better just going right to the algebra, and some find the picture to be helpful in understanding the situation. Your mileage may vary.

-Jordan Wolbrum
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Re: A bus from city M is traveling to city N at a constant speed   [#permalink] 21 Jul 2019, 23:06
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