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Lesson 20

UNIT FRACTIONS


In this Lesson, we will answer the following:

  1. What is a unit fraction?
  2. How can we express a whole number as a fraction with a given denominator?
  3. How do we change a mixed number to an improper fraction?
  4. What do we mean by the complement of a proper fraction?
  5. What will be the answer when we subtract a proper fraction from a whole number greater than 1?

 1.   What is a unit fraction?
 
  A fraction whose numerator is 1.

 

A unit is whatever we call one.  Each unit fraction is a part of number 1.  The denominator names that part.

1
2		   is one half of 1.

1
3		   is the third part of 1.

1
4		   is the fourth part of 1.

And so on.

We will see that every fraction is a sum of unit fractions.

  Example 1.    In the fraction  4
5		 , what number is the unit, and how many

of them are there?

Answer.  The denominator of a fraction names the unit -- the part of 1.  The numerator tells their number -- how many.

 In the fraction  4
5		 , the unit is  1
5		 .  And there are 4 of them.

 

4
5		  =  4 ×  1
5		  =   1
5		  +  1
5		  +  1
5		  +  1
5		 .
  Example 2.   Let  1
3		  be the unit, and count to 2 1
3		 .

Every fraction, then, is a sum of unit fractions.

2
3		  =  2 ×  1
3		  =  1
3		  +  1
3		 .
3
8		  =  3 ×  1
8		  =   1
8		  +  1
8		  +  1
8		 .

The symbols for all the numbers of arithmetic stand for a sum.

  Example 3.   Add   2
8		  +  3
8		 .
  Answer.   5
8		 .
2 eighths + 3 eighths are 5 eighths. The unit we are adding is  1
8		 .

This illustrates the following principle:

In addition and subtraction, the units must be the same.

We will see this in Lesson 24.  The denominator of a fraction has no other function but to name the unit.

7
9		  −  3
9 		  =   4
9		 .

Example 4.   1 is how many fifths?

  Answer.   1 =   5
5		  ("Five fifths.")
 
   
1
5		  is contained in 1 five times.

Similarly,

1 =  3
3		  =  4
4		  =  10
10

And so on.  We may express 1 as a fraction with any denominator.

  Example 5.   Add, and express the sum as an improper fraction:   5
9		  + 1.
  Answer.    5
9		  + 1 =  5
9		  +  9
9		  =  14
 9		 .

It was necessary to express 1 as so many ninths, because the things we add must have the same name.

 2.   How can we express a whole number as fraction
with a given denominator?
 
 
  Multiply the denominator by the whole number. Make that product the numerator.

  Example 8.    2 =  2 × 5
   5		  =  10
 5		 .
Since 1 =  5
5		 , then 2 is twice as many fifths:  2 =  10
 5		 .   3 =  15
 5		 .   4 =  20
 5		 .

And so on.

  Example 9.    6 =  ?
3
  Answer.    6  =   6 × 3
   3		   =   18
 3		 .
  Example 10.     How many times is  1
8		  contained in 5?  That is, 5 =  ?
8		 .
  Answer.   5  =   40
 8		 .

Let us now revisit the rule for changing a mixed number to an improper fraction (Lesson 19).  In fact, we will see why we have that rule
 3.   How do we change a mixed number to an improper fraction?
 
  Change the whole number to a fraction with the same denominator. Then add those fractions.
  Example 11.   3 5
8		  = 3 +  5
8		  =  24
 8		  +  5
8		  =  29
 8		 .
  Example 12.   5 2
7		  = 5 +  2
7		  =  35
 7		  +  2
7		  =  37
 7		 .

The complement of a proper fraction
 4.   What do we mean by the complement of a proper fraction?
  It is the proper fraction we must add in order to get 1.

  Example 13.    5
8		  + ? = 1
  Answer.   Since 1 =  8
8		 , then  5
8		  +  3
8		  =  1.
3
8		  is called the complement of   5
8		 .   3
8		  completes  5
8		  to make 1.

Equivalently, since finding what number to add  is subtraction:

1 −  5
8		   =   3
8		 .
  Example 14.    1 −  3
5		  =  2
5		 .
When we add   2
5		  to  3
5		 ,  we get  5
5		 , which is 1.
2
5		  is the complement of  3
5		 .

Example 15.   Compare

1 −  1
4		   and  6 −  1
4		 .
First, since 1 is  4
4		 , then
1 −  1
4		  =  3
4		 ,
  which is the complement of   1
4		 .

Look:

  since we are subtracting  1
4		  -- which is less than 1 -- from 6, the answer
  will fall beween 5 and 6.  It will be 5 3
4		 .  Again,  3
4		  is the complement of   1
4		 .

In other words:
 5.   What will be the answer when we subtract a proper fraction from a whole number greater than 1?
  It will be a mixed number which is one whole number less, and whose fraction is the complement of the proper fraction.
  Example 16. 5 −  1
3		  = 4 2
3

4 is one less than 5.  And  2
3		  is the complement of  1
3		 .
In fact, whenever we subtract  1
3		  from a whole number, the fractional
  part will be  2
3		 .
 
12 −  1
3		   =   11 2
3		 .
 
25 −  1
3		   =   24 2
3		 .
 
38 −  1
3		   =   37 2
3		 .

And so on.

  Example 17.   9 −  2
5		   =  8 3
5		 .

We could even check that by adding:

8 3
5		  +  2
5		  = 8 5
5		  = 9.

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

or

Continue on to the next Lesson.


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