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Lesson 6

THE MEANING OF SUBTRACTION

Mental Calculation


First, we will see the meaning of subtraction. Then we will see methods for achieving it easily and mentally. For, mental arithmetic is often the quickest way to do a problem.


In this Lesson, we will answer the following:

  1. What is the problem of "subtraction"?
    The complement to 10.
  2. How can we subtract a one-digit number from a two-digit number, mentally?

    Section2

  3. How can we find the difference by the endings?
  4. How can we find the difference between two-digit numbers?

    Section 3

  5. How do we complete 100?
  6. How do we find the difference by rounding off?

IN ADDITION we are given two numbers and we have to find their sum.

28 + 7 = ?

But in what is called the inverse of addition, we are given the sum and we have to find one of the numbers.

28 + ? = 35

That has been called "subtraction ."  It is the problem of finding the difference between two numbers.  It is the problem of finding what number we have to add to 28 to get 35.

28 + 7 = 35.

35 − 28 = 7.

The sign of subtraction is the horizontal dash (−), which we read "minus."

"35 minus 28 equals 7."

"The difference between 35 and 28 is 7."
 1.   What is the problem of "subtraction"?
 
  Larger − Smaller  =  Difference
 
  Smaller + Difference  =  Larger
 
  We are to name the number we must add to a Smaller number so that it equals a Larger.
That number we add is called the Difference of the two numbers.
We say that we are subtracting the Smaller number from the Larger.

The historical name subtraction -- literally, "taken away from below" -- referred to a specific written method, namely writing the smaller number (the subtrahend) under the larger (the minuend), and executing the method. However, that is not the only way of finding the difference of two numbers. These days, we could use a calculator! In any event, we should try to come up with a new name that emphasizes the problem of the inverse of addition.

We call the numbers the terms.  Thus when we write 35 − 28, the terms are 35 and 28.  We also call '35 − 28' a difference, even if we do not name the answer

If the terms are equal, of course, then their difference is 0.

5 − 5 = 0.

As in addition, we can only subtract quantities of the same kind, that is, that have the same name.

35 dollars − 28 dollars = 7 dollars.

Example 1.   How much is 149?

Answer.  9 plus what number is 14?

9 + 5 = 14.

9 + 5 = 14

149 = 5.

We can say that the difference of two numbers is the distance between them.  How far is it from 9 to 14?  A distance of 5.

Here is a simple example:

86 = 2  because  6 plus 2 = 8.

Knowing that, you would also know

8060  =  20   because 60 plus 20 is 80.
 
8460  =  24   because 60 plus 24 is 84.
 
812600  =  212   because 600 plus 212 is 812.

"Subtracting" is adding to the smaller number.

The complement to 10

The student should know all the ways of  "completing" 10.  For example,

103 = ?

3 plus what number makes 10?   7, of course.  7 is called the complement to 10 of 3.   7 "completes" 3 to make 10.

Example 2.   106 = 4.  In fact, whenever we take a 6 from any multiple of 10 -- 20, 30, 40, 50, 60 -- we always get a 4.

30 − 6 = 24

40 − 6 = 34

80 − 6 = 74

Similarly, each of the following differences ends in the complement to 10 of the number we are subtracting.

508 = 42

709 = 61

2003 = 197

$8.004¢ = $7.96 

These are not problems to write in a column and do some written method

Let us now look at subtracting from a two-digit number.  We can think of it as "taking away" the smaller number.

50 − 1 = 49.

"50 take away 1 is 49."

60 − 2 = 58.

"60 take away 2 is 58."

70 − 3 = 67.

"70 take away 3 is 67."
 2.   How can we subtract a one-digit number from a two-digit number, mentally?
523
 
  Break off from the smaller number the ones digit of the larger (2), and subtract to get a multiple of 10.
522 = 50.
 
  Then subtract the rest of the smaller number.
501 = 49.
 

Example 3.   Calculate mentally  83 − 5.

 Solution.   "83 minus 3 is 80; minus 2 is 78."

Example 4.   Calculate mentally  72 − 6.

 Solution.   "72 minus 2 is 70; minus 4 is 66."

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section.


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