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5.3
The Rational Numbers
Copyright © 2005 Pearson Education, Inc.
The Rational Numbers

The set of rational numbers, denoted by Q,
is the set of all numbers of the form p/q,
where p and q are integers and q  0.
Copyright © 2005 Pearson Education, Inc.
Slide 5-2
Fractions

Fractions are numbers such as:
1 2
9
, , and
.
3 9
53


The numerator is the number above the fraction
line.
The denominator is the number below the
fraction line.
Copyright © 2005 Pearson Education, Inc.
Slide 5-3
Reducing Fractions

In order to reduce a fraction, we divide both the
numerator and denominator by the greatest
common divisor.
72
Example: Reduce
to its lowest terms.

Solution: 72  72  9  8

81
81
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81  9
9
Slide 5-4
Mixed Numbers


A mixed number consists of an integer and a
fraction. For example, 3 ½ is a mixed number.
3 ½ is read “three and one half” and means
“3 + ½”.
Copyright © 2005 Pearson Education, Inc.
Slide 5-5
Improper Fractions


Rational numbers greater than 1 or less than -1
that are not integers may be written as mixed
numbers, or as improper fractions.
An improper fraction is a fraction whose
numerator is greater than its denominator.
An example of an improper fraction is 12/5.
Copyright © 2005 Pearson Education, Inc.
Slide 5-6
Converting a Positive Mixed Number to
an Improper Fraction


Multiply the denominator of the fraction in the mixed
number by the integer preceding it.
Add the product obtained in step 1 to the numerator of
the fraction in the mixed number. This sum is the
numerator of the improper fraction we are seeking.
The denominator of the improper fraction we are
seeking is the same as the denominator of the fraction
in the mixed
Copyright © 2005 Pearson Education, Inc.
Slide 5-7
Example

Convert
7
5
10
to an improper fraction.
7 (10  5  7) 50  7 57
5



10
10
10
10
Copyright © 2005 Pearson Education, Inc.
Slide 5-8
Converting a Positive Improper
Fraction to a Mixed Number


Divide the numerator by the denominator. Identify the
quotient and the remainder.
The quotient obtained in step 1 is the integer part of
the mixed number. The remainder is the numerator of
the fraction in the mixed number. The denominator in
the fraction of the mixed number will be the same as
the denominator in the original fraction.
Copyright © 2005 Pearson Education, Inc.
Slide 5-9
Example

Convert
236
7
to a mixed number.
33
7 236
21
26
21
5

5
7
The mixed number is 33 .
Copyright © 2005 Pearson Education, Inc.
Slide 5-10
Terminating or Repeating Decimal
Numbers



Every rational number when expressed as a decimal
number will be either a terminating or repeating
decimal number.
Examples of terminating decimal numbers 0.7, 2.85,
0.000045
Examples of repeating decimal numbers 0.44444…
which may be written 0.4,
and 0.2323232323... which may be written 0.23.
Copyright © 2005 Pearson Education, Inc.
Slide 5-11
Multiplication of Fractions
a c a  c ac
 

, b  0, d  0.
b d b  d bd

Division of Fractions
a c a d ad
   
, b  0, d  0, c  0.
b d b c bc
Copyright © 2005 Pearson Education, Inc.
Slide 5-12
Example: Multiplying Fractions

Evaluate the following.

a) 2  7
3 16
2 7
2  7 14
7




3 16 3  16 48 24
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
b)
 3   1
1 4    2 2 
  

 3   1 7 5
1 4    2 2   4  2
  

35
3

4
8
8
Slide 5-13
Example: Dividing Fractions


Evaluate the following.
a) 2 6

3 7
2 6 2 7
  
3 7 3 6
2  7 14 7



3  6 18 9
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
b) 5 4

8 5
5 4 5 5
 

8 5
8 4
5  5 25


84
32
Slide 5-14
Addition and Subtraction of Fractions
a b ab
 
, c  0.
c c
c
a b ab
 
, c  0.
c c
c
Copyright © 2005 Pearson Education, Inc.
Slide 5-15
Example: Add or Subtract Fractions

Add: 4  3
9 9


4 3 43 7
 

9 9
9
9

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Subtract:
11 3

16 16
11 3 11  3 8



16 16
16
16
1

2
Slide 5-16
Fundamental Law of Rational Numbers

If a, b, and c are integers, with b 
0, c  0, then
a a c a  c ac
  

.
b b c b  c bc
Copyright © 2005 Pearson Education, Inc.
Slide 5-17
Example:

Evaluate:

Solution:
7
9
 .
12 10
7
1  7 5  1 6

     
12 10  12 5   10 6 
35 6


60 60
29

60
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Slide 5-18
Next Steps





Read Examples 1-6, 11-15
Work Problems in text on
13-45, odds; 101-107, all
Do Online homework corresponding to this
section
Do Online quiz for Secs. 5.1-5.3
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Slide 5-19