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PARTS
OF NATURAL NUMBERS  2

Lesson 15  Section 2

We will now see how to take half of any number.
 5.   Which numbers are the even numbers?
 
  The multiples of 2:  2, 4, 6, 8, 10, etc.

To take half of an even number, decompose it mentally into two numbers whose half you know.  We saw that in the Section 1.

Example 1.   How much is half of 54?

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

Now we will see how to take half of an odd number.
 6.   What is an odd number?
 
  An even number plus a remainder of 1.

3 = 2 + 1

5 = 4 + 1

7 = 6 + 1

9 = 8 + 1

And so on.

1 itself is also called an odd number.
 7.   How much is half of 1?
 
  ½  ("One-half")

½ must be one half of something.  It is one half of 1.  (We assume here that we are referring to measurement rather than counting. Lesson 19.)

Half of $1, of course, is $.50
 8.   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.


 9.   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
 10.   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
 11.   How can we multiply by 15?
15 × 8
  Multiply by 10, and add half.

For, 15 = 10 + Half of 10.

Example 16.   15 × 8 = 80 + Half of 80

  Example 15.   15 × 8 = 80 + 40

  Example 15.   15 × 8 = 120.

Example 17.   15 × 42 = 420 + 210.

  Example 16.   15 × 42 = 630.

Example 18.   If you tip at the rate of 15%, and the bill is $24, how much do you leave?

Answer.  15% = 10% + Half of 10%

Answer.  15% = $2.40 + $1.20  (Lesson 3, Question 10.)

Answer.  15% = $3.60



 12.   How do we increase or decrease by a given part?
  To increase by a given part, add that given part; to decrease, subtract.

Example 19.   A scarf originally sold for $15.

a)  It now sells for a third more.  What is the present price?

Answer.   A third of $15 is $5.  It now sells for $5 more.  It sells for $20.

b)  It now sells for a third less.  What is the present price?

Answer.   $15 − $5 = $10.

Example 20.   A photgraph measures 8 inches by 12 inches.

a)  It will be enlarged by one quarter.  What are the new dimensions?

Answer.   A quarter of 8 is 2.  A quarter of 12 is 3.  The new dimensions are:

(8 + 2) inches by (12 + 3) inches = 10 inches by 15 inches.

b)  It will be reduced by one quarter. What are the new dimensions?

Answer.  (8 − 2) inches by (12 − 3) inches = 6 inches by 9 inches.


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

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