Subtracting Decimals with Borrowing

Date: 10/09/2001 at 20:18:08
From: Mario
Subject: Subtraction with six numbers

I just don't understand how to borrow when there are five or six 
numbers.

Example:

     536.30
  - 488.354
  ---------
     
Please help.

Date: 10/10/2001 at 15:21:05
From: Doctor Ian
Subject: Re: Subtraction with six numbers

Hi Mario,

The first thing you need to know is that you can always add zeros 
after the decimal point without changing anything:

  1.3 = 1.30 = 1.300 = 1.3000 = ...

So you can add zeros to get both numbers to the same number of decimal 
places:

      536.300
    - 488.354
   ----------

Okay, so now you can't subtract 4 from 0, so you have to 'borrow'. 

Borrowing is sort of like converting dollars to dimes, or dimes to 
pennies. For example, if you have 4 dollars and 2 dimes, that's the 
same as having 3 dollars and 12 dimes:

  4 dollars + 2 dimes = 3 dollars + 12 dimes

Make sure you understand why this is true. If you don't, nothing else 
about borrowing is going to make any sense at all. 

So if you have something like 

    35
  - 18
  ----

This is really the same as

     2   15
   - 1    8
   --------
     1    7

     ^    ^
     |    |
  dimes   |
          |
        pennies

Sometimes you have to do it more than once:

     Exchange           Exchange
     one $100           one $10
     for ten $10's      for ten $1's
     _____________    _____________
    /             \  /             \

  3 0 4       2   10   4        2  9  14
 -  2 9  ->  -     2   9  ->  -    2   9
 ------      -----------      ----------
                                2  7   5

But all this is kind of a drag to do, and it's easy to make mistakes.  
Fortunately, there is a simpler way to do subtraction - by doing 
addition instead!

Think about what it _means_ to subtract 18 from 35. It means there is 
some number that you can add to 18 to get 35. You can try to do that 
the standard way (lining up numbers and borrowing). But you can also 
try to do it by building up the missing amount. For example, adding 2 
to 18 gets you to 20:

  18 + 2 = 20

Adding 10 to 20 gets you to 30:

  18 + 2 + 10 = 30

Adding 5 more gets you to 35:

  18 + 2 + 10 + 5 = 35
       \________/

       This must be
     equal to 35 - 18         35 - 18 = 2 + 10 + 5 = 17

  
Let's try the second example:

  304
 - 29  add   1 to get to 30
 ----  add  70 to get to 100
       add 200 to get to 300
       add   4 to get to 304
           ---
           275  <--  Add the differences to get the total difference.

You can even do it with your problem:

      536.300
    - 488.354   add    .006  to get to 488.360
   ----------   add    .04   to get to 488.40
                add    .6    to get to 489
                add   1      to get to 490
                add  10      to get to 500
                add  36.3    to get to 536.3
                    -------
                     47.946  <---  This is how much you have to add
                                   to 488.354 to get to 536.300

There are teachers who would say that this is like counting on your 
fingers to solve a problem. I say: Apart from the fact that it's slow, 
there is nothing wrong with conting on your fingers!  

I would rather come up with a 'dumb' solution that I _know_ is right 
than follow a bunch of instructions that I don't understand to come up 
with an answer that I _hope_ is correct.  

At some point, when you're familiar enough with the 'dumb' method, 
you'll actually begin to see why the 'standard' method makes sense, at 
which point things like borrowing will seem perfectly obvious, rather 
than mysterious. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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