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

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?
3. What do we mean by the complement of a proper fraction?

Section 2

4. 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. 12 is one half of 1. 13 is the third part of 1. 14 is the fourth part of 1.

And so on.

 Example 1.    In the fraction 45 , 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 45 , the unit is 15 .  And there are 4 of them. Example 2.   Let 13 be the unit, and count to 2 13 . We see that every fraction is a multiple of some unit fraction:

 23 =  2 × 13 = 13 + 13 .
 35 =  3 × 15 = 15 + 15 + 15 .
 Example 3.   Add 28 + 38 .
 Answer. 58 .
 2 eighths + 3 eighths are 5 eighths. The unit is 18 .

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.

 79 − 39 = 49

Example 4.   1 is how many fifths?

 Answer. 55 . 15 is contained in 1 five times.

Similarly,

 1 = 33 = 44 = 1010

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: 59 + 1.
 Answer. 59 + 1 = 59 + 99 = 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 = 55 , 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 18 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. 58 + ? = 1
 Answer.   Since 1 = 88 , then 58 + 38 = 1

Equivalently, since finding what number to add  is subtraction,

 1 − 58 = 38 .
 38 is called the complement of 58 . 38 completes 58 to make 1.
 Example 12.    How much is 1 − 13 ?
 Answer.   1 − 13 = 23 .
 13 plus 23 = 33 , which is 1.
 Example 13.    1 − 25 = 35 .
 When we add 35 to 25 ,  we get 1.

 Example 14.   How much is 6 34 − 14 ?
 Answer.  6 24 .  The 6 is not affected.
 Example 15.   How much is 6 44 − 14 ?
 Answer.  6 34 .
 But 6 44 is 7.  That is,
 7 − 14 =  6 34

Look at the fact: We are subtracting 14 -- which is less than 1 -- from 7.  The answer
 therefore falls beween 6 and 7.  And 34 is the complement of 14 .

Compare

 1 − 14 : 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 − 13 = 4 23 4 is one less than 5.  And 23 is the complement of 13 .

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 33 .
 5 − 13 = 4 33 − 13 = 4 23 .
 Example 17.   9 − 25 =  8 35 .

We could check this by adding:

 8 35 + 25 = 9.

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

or

Continue on to the next Section.

www.proyectosalonhogar.com