Subtracting Fractions with Borrowing

Date: 12/17/2002 at 17:01:46
From: Hayley Hansen
Subject: Fractions

When you are subtracting a fraction, do you always take one away from 
the whole number?

 16 8/9
- 7 2/5

I don't know when to take a whole number away and when not to.

Date: 12/17/2002 at 22:04:02
From: Doctor Ian
Subject: Re: Fractions

Hi Hayley,

You only take one away if you need to.  

It's sort of like subtracting clock times. If you want to do this, 

    10:45
  -  8:15
  -------

you can subtract the minutes okay, 

    10:45
  -  8:15
  -------
       30

so you can just go ahead and subtract the hours, too:

    10:45
  -  8:15
  -------
     2:30

But if you have a problem with the minutes, 

    10:15
  -  8:45
  -------
    
then you have to 'break' one of the hours into 60 minutes:

     9:75
  -  8:45
  -------

and now you're okay again:

     9:75
  -  8:45
  -------
     1:30

So if you have something like 

   16 8/9
 -  7 2/5
 --------

you have to establish whether the top fraction is already large
enough. Since you're going to have to use a common denominator 
already, you can go ahead and change to that:

   16 40/45
 -  7 18/45
 ----------

So it looks like you're okay:

   16 40/45
 -  7 18/45
 ----------
    9 22/45

But if it had gone the other way, 

   16 18/45
 -  7 40/45
 ----------

you'd have to 'break' one of your ones into 45/45:

   15 63/45
 -  7 40/45
 ----------
    8 23/45

Does that make sense?  Now, just to be safe, I can go ahead and 'break
a one' even if I don't need to:

         16 8/9
       -  7 2/5
       --------
   
         15 17/9
       -  7  2/5
  =>   --------- 

         15 85/45
       -  7 18/45
  =>   ----------
          8 67/45

Then you can just 'unbreak' the one to get

   8 67/45 = 9 22/45

It depends on whether you'd rather think more and work less, or think
less and work more.  

Does that make sense? 

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