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CHAPTER
RPrealgebra Review
Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
R.1 Factoring and LCMs
R.2 Fraction Notation
R.3 Decimal Notation
R.4 Percent Notation
R.5 Exponential Notation and Order of Operations
R.6 Geometry
OBJECTIVES
R.2 Fraction Notation
Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a Find equivalent fraction expressions by multiplying by 1.
b Simplify fraction notation.c Add, subtract, multiply, and divide using fraction
notation.
The following are some examples of fractions:
This way of writing number names is called fraction notation. The top number is called the numerator and the bottom number is called the denominator.
3 13 7 3, , , ,
4 21 6 4
a a
b b
9
16 Numerator
Denominator
R.2 Fraction Notation
a Find equivalent fraction expressions by multiplying by 1.
Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
The arithmetic numbers are the whole numbers and the fractions, such as 8, 3/4, and 6/5. All these numbers can be named with fraction notation a/b, where a and b are whole numbers and b ≠ 0.
R.2 Fraction Notation
Fraction Notation
Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
For any number a,a + 0 = a.
Adding 0 to any number gives that same number.
12 + 0 = 12
R.2 Fraction Notation
The Identity Property of Zero (Additive Identity)
Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
For any number a,a 1 = a.
Multiplying any number by 1 gives that same number.
2 1 = 2
R.2 Fraction Notation
The Identity Property of 1 (Multiplicative Identity)
Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
.1
a
aFor any number a, a 0,
R.2 Fraction Notation
Equivalent Expressions for 1
Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
141
14
EXAMPLE3
4
9
9
3
4
93
4 9
3 9
4 9
27
36
SolutionSince 36 4 = ∙ 9, we multiply by 1, using :
R.2 Fraction Notation
a Find equivalent fraction expressions by multiplying by 1.
A Find a number equivalent to with a denominator of 36.
Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
1. 2. 3.
Solution1.
28
35
3
248
32
28
35
4 7
5 7
74
5 7
4
5 Removing a factor equal
to 1: 7/7 = 1
R.2 Fraction Notation
b Simplify fraction notation.
B Simplify.
(continued)
Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLESolution
2.
3.
3
241 3
8 3
31
8 3
1
8
Writing 1 allows for pairing of factors in the numerator and the denominator.
8
32
1 8
4 8
81
4 8
1
4
R.2 Fraction Notation
b Simplify fraction notation.
B Simplify.
Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Canceling is a shortcut that you may have used for removing a factor that equals 1 when working with fraction notation.
Canceling may be done only when removing common factors in numerators and denominators.
Canceling must be done with care and understanding.
R.2 Fraction Notation
b Simplify fraction notation.
Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Caution!The difficulty with canceling is that it is often applied incorrectly in situations like the following:
The correct answers are:
In each of the incorrect cancellations, the numbers canceled did not form a factor equal to 1. Factors are parts of products, but in 2 + 3, the numbers 2 and 3 are terms. You cannot cancel terms.
2 3
2
43;
1
4
1 15;
22
5
1
44
2 3 5 4 1 5 15 5; ;
2 2 4 2 6 54 18
Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a c a c
b d b d
To multiply fractions, multiply the numerators and multiply the denominators:
R.2 Fraction Notation
Multiplying Fractions
Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE12 14
28 21
41 7
7
2 14 2
28 2
3
1 73 4
Removing a factor equal to 1
34
4
7
737
2
2
7
Solution
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
C Multiply and simplify.
Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To add fractions when the denominators are the same, add the numerators and keep the same denominator:
a b a b
c c c
R.2 Fraction Notation
Adding Fractions with Like Denominators
Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE1. 2. 3.
Solution
1.
3 1
5 5
5 7
16 16
3 4
12 12
3 1
5 5
3 1
5
4
5
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
D Add and simplify.
(continued)
Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLESolution2.
5 7
16 16
5 7
16
12
16
34
4 4
3
4
3. 3 4
12 12
3 4
12
7
12
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
D Add and simplify.
Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
To add fractions when denominators are different:a) Find the least common multiple (LCM) of the
denominators. That number is the least common denominator, LCD.
b) Multiply by 1, using an appropriate notation, n/n, to express each number in terms of the LCD.
c) Add the numerators, keeping the same denominator.
d) Simplify, if possible.
R.2 Fraction Notation
Adding Fractions with Different Denominators
Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLESolutiona) The LCD is 10. Since 5 is a factor of 10, the LCM of 5 and
10 is 10.
1 3
5 10
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
E Add:
(continued)
Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE1 3
5 10
2 3 5 1.
10 10 10 2
Solutionb) We need to find a fraction equivalent to with a
denominator of 10:
c & d) We add:
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
E Add:
Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
1 3
5 10 2
2
1 3
5 10
EXAMPLE7 13
12 18
7 13
12 18 1
12 181
7 13
3 2
3
7 1
1 2
3
2 18
21 26 47
36 36 36
Think: 12 = 36. The answer is 3, so we multiply by 1, using 3/3.
Think: 18 = 36. The answer is 2, so we multiply by 1, using 2/2.
Solution: The LCD is 36. 12 = 2 2 3∙ ∙ 18 = 2 3 3∙ ∙LCM = 2 2∙ ∙ 3 3∙ , or 36
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
F Add:
Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
Solution: The LCM of 7 and 5 is 35, so the LCD is 35.
Find equivalent numbers with denominators of 35.
4 3.
5 7
4 3
5 7
7 5
7 5
4 3
5 7
Think: 5 = 35. The answer is 7, so we multiply by 1, using 7/7.Think: 7 = 35. The answer is 5, so we multiply by 1, using 5/5.
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
G Subtract:
(continued)
Slide 23Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
Solution
We subtract:
4 3.
5 7
28 15
35 35
28 15
35
13
35
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
G Subtract:
Slide 24Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
4 3
5 7
7 5
7 5
4 3
5 7
EXAMPLE5 1
.12 16
12 2 2 3
16 2 2 2 2
The LCM is 2 2 2 2 3 = 48
5 1 5 1
12 16 12 1
4 3
4 36
20 3
48 48
20 3
48
17
48
Solution: Determine the LCM of 12 and 16.
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
H Subtract:
Slide 25Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
Two numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other.
R.2 Fraction Notation
Reciprocals
Slide 26Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE6
75
9
1
4
8
91. 2. 3. 4.
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
I Find the reciprocal.
(continued)
Slide 27Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLESolution1. The reciprocal of is
2. The reciprocal of is
3. The reciprocal of is
4. The reciprocal of is
6
75
91
48
9
7
6.
9
5.
4.
9
8.
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
I Find the reciprocal.
Slide 28Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
a c a d
b d b c
To divide fractions, multiply by the reciprocal of the divisor:
R.2 Fraction Notation
Dividing Fractions
Slide 29Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE3 5
4 16
3 5
4 16
63
4
1
5
4
3 44
5
44
4
3
5
12
5
Multiply by the reciprocal of the divisor
Factoring and identifying a common factorRemoving a factor equal to 1
Solution
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
J Divide and simplify.
Slide 30Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
EXAMPLE
9 13
10 15
53 3 3
352 1
3 3
2 3
5 3
15
27
26
9 13
10 15
3
9
10
15
1 Multiply by the reciprocal of
the divisor
Factoring and identifying a common factor
Removing a factor equal to 1
Solution
R.2 Fraction Notation
c Add, subtract, multiply, and divide using fraction notation.
K Divide and simplify.
Slide 31Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.
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