Trains and Fractions

Date: 24 Dec 1994 23:26:52 -0500
From: Anthony D'Auria
Subject: !!!!!Can you solve this!!!!!!

I have a couple of problems that I had trouble with. See if you can solve 
them. Here they come....

1) A non-stop train leaves city A for city B at 60m.p.h. At the same 
time, another train leaves city B for city B at 50 m.p.h. How far apart 
are the trains one hour before they meet? (knowing how far apart they 
were when they started is irrelevant in order to solve this problem)

2) If to the numerator and denominator of the fraction 1/3 you add its 
denominator the fraction will double. Find a fraction which will triple 
when its denominator is added to its numerator and its denominator. Find 
out which will quadruple.

3) A train moving at 45 m.p.h. meets and is passed by a train moving at 
36 m.p.h. A passenger on the first train is holding a stop watch and 
notes that the second train takess 6 seconds to pass him. How long is the 
second train which was timed?

Please try to solve these problems and if you have any trouble, just 
write back and if you have an answer, please write back, I wuold like to 
know how you solved them.

Anthony D'Auria
dauriaa@voyager.bxscience.edu
THANKS

Date: 29 Dec 1994 12:23:43 -0500
From: Dr. Ken
Subject: Re: !!!!!Can you solve this!!!!!!

Hello there Anthony!

>1) A non-stop train leaves city A for city B at 60m.p.h. At the same 
>time, another train leaves city B for city B at 50 m.p.h. How far apart 
>are the trains one hour before they meet? (knowing how far apart they 
>were when they started is irrelevant in order to solve this problem)

Here's my question for you: is that a type-o in your statement of the
problem?  You say that the second train leaves city B for city B.  Well,
there could be two different arrangements of this problem (where the second
train goes from A->B or B->A), but they're not really all that different.
So I'll get you started.

If the two trains are going in opposite directions, then how fast are they
moving relative to one another?  Well, if one train is going 50 mph, and the
other is going 60 mph, then their relative speed is 110 mph.  So the
question becomes "if two objects are approaching each other at a speed of
110 mph, how far apart are they one hour before they meet?"

In the other configuration of the problem (where the trains are traveling in
the same direction), the relative speed of the trains will be 60 - 50 = 
10 mph.  But you can see that the question is essentially of the same type.

>2) If to the numerator and denominator of the fraction 1/3 you add its 
>denominator the fraction will double. Find a fraction which will triple 
>when its denominator is added to its numerator and its denominator. Find 
>out which will quadruple.

In this question, things become much simpler if you write an equation for
the condition which must be satisfied.  For instance, in the part of the
problem that's given, they say that if you add the denominator of a fraction
to both its numerator and its denominator, it will double.  So let n/d be
the fraction in question.  Then we have the equation

n + d    2n      And then
----- = ----     simplify    d^2 = 3nd
 2d      d       to get

So basically we can plug in something for d and then solve for n.  Note that
it looks like there will be a bunch of solutions for n and d.  But look
again.  What we're really looking for is n/d, right?  We're looking for the
fraction, not the specific values of n and d.  So notice that we can
simplify our equation further to get

              n/d = 1/3     (I divided through by 3d^2)

So that's our answer.  Note that the fractions 2/6, 3/9, etc. work too,
which I don't think is obvious from the statement of the problem.  Neat
stuff.

Anyway, the other two problems would get handled in a similar way.  Except
that now our equations would be 

n + d    3n              n + d    4n
----- = ----      and    ----- = ----
 2d      d                2d      d

I'll leave it to you to complete these.


>3) A train moving at 45 m.p.h. meets and is passed by a train moving at 
>36 m.p.h. A passenger on the first train is holding a stop watch and 
>notes that the second train takess 6 seconds to pass him. How long is the 
>second train which was timed?

Again, this is an exercise in relative speed.  And again, I'm not sure
whether you mean that the trains are going in the same direction, or
opposite directions, but the two cases will be handled in similar manners.

Basically, the trains are moving at a relative speed of either 81 mph or 
9 mph (that's 45+36 and 45-36).  So the question becomes "If an object is
moving at 81 mph (or 9 mph), how far will it travel in 6 seconds?"  What
you'll have to do here is some fancy conversion of seconds into hours, or
vice versa.

I hope this helps you see what's really happening in these problems.  Write
back if you need more help on them!

-Ken "Dr." Math