Transcript rational number - Cloudfront.net

Section 5.3
The Rational
Numbers
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Rational Numbers
Multiplying and Dividing Fractions
Adding and Subtracting Fractions
5.3-2
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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.
The following are examples of
rational numbers:
1
3
7
2
15
,
,  , 1 , 2, 0,
3
4
8
3
7
5.3-3
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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.
5.3-4
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Reducing Fractions
To reduce a fraction to its lowest
terms, divide both the numerator and
denominator by the greatest common
divisor.
5.3-5
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Example 1: Reducing a Fraction
to Lowest Terms
54
Reduce
to lowest terms.
90
Solution
GCD of 54 and 90 is 18
54 54  18
3


90 90  18
5
5.3-6
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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”.
5.3-7
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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
5.3-8
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Converting a Positive Mixed
Number to an Improper Fraction
1. Multiply the denominator of the fraction in
the mixed number by the integer
preceding it.
2. 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
number.
5.3-9
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Example 2: Converting Mixed
Numbers to Improper Fractions
Convert the following mixed numbers
to improper fractions.
3
4 1  3 4  3
7
a) 1



4
4
4
4
24  7
7
8 3  7
31

b) 3


8
8
8
8
5.3-10
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Converting a Positive Improper
Fraction to a Mixed Number
1. Divide the numerator by the denominator.
Identify the quotient and the remainder.
2. 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.
5.3-11
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Example 3: From Improper
Fraction to Mixed Number
Convert the following improper fraction
to a mixed number.
Solution
8
a)
5
5.3-12
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Example 3: From Improper
Fraction to Mixed Number
Solution
3
The mixed number is 1 .
5
5.3-13
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Example 3: From Improper
Fraction to Mixed Number
Convert the following improper fraction
to a mixed number.
225 Solution
b)
8
5.3-14
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Example 3: From Improper
Fraction to Mixed Number
Solution
1
The mixed number is 28 .
8
5.3-15
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Terminating or Repeating
Decimal Numbers
Every rational number when
expressed as a decimal number will
be either a terminating or a
repeating decimal number.
5.3-16
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Terminating or Repeating
Decimal Numbers
Examples of terminating decimal
numbers are 0.5, 0.75, 4.65
Examples of repeating decimal
numbers 0.333… which may be
written 0.3, 0.2323… or 0.23, and
8.13456456… or 8.13456.
5.3-17
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Example 5: Terminating Decimal
Numbers
Show that the following rational
numbers can be expressed as
terminating decimal numbers.
3
a)
= 0.6
5
23
c)
= 1.4375
16
13
b) 
= –0.65
20
5.3-18
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Example 6: Repeating Decimal
Numbers
Show that the following rational
numbers can be expressed as
repeating decimal numbers.
2
 0.6
a)
3
5
 1.138
c) 1
36
14
b)
 0.14
99
5.3-19
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Converting Decimal Numbers to
Fractions
We can convert a terminating or
repeating decimal number into a
quotient of integers.
The explanation of the procedure
will refer to the positional values to
the right of the decimal point, as
illustrated here:
5.3-20
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Converting Decimal Numbers to
Fractions
5.3-21
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Example 10: Converting a
Repeating Decimal Number to a
Fraction
Convert 12.142 to a quotient of
integers.
n  12.142
100n  1214.2
10  100n  10  1214.2
1000n  12142.2
5.3-22
1000n  12142.2
100n  1214.2
900n  10928
10,928 2732
n

900
225
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Multiplication of Fractions
The product of two fractions is found
by multiplying the numerators
together and multiplying the
denominators together.
a c
ac
ac
 

,
b d
bd
bd
5.3-23
b  0, d  0
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Example 11: Multiplying Fractions
Evaluate
3 7
21
37
a) 


5 8
40
58
  
  
 2   4 
2 4
b)     
 3  9 
3 9
8

27
 7   1  15 9
135
7
c) 1   2  


4
8 4
 8  4
32
32
5.3-24
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Reciprocal
The reciprocal of any number is 1
divided by that number.
The product of a number and its
reciprocal must equal 1.
1
3  1
3
5.3-25
3 5
 1
5 3
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1
6 
1
6
Division of Fractions
To find the quotient of two fractions,
multiply the first fraction by the
reciprocal of the second fraction.
a c
a d
ad
   
, b  0, d  0, c  0
b d
b c
bc
5.3-26
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Example 12: Dividing Fractions
Evaluate
5 3
5 4
20
54
a) 
 


7 4
7 3
21
7 3
 3 7
24
3  8 24
3 8
b)        


35
35
 5 8
5 7
5 7
5.3-27
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Addition and Subtraction of
Fractions
To add or subtract two fractions with a
common denominator, we add or
subtract their numerators and retain
the common denominator.
a b ab
 
, c  0;
c c
c
a b ab
 
, c0
c c
c
5.3-28
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Example 13: Adding and
Subtracting Fractions
Evaluate
1 3
13
1
4
a) 



8 8
8
2
8
19 5
19  5
14
7
b)




24 24
24
24
12
5.3-29
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Fundamental Law of Rational
Numbers
If a, b, and c are integers, with b ≠ 0,
and c ≠ 0, then
a a c
ac
  
b b c
bc
a
ac
and
are equivalent fractions.
b
bc
5.3-30
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Adding or Subtracting Fractions
with Unlike Denominators
When adding or subtracting two
fractions with unlike denominators,
first rewrite each fraction with a
common denominator. Then add or
subtract the fractions.
5.3-31
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Example 14: Subtracting Fractions
with Unlike Denominators
Evaluate
13 5

15 6
 13 2   5 5 

    
 15 2   6 5 
26 25


30 30
1

30
5.3-32
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Homework
P. 239 # 15 – 90 (x3)
5.3-33
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