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A R I T H M E T I C

Lesson 20

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?
What do we mean by the complement of a proper fraction?
Section 2
What kind of fraction does "out of" indicate?

1.	What is a unit fraction?

	A fraction whose numerator is 1.

A unit, recall, is whatever we call one. (Lesson 1.) Each unit fraction is a part of number 1.

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.

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.

Example 2. Let

1
3

be the unit, and count to 2

1
3

We see that every fraction is a multiple of some unit fraction:

2
3

= 2 ×

1
3

3
5

= 3 ×

1
5

Example 3. Add

2
8

3
8

Answer.

5
8

2 eighths + 3 eighths are 5 eighths. The unit 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. In any fraction, the denominator names the unit.

7
9

−

3
9

4
9

Example 4. 1 is how many fifths?

Answer.	5 5	.

1 5	is contained in 1 five times.

Similarly,

1 =

3
3

4
4

10
10

And so on. We may write 1 with any denominator. Which is to say, we may decompose 1 into any parts: Halves, thirds, fourths, fifths, millionths.

Example 5. Add, and express the sum as an improper fraction:

5
9

+ 1.

Answer.

5
9

+ 1 =

5
9

9
9

14
9

2.	How can we express a whole number as fraction?

	Multiply the whole number by the denominator. That product will be the numerator.

Example 8. 2 =

2 × 5
5

10
5

Since 1 =

5
5

, then 2 is twice as many fifths: 2 =

10
5

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

The complement of a proper fraction

3.	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 11.

5
8

+ ? = 1

Answer. Since 1 =

8
8

, then

5
8

3
8

Equivalently, since finding what number to add is subtraction,

1 −

5
8

3
8

is called the complement of

5
8

3
8

completes

5
8

to make 1.

Example 12. How much is

1 −

1
3

Answer. 1 −

1
3

2
3

1
3

plus

2
3

3
3

, which is 1.

Example 13. 1 −

2
5

3
5

When we add

3
5

2
5

, we get 1.

Example 14. How much is 6

3
4

−

1
4

Answer. 6

2
4

. The 6 is not affected.

Example 15. How much is 6

4
4

−

1
4

Answer. 6

3
4

But 6

4
4

is 7. That is,

−

1
4

= 6

3
4

Look at the fact:

We are subtracting

1
4

-- which is less than 1 -- from 7. The answer

therefore falls beween 6 and 7. And

3
4

is the complement of

1
4

Compare

1 −

1
4

In other words:

Whenever we subtract a proper fraction from a whole number greater than 1, the answer 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

On the other hand, we could say that we can only subtract thirds from thirds. Therefore we must create thirds by breaking off 1 from 5

and calling it

3
3

5 −

1
3

= 4

3
3

−

1
3

= 4

2
3

Example 17. 9 −

2
5

= 8

3
5

We could check this by adding:

3
5

2
5

= 9.

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

Continue on to the next Section.

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