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A R I T H M E T I C

Lesson 4

Mental calculation

We now begin the four operations of arithmetic:  Addition, subtraction, multiplication, and division.  The emphasis will continue to be on problems that do not require a calculator -- problems that you should not even have to write down!   Mental calculation is now arithmetic of the most practical kind.

In this Lesson, we will answer the following:

1. How do we add mentally by composing a multiple of 10?
2. What does it mean to add by endings?
3. How do we add mentally by units?

Section 2

4. How do we add by rounding off? And how do we compensate?

The student should now have mastered the elementary addition of one-digit numbers, and should be practicing the multiplication table.

For, once you know that

 5 + 4 = 9, then you would also know 50 + 40  = 90, 500 + 400  = 900, 5,000 + 4,000  = 9,000

These are not problems to write in a column.

 1. How do we add mentally by composing a multiple of 10? 27 + 5 Break up one number in order to regroupand compose a multiple of 10.

Example 1.     27 + 5  =  (27 + 3) + 2  =  30 + 2  =  32

Similarly,

38 + 4  =  (38 + 2) + 2  =  42

59 + 7  =  (59 + 1) + 6  =  66

96 + 8  =  (96 + 4) + 4  =  104

An educated person then does not have to take out a pencil, write in a

column and say,

"6 plus 8 is 14 -- write 4, carry 1.  etc., etc., etc."

Nor does an educated person count on her fingers.  Rather, an educated person knows elementary addition  and therefore that 96 + 8 is not very different from 6 + 8.  It must be 104.

These numbers that we compose --

10,  20,  30,  40,  50, etc.

-- are called the multiples of 10.

Example 2.   Since

9 + 6 = 15,

then

 90 + 60 = 150 \$.90 + \$.60 = \$1.50 \$.90 + \$.64 = \$1.54

A 9 plus a 6 always gives a "15."  This brings us to the practical technique for adding several numbers --

7 + 8 + 3 + 9 + 4

 2. What does it mean to add by endings? It means to know the digit in which the sum will end.

For example, since

8 + 4 = 12,

we would know that if we add any number ending in 8 to any number ending in 4, the sum will always end in 2.

28 + 4 = 32

38 + 4 = 42

68 + 4 = 72

Similarly, since

6 + 5 = 11

then

26 + 5 = 31

46 + 5 = 51

76 + 5 = 81

They all end in 1.  These are not problems to write down.

Example.   Add mentally -- or aloud -- from left to right.

 7 + 8 + 3 + 8 + 6 + 4 + 9

Say only the partial sum as you come to it.  Do not say,
"7 plus 8 is 15" -- say only "15."  Then look at 3 and say 18.  And so on.  Use your knowledge of the ending; even exaggerate saying it.

 7 + 8 + 3 + 8 + 6 + 4 + 9 Say: "15 18 26 32 36 45"

The following is a basic technique for mental addition:

 3. How do we add mentally by units? Add the higher units first -- from left to right.

435 + 461 = 896

First add the hundrededs, then the tens, then the ones.

Example 1.  Counting by 10's.    30 + 24  =  54

Here are other examples of mental calculations.  In each case, add the numbers as you read them, from left to right.  Add the 10's:

20 + 16 = 36

40 + 38 = 78

40 + 62 = 102

40 + 82 = 122

90 + 73 = 163

Example 2.     43 + 25

"Sixty --

43 + 25

-- eight."

Or, we could say,

"43 plus 20 is 63, plus 5 is 68."

Say, "150 plus 7 is 157."

Say only, "50 + 14 equals 64."

The art of mental calculation is to say as little as possible.  The last number you say is the answer.

"170 + 11 = 181"

Example 6.    23 + 32 + 25 + 12

Add up all the tens, then add on the ones.  Say each partial sum.  Say

The last number you say is the entire sum.

Example 7.    34 + 25 + 32

Example 8.    653 + 224

First add the hundreds, then the tens, then the ones.  Again, say each partial sum:

"800"

653 + 224

"870"

653 + 224

"877"

Example 9.    Three tracks on a CD have the following times:

10:34

6:25

8:07

What is the total time?

(10:34 means 10 minutes 34 seconds.  60 seconds = 1 minute.  Therefore, 72 seconds = 1 minute 12 seconds.  1:12.)

"16 plus 8 is 24 minutes."

"24:59 plus 7 is 24: 66."

The total time is 25 minutes 6 seconds.

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

or

Continue on to Section 2:

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