Think Before You Start

I ended the previous post by stating a problem from the 9th grade IMP math textbook called The Perils of Pauline. The girl is 3/8 of the way through a railroad tunnel when she hears a train from behind. She just has time to run back the way she came to reach the beginning of the tunnel when the train gets to the beginning, and she just has time to run forward to get to the end of the tnnnel when the train gets to the end. If the train goes 60 mph, how fast does Pauline run?

When a problem seems at all complicated, it is always a good idea to think about the problem before you start trying to express it mathemaically. In particular it is a good idea to make sure we understand what conditions we are given and what we are being asked to find

Condition A--She is 3/8 of the way into the tunnel when she hears the train.

Condition B--Running, she can just reach the beginning when the train does.

Condition C--Running, she can just reach the end of the tunnel when the train does.

Condition D--The train goes 60 mph

Find--How fast Pauline runs. From contest we may assume she will run the same speed in either direction.

Next we will see if we can re-express these conditions in a way might better help us solve the problem. Since the problem talks about her running in each direction we will re-express the first coondition as:


Condition A--When she hears the train, she is 3/8 of the way from the beginning and 5/8 of the way from the end of the tunnel.


It also makes sense to to look at the obvious mathematical implications of the next two conditions.


Condition B--Pauline can run through 3/8 of the tunnel in the time it takes the train to reach the beginning of the tunnel.


Condition C--Pauline can run through 5/8 of the tunnel in the time it takes the train to reach the end of the tunnel.

At this point what could have been a rather messy algebra problem can now be solved with just arithmetic. Looking at the difference between Conditions B and C we see that in C compared to to B, the train goes the extra distance represnted by the length of the tunnel, while the extra distance Pauline runs is 5/8 - 3/8 = 2/8 = 1/4 the length of the tunnel. Pauline runs 1/4 as fast as the train and 1/4 of 60 mph is 15 mph.

If we wish to solve this problem using algebra we must still think carefully about what we do.

We will need to express Condition B and Condition C algebraically. To do this we will not only need a variable such as P to represent Paula's running speed, we will need additional variables to represent the distances than Paula and the train travel. One way to do this would be to let L be the length of the tunnel and D be the distance the rain has to travel to reach the beginning of the tunnel.

With these variables we can set time equal to distance divided by rate--and if the train and Pauls could reach the beginning of the tunnel at the same time then

Condition B1 would be D/60 = (3/8)L/P

and if both could reache the end at the same time

Condition B2 would be (D+L)/60 = (5/8)L/P

At this point we have two equations and 3 unknowns which normally would not be enough to solve for any variables. But if we subtract B1 from B2 we can at least eliminate one variable and we get

L/60 = (2/8)L/P

Multiplying by 60P we are able to clear the fractions and we get

LP = (60)(2/8)L = 15L and dividing by L we get

P=15

An alternate approachh would involves clearing away the fraction in Conditions B1 and B2 first, and then subtracting one condition from the other and solving in much the same way that we did above.

Subtracting one condition from another is something we frequently find helpful in solving problems. We look at more at this technique in the next post.