Problem Solving

I'd just write out departure and arrival times for each train and look for the least possible wait times.

Train X: With just over 4 hours of travel time [1 (3/4) + 2(1/3)] the train leaving at 11am is the last possible one to catch to get to Z no later than 3:30pm
Departs Arrives
6am 7:45am
7am 8:45am
8am 9:45am
9am 10:45am
10am 11:45am
11am 12:45pm

Train Y: With 2 (1/3) hours of travel time, the train leaving at 1pm is the last possible one to catch to get to Z no later than 3:30pm
Departs Arrives
9am 11:20am
9:30am 11:50am
10am 12:20pm
10:30am 12:50pm
11am 1:20pm
11:30am 1:50pm
12:00pm 2:20pm
12:30pm 2:50pm
1:00pm 3:20pm

Train Z: With just over 4 hours of travel time from X, there is no way to get to Z any earlier than 10:20
Departs:
(8am, 8:45am, 9:30am, 10:15am)- Can ignore
11am
11:45am
12:30pm
1:15pm
2pm
2:45pm
3:30pm

OK, now take a look! The X train arrives 45 minutes past the hour and the Y train departs on the hour and on the half-hour, so the ideal shortest wait time would be 15 minutes (:45 to :00). Similarly, the Y train arrives at :20 and :50 past the hour and the Z train departs at various points :15, :30, :45 and :00 so the ideal shortest wait time would be 10 minutes (from :20 to :30 or from :50 to :00). So it looks like the shortest possible wait time is 25 minutes. Finally, check to see if that's possible by thinking backwards.

Could she take the 3:30 train from Z with minimum wait time? Sure! She could arrive at Y at 3:20. OK, could she get on that 1:00 train at Y with minimum wait time? Sure! she could get off the X train at 12:45 if she starts at 11am. Since there's no way she can wait less than 25 minutes, and this trip is possible, I'd say 25 minutes is your answer. Depart X at 11am. X: 11am - 12:45pm, Y: 1:00pm - 3:20pm, Z: 3:30pm.

Take note, that this is not your only option. She could also take X: 8am - 9:45am, Y: 10am-12:20pm, Z: 12:30 for that minimum wait time. So it's not like you have to just guess the right one. Sure, writing all this out took a few minutes, but I never worry about time when my pencil is moving. It's quicker than it seems. I only worry about how long a question is taking if I'm stuck staring at it.


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swerve wrote:A woman is planning a trip that involves 3 connecting trains that depart from Stations X, Y, and Z, respectively. The first train leaves Station X every hour, beginning at 6 a.m., and arrives at Station Y 1(3/4) hours later. The second train leaves Station Y every half hour, beginning at 9 a.m., and arrives at Station Z 2(1/3) hours later. The third train leaves Station Z every 45 minutes, beginning at 8 a.m. What is the least total amount of time the woman must spend waiting between trains if all trains depart and arrive on schedule and if she arrives at Station Z no later than 3:30 p.m.?

A. 15 minutes
B. 25 minutes
C. 1 hour 15 minutes
D. 1 hour 40 minutes
E. 4 hours 30 minutes

We see that the total wait time is the total time waiting at stations Y and Z assuming that she doesn't have to wait at station X. Let's also let the trains leaving at stations X, Y and Z be trains X, Y and Z, respectively.

Assuming she arrives at station X exactly on the hour, except for arriving at 6 am at station X, she must wait 15 minutes at station Y for train Y. For example, if she catches train X at station X at 7 am, she will arrive at station Y at 8:45 am and catch train Y at 9 am. If she catches train X at station X at 8 am, she will arrive at station Y at 9:45 am and catch train Y at 10 am.

We see that the minimum wait time at station Y is 15 minutes. Now let's see how we can minimize the wait time at station Z. Since train Z leaves station Z every 45 minutes beginning at 8 am, train Z leaves station Z at 8 am, 8:45 am, 9:30 am, 10:15 am, 11 am, 11:45 am, 12:30 pm, etc. Also, since train Y takes 2 hours and 20 minutes to arrive at station Z and it leaves station Y every 30 minutes beginning at 9 am, train Y will arrive at station Z at 11:20 am, 11:50 am, 12:20 pm, etc. We see that the minimum wait time is 10 minutes if she can arrive at station Z at 12:20 pm and take train Z at 12:30 pm.

Thus the minimum total wait time is 15 + 10 = 25 minutes.

Answer: B