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PARTS
OF NATURAL NUMBERS  2

Lesson 15  Section 2

We will now see how to take half of any number.
 7.   Which natural numbers are called the even numbers?
 
  Numbers that can be divided into two equal parts.

Thus 8 is an even number, because we can divide it "evenly" into two 4's.

4 is half of 8.

9 is not an even number.  There is an "odd" 1,

and therefore we call 9 an odd number.  We cannot take half of a natural number that is odd.  We cannot take half of 9 people

The even numbers, then, are the numbers divisible by 2:  2, 4, 6, 8, and so on.  The odd numbers are 1, 3, 5, 7, and so on.

If you do not know half of some even number, you can find it by decomposing it mentally into two even numbers whose half you do know. We saw how to do that in Section 1, Example 8.

Example 1.   How much is half of 54?

  Answer.   Half of 54 = Half of 50 + Half of 4
 
  = 25 + 2
 
  = 27.

As for odd numbers, note that every odd number is equal to an even number plus 1.

3 = 2 + 1.

5 = 4 + 1.

7 = 6 + 1.

9 = 8 + 1.

And so on.

Again, if a natural number is odd, then we cannot take half of it. Therefore, we will now assume that we are not necessarily dealing with natural numbers, and that 1 refers to some unit that has a half, such as 1 minute, 1 centimeter, 1 pound, and so on. (Lesson 19.)
 8.   How much is half of 1?
 
  ½  ("One-half")

½ must be one half of something.  It is one half of 1.

Half of $1, of course, is $.50.
 9.   How can we take half of an odd number?
Half of 7
  Take half of its even part, and add half of 1.

Example 2.   How much is half of 7?

Answer.  7 = 6 + 1.  Half of 6 is 3;  half of 1 is ½; therefore half of 7
is 3½.

Half of $7.00 is $3.50.

Example 3.   A scarf that normally sells for $17 is being sold at half price.  What do you pay?

Answer.  17 = 16 + 1.  Half of 16 is 8.  Therefore, half of $17 is $8.50.

Example 4.   How much is half of 26?

Answer.  If you do not know, then you can break 26 up as

20 + 6.

Half of 20 is 10.  Half of 6 is 3.  Therefore half of 26 is

10 + 3 = 13.

Example 5   How much is half of $27.40?

Answer.  We will take half of $27 and add it to half of $.40.

Since 27 = 26 + 1, then

Half of $27 is $13.50.

Half of $.40 is $.20.

Therefore, half of $27.40 is $13.70.

Example 6.   How much is half of 70?

Answer.  70 = 60 + 10.  Half of 60 is 30; half of 10 is 5; therefore half of 70 is 35.

Example 7.   How much is half of $9.70?

Answer.  Half of $9.00 (8 + 1) is $4.50.  Half of $.70 is $.35.  Therefore half of $9.70 is $4.85.

Example 8.   The famous number π ("pi") is approximately 3.14.  How

  much is half of π; that is,  π
2		 ?

Answer.  Ignore the decimal point in 3.14. Then half of 300 is 150; half of 14 is 7; therefore half of 314 is 157.  On replacing the decimal point,

π
2		   =  1.57.


 10.   How can we take a fourth or 25% of a number?
 
  Take half of half.

Four fourths = Two halves

A fourth is half of half.

Example 9.   How much is a fourth, or a quarter, of 60?

Answer.  Half of 60 is 30.  Half of 30 is 15.

Example 10.   How much is 25% of 180?

Answer.  25% means a fourth.  (Lesson 14.)  Half of 180 is 90.  Half of 90 is 45.

Example 11.   How much is 25% of 112?

Answer.  Half of 112 = Half of 100 + Half of 12 = 56.

Answer.  Half of 56 = Half of 50 + Half of 6 = 25 + 3 = 28.

Example 12.   How much is 25% of $9.60?

Answer.  Half of $9.60 = $4.50 + $.30 = $4.80.  Half of $4.80 = $2.40
 11.   How can we multiply by 5?
  5 × 123
  Take half of the multiplication by 10.

Since 5 is half of 10, then 5 times a number will be half of 10 times that number.


Example 13.   5 × 123 = Half of 10 × 123

  Example 13   5 × 123 = Half of 1230  (Lesson 3, Question 1.)

  Example 13.   5 × 123 = 615.

Example 14.   5 × $1.50 = Half of $15

  Example 14.   5 × $1.50 = $7.50.

Example 15.   5 × $46.80 = Half of $468

  Example 15.   5 × $46.80 = $234.
 12.   How can we take 5% of a number?
 
  Take half of 10%!

Example 16.   How much is 5% of $162?

Answer.   10% is $16.20. (Lesson 3, Question 8.)  Therefore, 5% is $8.10.

Anyone who would write .05 and multiply, does not understand percent.

Example 17.   How much is 5% of $475?

Answer.   10% is $47.50.   Therefore, 5% is $23.50 + $.25 = $23.75.
 13.   How can we multiply by 15?
15 × 8
  Multiply by 10, and add half.

For, 15 = 10 + Half of 10.

Example 18.   15 × 8 = 80 + Half of 80

  Example 18.   15 × 8 = 80 + 40

  Example 18.   15 × 8 = 120.

Example 19.   15 × 42 = 420 + 210.

  Example 19.   15 × 42 = 630.
 14.   How can we take 15% of a number?
 
  Take 10% and add half!

Example 20.   If you tip at the rate of 15%, and the bill is $40, how much do you leave?

  Answer.  15% = 10% + Half of 10%
 
  = $4.00 + $2.00
 
  = $6.00.

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

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