AM1101 Math Essentials Class Notes Module Two Module 2 ...

1

AM1101 Math Essentials

Class Notes

Module Two

Module 2, Section 1: Understanding Fractions

Outcomes:

Define a numerical fraction.

Identify the parts of a fraction and tell what each part of the fraction represents.

Use a fraction to represent part of a whole.

Draw a sketch to represent a fraction.

Use fractions to represent real-life situations.

nDef : A numerical fraction is the quotient (division) of two whole numbers.

E.g.: 1 3 9

; 1: 2; ; 3: 4; ; 9 : 42 4 4

Parts of a Fraction

A fraction has two main parts. The top of the fraction is

called the numerator. The bottom of the fraction is called

the denominator.

The denominator represents the number of equal parts in the whole. The numerator represents the

number of parts of the whole that we want.

Using a Fraction to Represent Part of a Whole

Each of the three diagrams below represents 1 out of 4 or 1

4

2

E.g: Use a fraction to represent the shaded part of the whole.

The whole is broken into 7 equal parts so the denominator (bottom) is 7. Two parts are shaded so the

numerator (top) is 2. The fraction is

2

7


The whole is broken into 6 equal parts so the denominator (bottom) is 6.

One part is shaded so the numerator (top) is 1. The fraction is

1

6


The whole is broken into 5 equal parts so the denominator (bottom) is 5. Three parts are shaded so the

numerator (top) is 3. The fraction is

3

5

E.g.: Draw a sketch to illustrate 2

3.

The whole is broken into 3 equal pieces and 2 are shaded.


8.


3


9.


Applications Involving Fractions

E.g.: Twelve students are enrolled in the CCA&T program at CNA in

Bonavista. Seven of the students are male.

(a) What fraction of the students are

male?

7 of 12 or 7

12

(b) What fraction of the students are

female?

5 of 12 or 5

12

E.g.: There are 21 lengths of plastic pipe in the plumbing shop at CNA in

Bonavista. 9 lengths have an inside diameter of 3 inches, 5 lengths have an

inside diameter of 1

12

inches and the remainder have an inside diameter of 1

14

inches.

(a) What fraction of the pipe has

an inside diameter of 3 inches?

9 of 21 or 9

21

(b) What fraction of the pipe has

an inside diameter of 1

12

inches?

5 of 21 or 5

21

(c) What fraction of the pipe has

an inside diameter of 1

14

inches?

21 – 9 – 5 = 7

7 of 21 or 7

21

Watch https://www.youtube.com/watch?v=CA9XLJpQp3c

Do #’s 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 39, 43, pp. 124-125 text in your

PB.

https://www.youtube.com/watch?v=CA9XLJpQp3c

4

Module 2, Section 2: Simplifying Fractions

Outcomes:

Define equivalent fractions.

Reduce a fraction to lowest terms.

Determine if two fractions are the equal.

nDef : Equivalent fractions are fractions that have the same value.

1 2 and

3 6

1 3 and

2 6

4 16 and

5 20

Making Equivalent Fractions

1 2

3 6

1 3

2 6

4 16

5 20

In each example in the table above, what number do you multiply the first fraction by to get the second

fraction?

2 1

6 3

3 1

6 2

16 4

20 5

In each example in the table above, what number do you divide the first fraction by to get the second

fraction?

Determining if Two Fraction are Equivalent

Recall that equivalent fractions are fractions that have the same value. There is a quick way to determine

if two fractions are equal if you know your multiplication facts. It’s called the Equality Test for

Fractions

Two fractions are equivalent (equal) if the products of the diagonals (cross product) are equal. If the

cross products are NOT equal, the fractions are NOT equivalent (equal).

5

E.g.:

E.g.:

E.g.: Does 3 9

5 15 ?

Since 3 15 45 and 5 9 45 the fractions are equivalent

(equal).

Do #’s 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, pp. 133-134 text in your PB.

Reducing Fractions

Reducing a fraction means rewriting the fraction as an equivalent fraction so that the numerator and the

denominator can be divided by 1 but not by 2 or 3 or 4 or 5 or ….

