chapter r prealgebra review slide 2copyright 2011, 2007, 2003, 1999 pearson education, inc....

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



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




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.


Fraction Notation


For any number a,a + 0 = a.

Adding 0 to any number gives that same number.

12 + 0 = 12


The Identity Property of Zero (Additive Identity)


For any number a,a 1 = a.

Multiplying any number by 1 gives that same number.

2 1 = 2


The Identity Property of 1 (Multiplicative Identity)


.1

a

aFor any number a, a 0,


Equivalent Expressions for 1


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 :



A Find a number equivalent to with a denominator of 36.


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


b Simplify fraction notation.

B Simplify.

(continued)


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



B Simplify.


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.




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


a c a c

b d b d

To multiply fractions, multiply the numerators and multiply the denominators:


Multiplying Fractions


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


c Add, subtract, multiply, and divide using fraction notation.

C Multiply and simplify.


To add fractions when the denominators are the same, add the numerators and keep the same denominator:

a b a b

c c c


Adding Fractions with Like Denominators


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



D Add and simplify.

(continued)


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



D Add and simplify.


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.


Adding Fractions with Different Denominators


EXAMPLESolutiona) The LCD is 10. Since 5 is a factor of 10, the LCM of 5 and

10 is 10.

1 3

5 10



E Add:

(continued)


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:



E Add:


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



F Add:


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.



G Subtract:

(continued)


EXAMPLE

Solution

We subtract:

4 3.

5 7

28 15

35 35

28 15

35

13

35



G Subtract:


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.



H Subtract:


Two numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other.


Reciprocals


EXAMPLE6

75

9

1

4

8

91. 2. 3. 4.



I Find the reciprocal.

(continued)


EXAMPLESolution1. The reciprocal of is

2. The reciprocal of is



6

75

91

48

9

7

6.

9

5.

4.

9

8.



I Find the reciprocal.


a c a d

b d b c

To divide fractions, multiply by the reciprocal of the divisor:


Dividing Fractions


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



J Divide and simplify.


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



K Divide and simplify.


chapter r prealgebra review slide 2copyright 2011, 2007, 2003, 1999 pearson education, inc....

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