Kevin took a train at Figsdale with the intention of going to Garsville (which is located 8 kilometers from Figsdale). At the precise instant that the train started leaving Figsdale, the minute hand of his 12 hour analog wristwatch was located exactly on a minute mark.

Kevin looked at his watch again at the instant that the train reached Garsville, when he observed that the minute hand and the hour hand were exactly coincident, but the minute hand was not situated on a minute mark.

Assuming that that the train moved at a constant speed and its speed, in kilometers per hour, is an odd integer greater than 60 and the train left Figsdale between 6:00 a.m. and 6:00 p.m. inclusively, what is the speed of the train?

Determine precisely the respective times when the train left Figsdale and when the train reached Garsville.

Note: Each hour mark is also deemed as a minute mark.

There are 11 times between 6am and 6pm where the hour and minute hands are coincident.  There occur at intervals of 12/11 hours.  Thus the possible times for the trains arrival are
6:32:43.6363 am
7:38:10.9090 am
8:43:38.1818 am
9:49:5.45454 am
10:54:32.72727 am
12:00:00 pm
1:05:27.27273 pm
2:10:54.54545 pm
3:16:21.81818 pm
4:21:49.09091 pm
5:27:16.36364 pm

Now I simply started by calculating the number of minutes seconds the train would take for each of the possible speeds starting at 61 kph and looked for one where the number of seconds exactly matches one of the seconds above.  This happens for 165kph which means the train takes 2min 54.54545 sec to make the trip.  Thus it had to arrive at 2:10:54.54545pm and thus left at 2:08pm.

So the answer is that traveling at 165kph the train leaves at
2:08:00 pm
and arrives at
2:10:54.54545 pm

Posted by Daniel on 2009-08-22 14:17:11