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### 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/
```
Associated Topics:
Elementary Fractions
Elementary Place Value
Elementary Subtraction

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