E.g.: 4

9is reduced because 1 is the only number that divides into 4 and 9.

E.g.: 6

9is not reduced because 3 divides into 6 and into 9.

6

To reduce fractions, it helps if you know the divisibility rules.

Divisibility Rules

E.g.: Reduce 9

15

Using the divisibility rules, we can see that 3 is the biggest number that divides into both 9 and 15.

9 3 3

15 3 5

E.g.: Reduce 16

24


16 8 2

24 8 3

7

E.g.: Reduce 16

24


16 8 2

24 8 3

E.g.: Reduce 30

42

30 2 15

42 2 21

15 3 5

21 3 7

E.g.: Reduce 45

120

45 5 9

120 5 24

9 3 3

24 3 8

E.g.: Reduce 20

80

20 10 2

80 10 8

2 2 1

8 2 4

Do #’s 43, 45, 47, 49, 59, 61, 63, 65, 67, pp. 132-133 text in your PB.

8

Module 2, Section 3: Converting Between Improper Fractions & Mixed Numbers

Outcomes:

Define and give examples of a proper fraction, an improper fraction, & a mixed number.

Convert an improper fraction to a mixed number.

Reduce a mixed number to lowest terms.

Reduce an improper fraction to lowest terms.

nDef : A proper fraction is a fraction in which the numerator (top) is less than the denominator

(bottom). The value of the fraction is always less than one.

E.g.: 23 7 1000

, ,80 9 1001

nDef : An improper fraction is a fraction in which the numerator (top) is more than the denominator

(bottom). The value of the fraction is always more than one.

E.g.: 80 9 1001

, ,23 7 1000

nDef : A mixed number is the sum (add) of a counting number 1, 2, 3, 4, and a proper fraction.

E.g.: 1 1

1 or 12 2

E.g.:3 3

10 or 104 4

Watch https://www.youtube.com/watch?v=17IgK9b6P2M

https://www.youtube.com/watch?v=17IgK9b6P2M

9

Changing from a Mixed Number to an Improper Fraction

E.g.: Use the diagram to change each mixed number to an improper fraction.

Mixed Number & Diagram Improper Fraction

12

4

31

4

53

6

33

4

10

Changing from a Mixed Number to an Improper Fraction Shortcut (See top of p. 136 text)

E.g.: Change each mixed number to an improper fraction.

Mixed Number Improper Fraction

12

4 2 4 1 8 1 9

4 4 4

31

4

1 4 3 4 3 7

4 4 4

53

6 3 6 5 18 5 23

6 6 6

33

4

3 4 3 12 3 15

4 4 4

Watch https://www.youtube.com/watch?v=qk2oP6FZ6HA (0 to 8:33). This video shows another

method that can be used to convert a mixed fraction into an improper fraction.

Do #’s 3, 5, 7, 9, 11, 13, 21, 23, 25, 27, 29, p. 139 text in your PB.

Changing from an Improper Fraction to a Mixed Number

Method 1: Changing an improper fraction into a mixed number can be done using long division of

whole numbers.

E.g.: Convert 12

7 to a mixed number.

First rewrite 12

7 as 7 12 and divide

So 12 5

17 7

Method 2: Changing an improper fraction into a mixed number can also be done using whole fractions.

E.g.: Convert 12

7 to a mixed number.

First rewrite 12

7 as

7 5 7 5

7 7 7

Change the whole fraction 7

7 to a whole number.

7 5 7 5 5 51 1

7 7 7 7 7

1

7 12

7

5R

https://www.youtube.com/watch?v=qk2oP6FZ6HA

11

E.g.: Convert 32

9 to a mixed number using Method 1.

First rewrite 32


So 32 5

39 9

E.g.: Convert 32


First rewrite 32

9 as

9 9 9 5 9 9 9 5

9 9 9 9 9

Change each whole fraction to a whole number.

