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Lesson 5  Section 2

MORE ELEMENTARY ADDITION

Mental calculation by
Rounding off  or  Regrouping

Back to Section 1

A basic property of addition is the following:

A sum will not change if we both add and subtract the same number,
or if we add a number to one term and then subtract it from another.

18 + 7  =  25
 
18 + 7 + 2 − 2  =  25
 
(18 + 2) + (7 − 2) = 20 + 5  =  25

For that reason, subtraction is called the inverse of addition.  Subtracting undoes the effect of adding.

This property leads to the technique of rounding off.


 4.   How do we add mentally by rounding off?
 
256 + 98
 
  Round off, or approximate, one or more numbers to the nearest whole number or multiple of 10.
To compensate, subtract from your answer whatever you added to each number to round it off.
 
    Example 1.    256 + 98.
 
Say:  " 256 plus 100 is 356, minus 2 is 354."

We rounded off 98 to 100 by adding 2. To compensate we had to subtract 2.

    Example 2.    $3.99 + $4.99 + $5.99.
 
Say:  " $4 plus $5 is $9, plus 6 is $15.  Minus 3 cents is $14.97."

We rounded off each one to the next whole dollar.  But then we had to subtract 3 cents -- because we added 1 cent to each number to round it off.

Before looking at this next example, can you start at 100 and count backwards by 5's?

"100,  95,  90,  85,  80,  75,  70,"  and so on.

Example 3.   You buy three items that cost $1.95 each.  How much do you pay?

Answer.  Round off to $2.00.  Say:

"Three times $2.00 is $6.00;  minus 15 cents is $5.85."

Why do we take off 15 cents?  Because we added 5 cents to each $1.95.  And $1.00 minus 15¢ is 85¢.

Example 4.    34 + 48

Example 4.     "34 + 50 is 84, minus 2 is 82."

Example 5.   Regrouping.

39 + 26

Round off 39 to 40.  But instead of subtracting 1 from the answer, take 1 from 26.  Say

"40  +  25 = 65."

Example 6.    57 + 25 = 60 + 22 = 82.

Take 3 from 25 and regroup it with 57.

Example 7.   $3.58 + $2.35

Technique.   With dollars and cents, the technique is to add the cents first:

"60¢ + 33¢ = 93¢"

Now add the whole dollars (3 + 2):

"plus 5 is $5.93."

Example 8.   One item cost  $3.79  and another,  $6.49.  How much do you pay?

  Add the cents first.

"80¢ + 48¢ = $1.28";

now add the whole dollars (3 + 6):

"plus $9.00 is $10.28."


Or, add the cents by rounding them both:

$3.79 + $6.49

"80¢ + 50¢ is $1.30;

plus $9.00 is $10.30;

minus 2¢ is $10.28."

With a little practice, this is faster than a calculator!

We should point out that by mental calculation, we do not mean visualizing what you would write.  (79 + 49.  "9 + 9 is 18;  write 8, carry 1. . .")  In mental calculation you really count -- and the last number you say is the answer.

What is more, these are not "tricks."  Counting is not a trick.  It is the written method that is the trick -- to get the right answer.

Example 9.   Simple regrouping.   Add   4 + 59 + 26

It is not necessary to add the numbers as they appear.  We may regroup them as we please.  For example,

"4 + 26 = 30;  plus 59 is 89."

Example 10.      6 + 2 + 4 + 5 + 8.

Pick out the 10's:

6 + 2 + 4 + 5 + 8  = 10 + 10 + 5
 
   =  25.

Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

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