Borrowing in Subtraction in Various Contexts

Date: 10/25/2006 at 23:48:36
From: Evelyn
Subject: 12 5/11 - 2 6/11

How can you subtract 12 5/11 - 2 6/11?  You can't take 6 apples from 5!

Date: 10/26/2006 at 10:09:57
From: Doctor Ian
Subject: Re: 12 5/11 - 2 6/11

Hi Evelyn,

You're right, but you CAN take 6 apples from 16.  What do I mean by that? 

Well, note that 

    12 5/11

  = 12 + 5/11                This is what we MEAN by a mixed number

  = 11 + 1 + 5/11            Because 12 = 11 + 1

  = 11 + 11/11 + 5/11        Because 1 = 11/11

  = 11 + 16/11               Add the fractions

Were you able to follow that?  If so, note that your subtraction can
be written this way:

    11 16/11
  -  2  6/11
  ----------

Can you do it now?

Note that this is EXACTLY the same thing we do with a "normal"
subtraction, like

    43
  - 17
  ----

where we "borrow" or "regroup", trading one group of 10 for 10
"groups" of 1, so we can complete the subtraction in each column:

       1
    3   3      40 + 3 is the same as 30 + 13
  - 1   7
   ------
    2   6
        |____  13 - 7 = 6


It's just the same idea, appearing in a different situation.  You can
use the same idea again when dealing with times, e.g., 

    4:22         3:82       4 hours + 22 minutes is the
  - 1:40   ->  - 1:40       same as 3 hours + 82 minutes. 
  ------       ------
                 2:42

or other units, 

    4 gallons, 2 quarts        3 gallons, 6 quarts
  - 1 gallon,  3 quarts  ->  - 1 gallon,  3 quarts
  ---------------------      ---------------------
                               2 gallons, 3 quarts

So this single idea turns out to be pretty useful!  The trick is
remembering that it IS the same idea, and not a bunch of unrelated
techniques. 

Does that make sense?  Let me know if you need more help.

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