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Lesson 21

UNIT FRACTIONS


A unit -- "one" -- is the source of a number of anything.
We count units.


 1.   What is a unit fraction?
 
  A fraction with 1 as its numerator.
 

 

Each unit fraction is a part of number 1.  Half of 1, a third, a fourth, and so on.

Here is how we count  1
5
's. "One fifth, two fifths, three fifths," and so on.

Every fraction is thus a number of unit fractions.

 

In the fraction  3
5
, the unit is  1
5
.  And there are 3 of them.
3
5
 =  1
5
 +  1
5
 +  1
5
.

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

  Example 1.   In the fraction  5
6
 , what number is the unit, and how many

of them are there?

  Answer.   The unit is  1
6
.  And there are 5 of them.
5
6
 = 5 ×  1
6
 =  1
6
 +  1
6
 +  1
6
 +  1
6
 +  1
6
.
  Example 2.   Let  1
3
 be the unit -- and count to 2 1
3
.

Again, every fraction is a sum -- a number -- 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 of units.

  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;
whatever we add or subtract, we must be able to call them by the same name.

We will see this in Lesson 25.  The denominator of a fraction has no other function but to name the unit.

5
9
 −  3
9
  =   2
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
.
  Example 11.    Add:    5
3
 + 4.
  Answer.    5
3
 + 4 =   5
3
 +  12
 3
 =  17
 3
 = 5 2
3
.

Let us now revisit the rule for changing a mixed number to an improper fraction (Lesson 20).  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 12.   3 5
8
 = 3 +  5
8
 =  24
 8
 +  5
8
 =  29
 8
.
  Example 13.   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 14.    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 15.    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 16.   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 17. 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 18.   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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