2-1

2-2

Chapter 2Chapter 2

FractionsFractions

2-3

• Recognize the three types of fractions

• Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions

• Convert fractions to lowest and highest terms

Fractions#2#2Learning Unit ObjectivesTypes of Fractions and Conversion Procedures

LU2.1LU2.1

2-4

• Add like and unlike fractions

• Find the least common denominator (LCD) by inspection and prime numbers

• Subtract like and unlike fractions

• Add and subtract mixed numbers with the same or different denominators

Fractions#2#2Learning Unit ObjectivesAdding and Subtraction of FractionsLU2.2LU2.2

2-5

• Multiply and divide proper fractions and mixed numbers

• Use the cancellation method in the multiplication and division of fractions

Fractions#2#2Learning Unit ObjectivesMultiplying and Dividing FractionsLU2.3LU2.3

2-6

Types of Fractions

3, 4, 12, 1115 8 26 35

19, 9, 13, 4219 4 10 29

5, 3, 2, 18 17 9 8

2 9 15 4

Proper

Mixed Numbers

ImproperNumerator

Denominator

2-7

Converting Improper Fractions to Whole or Mixed Numbers

2 Steps1. Divide the numerator

by the denominator2. a.If you have no

remainder, the quotient is a whole number

2 b.If you have a remainder, the quotient is a mixed number

2424

21 1 4 4

5 R 14 21 20 1

= 5

= 1

2-8

Converting Improper Fractions to Whole or Mixed Numbers

3 Steps1. Multiply the denominator of the

fraction by the whole number.2. Add the product from Step 1 to

the numerator of the old fraction.

3 Place the total from Step 2 over the denominator of the old fraction to get the improper fraction.

(8 x 6) = 48

(8 x 6) = 4848 + 1 = 49

49 8

1 8

6

2-9

Reducing Fractions to Lowest Terms by Inspection

Find the lowest whole number that will divide evenly into the numerator and denominator

24 24 / 6 430 30 / 6 5

= =

2-10

Finding the Greatest Common Divisor

Step 1. Divide the numerator into the denominator

212 30 24 6

26 12 12 0

Step 2. Divide the remainder in Step 1 into the divisor of Step 1

12 / 6 230 / 6 5

=

Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0

1230

2-11

Divisibility Tests

2 3 4 5 6 10Last digit is 0,2,4,6,8

Sum of the digits is divisible by 3

Last two digits can be divided by 4

Last digit is 0 or 5

The number is even and 3 will divide into the sum of the digits

The last digit is 0

12 6

14 7

36 12

69 23

140 1(40)

160 1(60)

15 3

20 4

12 2

18 3

90 9

100 103 + 6 = 9 / 3 = 3

6 + 9 = 15 / 3 = 5

35 7

40 8

= = = = = =

=

2-12

Raising Fractions to Higher Terms When Denominator is Known

2 Steps1. Divide the new denominator by

the old denominator to get the common number that raises the fraction to higher terms.

2. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator.

47 28 28 0

4 x 4 = 16

4 = ? 7 28

1628

2-13

Least Common Denominator (LCD)

• The smallest nonzero whole number into which ALL denominators will divide evenly.

5 1620 10

40

60

20

+

What is the least common

denominator?

2-14

Adding Like Fractions• Add the numerators

and place the total over the denominator

• If the total of your numerators is the same as your original denominator, convert your answer to a whole number; if the total is larger than your original denominator, convert your answer to a mixed number

2 3 59 9 9

+ =

-b

5 6 11 29 9 9 9

+ = = 1

2-15

Adding Unlike Fractions4 Steps

1. Find the LCD 2. Change each fraction to a like fraction with the

LCD.3. Add the numerators and place the total over

the LCD.4. If necessary, reduce the answer to lowest terms.

1 1 1 13 8 9 12

+++

24 + 9 + 8 + 6 = 4772 72 72 72 72

2-16

Prime Numbers

A whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number.

Examples

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43

2-17

Adding Mixed Numbers

3 Steps

1. Add the fractions.2. Add the whole numbers.3. Combine steps 1 & 2. Be sure you do not have

an improper fraction in your final answer. If necessary, reduce the answer to lowest terms.

7 720 20

3 125 20

1 54 20

24 4 20 20 17 . 4 1 20 5

4 4

6 6

7 7

1

18 18

=

=

=

+ +

Step 1

Step 2

Step 3

2-18

Subtracting Like Fractions

• Subtract the numerators and place the total over the denominator

• If necessary, reduce the answer to lowest terms

7 1 6 / 2 312 12 12 / 2 6

=- =

2-19

Subtracting Unlike Fractions

5 2 8 64

-

5 40 8 64 2 - 2 64 64 38 = 19 64 32

4 Steps

1. Find the LCD 2. Raise the fraction to its equivalent value.3. Subtract the numerators and place the

answer over the LCD.4. If necessary, reduce the answer to lowest

terms.

-

2-20

Subtracting Mixed Numbers

1 4 2 8 - 3 - 3 8 8 1 8

3 Steps1. Subtract fractions, making sure to

find the LCD. 2. Subtract whole numbers.3. Reduce the fractions to lowest

terms.

When Borrowing is Not Necessary

6 6

6

2-21

Subtracting Mixed Numbers

1 2 6 2 4 4 3 3 3 4 4 4 3 4

4 Steps1. Make sure the fractions have the LCD.2. Borrow from the whole number.3. Subtract whole numbers and fractions.4. Reduce the fractions to lowest terms.

When Borrowing is Necessary

3 3

1

-1 -1 -1

2

2-22

Subtracting Unlike Fractions

5 2 8 64

-

5 40 8 64 - 2 2 6464 38 = 19 64 32

4 Steps1. Find the LCD 2. Raise the fraction to its equivalent value.3. Subtract the numerators and place the

answer over the LCD.4. If necessary, reduce the answer to lowest

terms.

-

2-23

Multiplying Proper Fractions

2 Steps

1. Multiply the numerator and the denominators

2. Reduce the answer to lowest terms

5 2 4 40 202 5 7 70 35

= =x x

2-24

Multiplying Mixed Numbers

Convert the mixed numbers to

improper fractions

Multiply the numerator and denominators

1 1 7 3 7 13 2 3 2 2 2

2 3=X 1 X = =

1

1

Reduce the answer to lowest terms

2-25

Dividing Proper Fractions

1 2 1 3 38 3 8 2 16

= =X

Invert (turn upside down) the divisor (the second fraction)

Multiply the fractions

.

.

Reduce the answer to lowest terms

2-26

Dividing Mixed Numbers

Convert all mixed numbers to

improper fractions

Invert the divisor and multiply

3 5 35 6 105 34 6 4 17 34 34

8 =X 2 X = =

Reduce the answer to lowest terms

3