Using Addition and Subtraction to Check Answers

Date: 02/25/2003 at 16:23:01
From: Chelsea
Subject: Addition and subtraction

How can you use addition to check subtraction? How can you use 
subtraction to check addition. Why does this work?

Date: 02/25/2003 at 17:05:16
From: Doctor Peterson
Subject: Re: Addition and subtraction

Hi, Chelsea.

Addition and subtraction are opposite operations, so you can use one 
to check the other. For example, these two facts mean the same thing:

    5 + 6 = 11

    11 - 6 = 5

The first says, if we start with 5 and add 6, we get 11. The second 
says, if we start with 11 and take away 6, we get 5. We're just 
playing the same movie forward and backward. This is always true; we 
can say it this way:

    if A + B = C, then C - B = A

So if we do a big subtraction problem, like

    7549 - 3958 = 3581

we can check it by adding the 3958 to the 3581:

    3581 + 3958 = 7539

Ah! We see that our answer was wrong, because we didn't get 7549 back. 
We even have a good idea which digit of the answer is wrong. (Unless, 
of course, we actually did the addition wrong!)

You can think of it this way: Suppose I had $7549 in a bag, and pulled 
out $3958 and gave it to you. Later I count my money and find I have 
$3581. I can check whether everything adds up by adding my $3581 and 
your $3958 together. The total ought to be the amount I had to start 
with. When I find that it is $10 less than I expected, I know that 
either someone counted wrong, or I lost $10 somewhere. I don't know 
for sure whom to blame, but I know I need to investigate the problem. 
And that's what this kind of check does for you.

The way I would do the check is like this:

             |   7549  ^
    subtract | - 3958  | add
             | ------  |
             v   3581  |

I just add upward, as if it were

                 3581
               + 3958
               ------
                 7549?

When I see that the 4 isn't right, I stop and do that column of the 
subtraction again.

People generally find subtraction harder (and easier to make mistakes 
in) than addition, so we most often use addition to check subtraction; 
but you can check addition by doing the opposite subtraction if you 
find that necessary. More often you would check addition by adding in 
a different order.

If you have any further questions, feel free to write back.

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