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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.
 

unit fractions 

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

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

Every fraction is thus a number of unit fractions.

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 fraction, 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
.
unit fractions

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 1-4 .

This illustrates the following principle:

We can only add or subtract things that have the same name,
which we call the unit.
In the Example above , the name of what we are adding is "Eighths."

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.")

unit fractions

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 whatever we add must have the same name.

unit fractions

unit fractions

 2.   How can we express a whole number as fraction
with a given denominator?
 
unit fractions
 
  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?
mixed number
 
  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?
unit fractions
  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:

unit fractions

since we are subtracting 1-4 -- which is less than 1 -- from 6, the answer will fall beween 5 and 6.  It will be 54-4.

Again, 4-4 is the complement of 1-4 .


 5.   What will be the answer when we subtract a proper fraction from a whole number greater than 1?
unit fractions
  It will be the mixed number that is one whole number less, and whose fraction is the complement of the proper fraction.
 
  Example 17. 5 −  1
3
 = 4 2
3

unit fractions

4 is one less than 5.  And  2
3
 is the complement of  1
3
.

In fact, whenever we subtract 1-4 from a whole number, the fractional part will be 1-4 .

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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