Frank is walking across a one mile long railroad bridge ignoring his mother's commands and the signs on the bridge.   He has walked 1/4 of a mile when he looks back and sees a train coming.   The train is 3/4 of a mile from the beginning of the bridge.   Frank's top running speed is a six minute mile.   Which way should he run?

Determine the answer to the question if the train is coming at the following speeds:
(1) 25 mph
(2) 35 mph
(3) 20 mph

Be complete and show your work.

Extra Credit:
Determine all possible integer train speeds which result in Frank's
(a) escaping in both directions
(b) escaping only by running toward the train and
(c) escaping in neither direction

Solution to the Problem:

(1) Run toward the train.
(2) He will not escape either way so it doesn't matter.
(3) He will escape either way so it doesn't matter.

(a) All speeds 23 mph or less ( 1, 2, 3, 4, ... 23)
(b) 24, 25, 26, 27, 28, 29
(c) All speeds above 30 mph (31, 32, ...)

You may want to sketch a diagram of the bridge and the location of the train.
If Frank runs toward the train he must cover 1/4 mile before the train covers 3/4 mile.
If Frank runs away from the train, he must cover 3/4 mile before the train covers 7/4 mile.
Use the fact that Rate x Time = Distance.
You must first convert a six minute mile to miles per hour.
Use dimensional analysis to do this:
 1 mile        60 minutes     10 miles   
----------  x  ------------ = ---------  = 10 mph
6 minutes       1 hour          1 hour
(1) If Frank runs toward the train, it will take him 1/40 hour to get off the bridge (1/4 mile / 10 mph)
      Meanwhile, it will take the train 3/100 hour to get to the bridge (3/4 mile / 25 mph).
      Since 1/40 = 5/200 and 3/100 = 6/200, Frank will escape by running toward the train.

      If Frank runs away the train, it will take him 3/40 hour to get off the bridge (3/4 mile / 10 mph)
      Meanwhile, it will take the train 7/100 hour to get to the end of the bridge (7/4 mile / 25 mph).
      Since 3/40 = 15/200 and 7/100 = 14/200, Frank would not escape by running away from the train.

(2) If Frank runs toward the train, it will take him 1/40 hour to get off the bridge (1/4 mile / 10 mph)
      Meanwhile, it will take the train 3/140 hour to get to the bridge (3/4 mile / 35 mph).
      Since 1/40 = 7/280 and 3/140 = 6/280, Frank will not escape by running toward the train.

      If Frank runs away the train, it will take him 7/40 hour to get off the bridge (3/4 mile / 10 mph)
      Meanwhile, it will take the train 7/140 hour to get to the end of the bridge (7/4 mile / 35 mph).
      Since 3/40 = 21/280 and 7/140 = 14/280, Frank would not escape by running away from the train.

(3) If Frank runs toward the train, it will take him 1/40 hour to get off the bridge (1/4 mile / 10 mph)
      Meanwhile, it will take the train 3/80 hour to get to the bridge (3/4 mile / 20 mph).
      Since 1/40 = 2/80, Frank will escape by running toward the train.

      If Frank runs away the train, it will take him 3/40 hour to get off the bridge (3/4 mile / 10 mph)
      Meanwhile, it will take the train 7/80 hour to get to the end of the bridge (7/4 mile / 20 mph).
      Since 3/40 = 6/80, Frank would escape by running away from the train.

Let x = rate of the train.
If Frank runs toward the train, it takes him 1/40 hour to get off the bridge and it takes the train 3/4x hours to get to the beginning of the bridge.
Set these equal to get x = 30 mph.
So, when x < 30 mph, Frank can escape by running toward the train.

If Frank runs away from the train, it takes him 3/40 hour and it takes the train 7/4x hours to reach the end of the bridge.
Set these equal to get x = 23.33 mph.
So, when x < 23.33, Frank get escape by running away from the train.

(a) So, Frank can escape in either direction if the train is traveling at less than 23.33 mph.
(b) Frank can escape only by running toward the train if the train is moving between 23.33 and 30 mph.
(c) Frank cannot escape if the train is moving at a speed > 30 mph.   Actually, if the train is moving at 30 mph. it is a tie and Frank loses.


Correctly solved by:

1. Colin (Yowie) Bowey ** Beechworth, Victoria, Australia
2. Veena Mg ** Bangalore, Karnataka, India
3. Ivy Joseph ** Pune, Maharashtra, India
4. Ashton Eldredge Mountain View High School,
Mountain View, Wyoming
5. John Simmons ** Memphis, Tennessee

** Solved extra credit