Strategies for Solving Word Problems with Fractions and Units

Date: 08/02/2007 at 09:21:19
From: Jamie
Subject: word problems with fractions

On a road map with a scale of 1/4 inch per 10 miles, the highway from 
Waukee to Winterset is 1 3/8 inches long.  How many miles long is 
this highway?

I can't figure out if I need to multiply, divide, add or subtract. 
For example:

1 3/8 * 1/4 = 11/8 * 1/4 = 11/8 * 2/8 = 22/8 = 2 6/8
  
I get stuck because the answer is a whole number with no fraction.  I 
tried subtracting and adding as well but I cannot get a whole number 
answer.

Date: 08/02/2007 at 10:00:54
From: Doctor Ian
Subject: Re: word problems with fractions

Hi Jamie,

When you don't know what to do, sometimes it's helpful to use simpler
numbers to help you see what's going on.

For example, we might ask:

  On the map, the highway from A to B is 1/4 of an inch long. 
  How long is the highway?

It's easy to see that, since 1/4 of an inch corresponds to 10 miles,
the answer must be: 10 miles.

Okay, how about this one?

  On the map, the highway from A to B is 2/4 of an inch long. 
  How long is the highway?

It's got to be twice as long as the last one, so it's 2*10 = 20 miles.

If we do this enough times, we can see that if we can write the
distance as "something over 4", we can just multiply the something by
10 to get the distance we want.

Does that make sense?  Now, suppose we have something where the
numbers don't work out so nicely, like

  On the map, the highway from A to B is 2/5 of an inch long. 
  How long is the highway?

We'd like to write 2/5 as "something over 4".  To do that, we set up a
proportion,

  2   ?
  - = -
  5   4

and we can solve that to find that

  2 * 4   
  ----- = ?
    5   

      8
      - = ?
      5

    1.6 = ?

(If you haven't learned how to solve a proportion like that, see

  Flipping and Switching Fractions
    http://mathforum.org/library/drmath/view/58193.html 

and let me know if it's not clear.) 

That is, 

   2   1.6
   - = ---
   5    4

So the highway must be 1.6*10 = 16 miles long.  We could also do this
with a fraction:

   2   8/5
   - = ---
   5    4

so the highway is 8/5*10, or 16 miles long.

Does this make sense?  If so, let's try it with your numbers: 
1 3/8 is the same as 11/8, so we have

  11   ?
  -- = -
   8   4

and we can find that

  11   5 1/2
  -- = -----          Just divide both the numerator and denominator
   8     4            by 2, to get an equivalent fraction.

So the length of the highway is 5 1/2 * 10 = 55 miles long. 

So far, so good? 

Note that including units in your calculations can also help you see
when you're going astray.  You wrote

  1 3/8 * 1/4 = 11/8 * 1/4 = 11/8 * 2/8 = 22/8 = 2 6/8 

But what are the units on those figures?  1 3/8 is the number of map
inches, and 1/4 is the number of map inches corresponding to 10 miles.
So we really have

  1 3/8 map inches * 1/4 map inches per 10 miles = ...

which gives you units of (map inches) squared per 10 miles... which
isn't what you want.  You'd like to end up with miles.  Now, suppose
we divide, instead:

  1 3/8 map inches
  -----------------------------
    1/4 map inches per 10 miles

That looks pretty messy, but it's the same sort of thing we do when we
divide a distance by a rate to get a time:

   90 miles
   ----------------- = 3 hours
   30 miles per hour

So we end up with

   1 3/8
   ----- (units of 10 miles)
     1/4

Now, do you see how important the units are?  If we leave them out, it
would be easy to think that 1 3/8 divided by 1/4 is the number of
MILES in the highway.  But it's not!  It's the number of UNITS OF 10
MILES, which is a very, very different thing.  So we get

     1 3/8
     ----- (units of 10 miles)
       1/4

  =  1 3/8 * 4/1 (units of 10 miles)

  = 11/8 * 4/1 (units of 10 miles)

  = 11/2 (units of 10 miles)

  = 5 1/2 (units of 10 miles)

  = 5 1/2 * 10 miles

  = 55 miles

which is what we got doing it the other way. 

So there are two lessons here, that you want to remember long after
you've forgotten about this particular problem:

  1) When in doubt, try simpler numbers to see if you can spot
     a pattern you can use for any numbers.

  2) Include units in your calculations as a way of helping
     you spot nonsense calculations. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/04/2007 at 09:51:42
From: Jamie
Subject: Thank you (word problems with fractions)

Thank you very much!  I get it now.