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Trains, Times, and Tunnel Vision: Thinking beyond Formulas

Date: 06/28/2010 at 08:29:25
From: Jenny
Subject: solving equations using travel time

What is the formula for finding the answer to this type of problem?

If John leaves at 9:00 traveling 45 mph, and Bob leaves at 10:00 traveling
60 mph, at what time will they both meet? 

I started by making this table:

         John            |            Bob
   10:00      45 miles   |     10:00       0 miles
   11:00      90         |     11:00      60
   11:30     112.5       |     11:30      90

But I want to find the formula for an algebraic equation so I don't have
to use guess and check.

Date: 06/28/2010 at 09:39:21
From: Doctor Ian
Subject: Re: solving equations using travel time

Hi Jenny,

The thing is, most problems don't have 'formulas' you can use to solve

A formula is a way of expressing the solution to a problem that comes up
all the time. Once you get past grade school, most of the math problems
you'll be asked to solve are designed to be unusual in some way, precisely
so that you _can't_ just use formulas to solve them.

And that's because most problems in real life aren't going to be ones
you've seen before. The important skill you're supposed to be learning and
practicing in your math classes isn't remembering formulas so you can plug
in numbers and crank out answers. It's supposed to be learning how to deal
with a problem that you've never seen before, and don't know how to

Anyway, one way to approach just about any problem is to see if you can
change the way you think about the problem a little, so that it looks more
like something you know how to solve.

For example, suppose you were given something like this:

  Bob leaves at 10:00, traveling 15 miles per hour.
  At what time will he have traveled 45 miles?

Would you know how to solve that? If so, and if we can change your
original problem to look like this one, then you're golden.

How can we do that? Well, instead of thinking about John and Bob traveling
relative to the ground at the same time, you can think about Bob traveling
relative to John.

That is, for each hour they travel, Bob gets 15 miles closer to John. His
speed relative to John is 15 miles per hour. Do you see why?

So now we just need to know: when Bob starts moving, how far away is John?
And we do know that: traveling at 45 miles per hour with a head start of
1 hour, he's 45 miles away.

And so in fact, the simplified problem above is equivalent to the original
problem, but easier to think about -- and easier to solve.

Does this make sense?

So that's one way to think about it. If you're comfortable with systems of
equations (which your subject line suggests you are), you can try that,
too. If we let t represent the number of hours since John leaves, then we
can write the distance that each person has traveled as a function of
time. For John, this is just

  d = 45 mph * t

What about for Bob? His equation will look like

  d = 60 mph * (t - 1)

Do you see why? (Let me know if you don't. This is a pretty important
thing to understand.) 

We can check that by thinking about some obvious times, like t = 1 and
t = 2. When t = 1 (the first row of your table), John has gone 45 miles,
and Bob hasn't started moving. When t = 2 (the second row), John has gone
90 miles, and Bob has gone 60 miles. Those seem pretty reasonable, so we
can have some confidence in our equations. 

So now you have two linear equations, with two unknowns:

  d = 45t

  d = 60(t - 1)

As I said, if you already know how to deal with this kind of system, then
at this point, you've done all the hard work in solving the problem, and
the rest is just busywork.

There are other ways you might approach it, too. In fact, it's hardly ever
the case that a problem can be solved in _only_ one way.

Which leads me to another reason to not rely on formulas: even when a
formula for something _does_ exist, sometimes it turns out to be the long
way around the block, and there are much easier routes to the solution if
you just take a little time up front to look for them. 

Relying on formulas is a little like only knowing one way to get home. Say
you live on Manhattan Island, and the only way you know to get there is
the George Washington Bridge. Then suppose one day the bridge is closed
for repairs. What do you do? Your 'formula' doesn't work, so you'd better
be able to think of something else! Are there other bridges? Any tunnels?
Can you take a ferry? Could you fly into an airport? Rent a boat? A jet
ski? A canoe? Could you swim there?

(Just how badly you need to get there determines the range of methods
you'd be willing to consider!)

If you think of math in terms of memorizing formulas for specific 
problems, then you really are missing the whole point of things -- and 
you're almost certainly having less fun than you could be having!

I hope this helps. Let me know if you'd like to talk more about this, or
anything else. 

- Doctor Ian, The Math Forum 
Associated Topics:
High School Linear Equations
Middle School Equations
Middle School Word Problems

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