FractionsFractions

Math 173 DFMath 173 DFFall 2008Fall 2008R. RainaR. Raina

OverviewOverview

►Fraction BasicsFraction Basics►Types of FractionsTypes of Fractions►Simplifying FractionsSimplifying Fractions►Multiplying and DividingMultiplying and Dividing►Adding and SubtractingAdding and Subtracting►Prime Factorization TechniquePrime Factorization Technique►Converting from Fractions to DecimalsConverting from Fractions to Decimals

The Basics: The Basics:

a

b

A fraction has a A fraction has a numerator,numerator, a a denominator,denominator, and a and a fraction line.fraction line.

If a whole quantity is divided into parts, each of those parts is called a fraction of the whole.

Since division by zero is not permitted, it should be understood in our work with fractions that the denominator cannot be zero.

Numerator

Denominator

Fraction Line

Example: There are 5 people in a room, two men and three women.

Thus of the people are men, and are women.

2

5

3

5

Common vs. Algebraic Common vs. Algebraic FractionsFractions

► Common Fraction:Common Fraction: A fraction whose numerator and A fraction whose numerator and

denominator are both integersdenominator are both integers

►Algebraic FractionAlgebraic Fraction A fraction whose numerator and/or A fraction whose numerator and/or

denominator contain denominator contain literalliteral quantities. quantities.

Examples:

2 -7 52

3 10 -109

Examples:

2

- 3 10

5

x x

y y

Proper, Improper and Mixed Proper, Improper and Mixed FractionsFractions

► A A properproper common fraction is one whose common fraction is one whose numerator is smaller than its denominator. numerator is smaller than its denominator.

1 19 1

3 20 2

► A A mixed numbermixed number is the sum of an integer is the sum of an integer and a fraction. and a fraction.

► A A improperimproper common fraction is one whose numerator is common fraction is one whose numerator is larger than its denominator.larger than its denominator.

1 19 15 7 15

3 20 2

16 159 31

3 20 2

Changing Improper Fractions to Changing Improper Fractions to Mixed Form Mixed Form

► To change an improper fraction to a mixed number:To change an improper fraction to a mixed number:- Divide the denominator into the numerator.- Divide the denominator into the numerator.- Write the remainder over the denominator. - Write the remainder over the denominator.

30

4

30 ÷ 4 = .5

7× 4 = 28

30

7

- 28 = 2=

27

4

Changing Mixed Numbers to Changing Mixed Numbers to Improper FractionsImproper Fractions

► To change a mixed number to an improper fraction:

- Multiply the whole number by the denominator and add this number to the numerator.

- Write the sum over the denominator.

27

42

= 7 +4

28 2= +

4 4

30=

4

Simplifying a Fraction Simplifying a Fraction

► Reducing to lowest terms: Reducing to lowest terms: dividing both numerator and denominator by any dividing both numerator and denominator by any

factor that is contained in both. factor that is contained in both.

► Changing signs:Changing signs: any two of the three signs of a fraction may be any two of the three signs of a fraction may be

changed without changing the value of a fraction.changed without changing the value of a fraction.

3

6

20

8

42

121

=2

5=

2

1= 2

3

1 -1 -1 1- = - = =2 -2 2 -2

3 -3 -3 3= = - = -

4 -4 4 -4

Simplify the following fractions:Simplify the following fractions:

6=

8

15=

6

48=

76

33 =

9

3

4

24

3812

=19

5

2

13

3

Multiplication and DivisionMultiplication and DivisionWhat are we doing?What are we doing?

1 1×

2 2

1

4

1 1÷

2 4 2

MultiplicationMultiplication

► Fraction Multiplication (Steps):Fraction Multiplication (Steps):1) Change mixed numbers to improper fractions.1) Change mixed numbers to improper fractions.

2) Change whole numbers to fractions by dividing by 2) Change whole numbers to fractions by dividing by one.one.3) Multiply all numerators together – this number is the 3) Multiply all numerators together – this number is the

numerator of your answer.numerator of your answer.4) Multiply all denominators together – this number is 4) Multiply all denominators together – this number is the the denominator of your answer.denominator of your answer.5) Simplify you answer. 5) Simplify you answer.

