Tricky Problem?

Question

Please help me answer the following problem.

A city, in the shape of a '+' sign, has 5 bus stations, one at each end of the cross and the depot in the middle. The east station is A, north is B, west is C and south is D, while the middle depot is O. All buses travel from O to A and back to O, then immediately O to B and back etc. When the bus travels from O to D and back, it instantly starts again and travels to A, continuing the course repetitively for 9 hours. The time it takes to travel to O to A and back is 11min, O to B and back is 4 min, O to C and back is 7 min while O to D and back is 4 min. A second bus leaves the depot O at 7:37, traveling the exact same route as the first bus, also for 9 hours. How many times between 8:00 and 3:00 do the buses meet?

Thanks for helping me.
If it is not clear enough, state it in your answer and I will add the details, or email me at my yahoo address.

Sorry, the first bus leaves at 7:00

Answer 1

The two buses will pass each other 49 times.
The way I did it was to calculate the total amount of time they met each other in a full trip. As a full trip takes 26 min (11 + 4 + 7 + 4), we can now calculate how often they meet in 26 min.
1) Starting from 7:00, bus 1 starts its journey. At 7:26, it has finished 1 round.
2) 11 mins later, at 7:37, bus 1 arrives back to O from A just as bus 2 starts its first journey to A. This is 1 pass. (11 min has passes)
3) Another 11 mins later, at 7:48, bus 1 has finished two stops – B and C, and returns back to O just as bus 2 finishes its first journey to A and back. This is the second pass. (22 min has passes)
4) 4 mins later, 7:52, they have both finished one stop (D and B respectively) and return to O at the same time. This is the 3rd pass. (26 min has passed – Bus 1 has completed its 2nd journey while bus 2 is ½ way through its 1st journey).
5) This means that every 26 min from now on, they will pass 3 times until the last stop before 3:00.
6) 11 min later, it is 8:03, where we can now start counting every pass they make until 3:00.
7) Since they will pass every 26 min starting 8:03, the list below show the next times.
8:03, 8:29, 8:55, 9:21, 9:47, 10:13, 10:39, 11:05, 11:31, 11:57, 12:23, 12:49, 1:15, 1:41, 2:07, 2:33 and 2:59.
8) They pass 3 times from 8:03 to 2:33, but at 2:59, they pass only once, as 1 min later, bus 1 finishes its journey for the day. Plus, if they were to continue, the next pass would be out of the 3:00 time limit.
9) That means they pass 3 times for 16 journeys and 1 time at the end. This is equal to 3 x 16 + 1, which equals 49.

Answer 2

Easiest way to visualize this is to draw your plus & put an 11 by A, 4 by B, 7 by C & 4 by D.

Total round trip = 11+4+7+4=26 minutes, so what ever happens will repeat every 26 minutes.

At 7:26, Bus 1 (B1) is starting second route. 11 minutes later, 7:37, it's back at O just as B2 is leaving, so they pass once.

11 minutes later, 7:48, B1 is at O coming from C & B2 is at O coming off A, that's the second pass.

4 minutes later, 7:52, they are both at O again, third pass.

11 minutes later, 8:03, they are both at O (B1 coming off A, B2 coming off D). Note they are in their starting positions again. This means that each 26 minutes they pass 3 times (we don't count 8:03 as a pass in that first 26 minutes because it is the first pass of the second 26 minutes).

Their cycle is every 26 minutes, so we have them at:
8:03
8:29
8:55
9:21
9:47
10:13
10:39
11:05
11:31
11:57
12:23
12:49
1:15
1:41
2:07
2:33
2:59

The first 16 cycles, they pass 3 times each. But the last one, 2:59, they pass only once.

Therefore they pass 16x3+1=49 times

Answer 3

You didn't include when the first bus left.