S k i l l
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A R I T H M E T I C

Lesson 21

UNIT FRACTIONS

In this Lesson, we will answer the following:

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

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.

Here is how we count

1
5

's. "One fifth, two fifths, three fifths," and so on.

Every fraction -- every division of 1 -- 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

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

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

3
8

= 3 ×

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;
we must 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.

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 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 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

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

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:

3
5

2
5

= 8

5
5

= 9.

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

Continue on to the next Lesson.

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