How do we do it?How do we do it?

4 3× =

9 54×3

=9×5

12

454

=15

DivisionDivision

► Dividing Fractions (Steps)Dividing Fractions (Steps)1) Change mixed numbers to improper fractions.1) Change mixed numbers to improper fractions.2)Change whole numbers to fractions by dividing by 2)Change whole numbers to fractions by dividing by one.one.3) Invert and Multiply3) Invert and Multiply

(Take the reciprocal of the fraction after the sign, (Take the reciprocal of the fraction after the sign, Change the division sign to a multiplication sign.)Change the division sign to a multiplication sign.)

5) Multiply the two fractions together5) Multiply the two fractions together

How do we do it?How do we do it?

4 3÷ =

9 54 5

× =9 3

4×5=

9×3

20

27

Adding and SubtractingAdding and Subtracting

1 3+ =

3 5

+4 1

+ =9 3 =

7=

9

+ =

3 1- =

5 3

14=

15

+ =4

=15

5

15

9

15

14

15

5

15

9

154

15

► If the fractions have a common denominator:If the fractions have a common denominator:- Add the numerators together.- Add the numerators together.- Keep the common denominator. - Keep the common denominator. - Simplify if required. - Simplify if required.

► If the fractions do not have a common denominator:If the fractions do not have a common denominator:

- Find the lowest common denominator (LCD) – use - Find the lowest common denominator (LCD) – use prime factorization.prime factorization.

- Find the equivalent fraction with the chosen - Find the equivalent fraction with the chosen denominator. denominator.

- Add the fractions. Simplify if required. - Add the fractions. Simplify if required.

AddingAdding

3 1+

8 83 +1

=8

4=

81

=2

2 3+

3 4

8 9= +

12 12

8 + 9=

12

17 5= = 1

12 12

SubtractingSubtracting

► If the fractions do not have a common denominator:If the fractions do not have a common denominator:

- Find the lowest common denominator (LCD) – use prime - Find the lowest common denominator (LCD) – use prime factorization.factorization.

- Find the equivalent fraction with the chosen - Find the equivalent fraction with the chosen denominator. denominator.

- Subtract the fractions. Simplify if required. - Subtract the fractions. Simplify if required.

► If the fractions have a common denominator:If the fractions have a common denominator:- Subtract the numerators together.- Subtract the numerators together.- Keep the common denominator. - Keep the common denominator. - Simplify if required. - Simplify if required.

3 1- =

8 8

3 -1=

82

8

1=

4

2 3- =

3 48 9

- =12 12

8 - 9=

12-1

12

A Useful Technique – Prime A Useful Technique – Prime FactorizationFactorization

Ex) Find the LCM of 18 and 24.Ex) Find the LCM of 18 and 24.► Find the prime factors of each number, and line Find the prime factors of each number, and line

them up vertically. If a number appears in a them up vertically. If a number appears in a column more than once, move only one down. column more than once, move only one down.

18 = 2 3 3

24 = 2 2 2 3

2 2 2 3 3 72

Decimals and Fractions Decimals and Fractions

► To change a fraction to an equivalent decimal, To change a fraction to an equivalent decimal, divide the numerator by the denominator. divide the numerator by the denominator.

► To change a decimal number to a fraction:To change a decimal number to a fraction: Write a fraction with the decimal the numerator and 1 in Write a fraction with the decimal the numerator and 1 in

the denominatorthe denominator Multiply by a multiple of 10 that will make the numerator a Multiply by a multiple of 10 that will make the numerator a

whole number. whole number. Reduce to lowest terms. Reduce to lowest terms.

►To express a To express a repeatingrepeating decimal as a fraction: decimal as a fraction:

____Example: Express 0.3232 as a fractionExample: Express 0.3232 as a fraction

Let x = 0.3232 (1)

Then 100x = 32.32 (2)

Subtract: (2) - (1)

100x = 32.32

- x = 0.3232

99x = 32

99x = 32

32 x =

99

32 0.3232 =

99Therefore