An algebraic method for
eliminating borrowing.

Almost all errors in subtraction involve the step in which we borrow and carry.

Therefore we should follow the all important rule of "change something difficult into something easy".

If we don't want to make mistakes, we should try to avoid borrowing and carrying in subtraction if at all possible.

Once again, the easiest number to subtract is zero.   It never requires us to borrow.  Therefore we need to try to end our second number, in a zero.  Let's look at an example:

 53 -28 If we add 2 to the 28, we get 30.  It is easy to subtract 30 from 53.  We get 23 mentally, but this answer is 2 units too small because we took away MORE   than the problem intended. To compensate for this we simply ADD two onto the answer. 23 + 2 = 25. So  53 - 28 can be done mentally as 53 - 30 + 2 = 25

The algebra for this is very straightforward.
You start with a - b and add some number n to the b, which gives a - b + n.

So you can see the answer is now too large by the amount of n.

To compensate, we simply subtract the n from the answer, which looks something like this:
(a - b + n) - n.  Since n - n = 0 we are now balanced with the correct difference, and no borrowing was necessary.

*NOTE...It might be said that this method will not work for really long numbers being subtracted, and that indeed the old borrowing and carrying method is superior for those problems. We would agree.

However, it is the small subtractions you will meet everyday that can be done quickly with this.

Once you are handy with this method perhaps 90% of your subtractions will trouble you no more.

Let's try these:

 35 - 17 54 - 39 83 - 66 145 - 78 \$5.00-\$.89 \$1.95-\$.98 134 - 65 254 - 199

Take a sheet of notebook paper and fill one side of it
with examples of this type of subtraction.
Write out what you are adding in  each case like this:

54 - 39 = (54 - 40) + 1 = 15

\$1.95 - \$.98 = (\$1.95 - \$1) + \$.02 = \$.97

Your goal is to be able to write the problem and
get the answer mentally without writing the middle part.

Like all mental math tricks, the more you practice this one
the faster you will get.  Soon you will be amazing people.

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