9 9 9 5 5 51 1 1 3

9 9 9 9 9 9

E.g.: Convert 151


First rewrite 151


So 151 7

436 36

E.g.: Convert 67


First rewrite 67

13 as

13 13 13 13 13 2 13 13 13 13 13 2 2 21 1 1 1 1 5

13 13 13 13 13 13 13 13 13

Watch https://www.youtube.com/watch?v=qk2oP6FZ6HA (8:33 to end).

Do #’s 31, 33, 35, 37, 39, 41, 49, 51, 55, 57, p. 139 text in your PB.

3

9 32

27

5R

4

36 151

144

7R

https://www.youtube.com/watch?v=qk2oP6FZ6HA

12

Reducing a Mixed Number

Reduce the fractional part only, if necessary.

E.g.: Reduce 6

48

6 2 3

8 2 4

so,

6 34 4

8 4

E.g.: Reduce 15

890

15 5 3 3 3 1

90 5 18 18 3 6

so,

15 18 8

90 6

E.g.: Reduce 15

1170

15 5 3

70 5 14

so,

15 311 11

70 14

Do #’s 59, 61, 63, p. 139 text in your PB.

Reducing an Improper Fractions

Use the same divisibility rules that were used to reduce proper fractions, use whole fractions, or use long

division.

E.g.: Reduce 25

5using divisibility rules.

25 5 55

5 5 1

13

E.g.: Reduce 30

18 using divisibility rules.

30 2 15 15 3 5

18 2 9 9 3 3

E.g.: Reduce 2150

1000using whole fractions.

2150 1000 1000 150 1000 1000 150 150 150 1501 1 2 2

1000 1000 1000 1000 1000 1000 1000 1000

Now reduce 150

1000

150 10 15 15 5 3

1000 10 100 100 5 20

so,

2150 150 32 2

1000 1000 20

E.g.: Reduce 372

78using long division.

372 604

78 78

Now reduce 60

78

60 2 30 30 3 10

78 2 39 39 3 13

so,

372 60 104 4

78 78 13


4

78 372

312

60R

14

Module 2, Section 4: Multiplying Fractions & Mixed Numbers

Outcomes:

Multiply proper fractions.

Multiply improper fractions.

Multiply proper & improper fractions.

Multiply a whole number by a fraction.

Multiply mixed numbers.

Multiplying Two Proper Fractions

E.g.: What is 1

2of

3

7?

1. First we draw a fraction strip for 3

7.

2. Then we cut the fraction strip into 2 equal pieces.

3. Finally, we take 1 of those 2 equal pieces to get

the answer of 3

14.

E.g.: What is 1

3of

5

8?

First we draw a fraction strip for 5

8.

Then we cut the fraction strip into 3 equal pieces.

Finally, we take 1 of those 3 equal pieces to get the

answer of 5

24.

15

E.g.: What is 3

4of

5

6?

First we draw a fraction strip for 5

6.

Then we cut the fraction strip into 4 equal pieces.

Finally, we take 3 of those 4 equal pieces to get

the answer of 15

24.

Multiplying Two Proper Fractions (Shortcut)

The short cut to multiplying two proper fractions is to multiply across the top and across the bottom.

E.g.: What is 1

2of

3

7?

1 3 1 3 3of

2 7 2 7 14

E.g.: What is 1

3of

5

8?

1 5 1 5 5of

3 8 3 8 24

E.g.: What is 3

4of

5

6?

3 5 3 5 15of

4 6 4 6 24

Sometimes the fraction that you get for your answer will need to be reduced.

E.g.: Simplify 7 16

8 21

7 16 7 16 112 112 56 2

8 21 8 21 168 168 56 3

OR 1

7 16 7

8 21

2

161

83

21

1 2 2

1 3 3

16

E.g.: Simplify 8 5 3

7 12 10

8 5 3 8 5 3 120 120 10 12 12 12 1

7 12 10 7 12 10 840 840 10 84 84 12 7

Watch https://www.youtube.com/watch?v=qmfXyR7Z6Lk

Do #’s 1, 3, 5, 7, 9, 11, 17, 19, p. 145 text in your PB.

Multiplying a Whole Number and a Fraction

Write the whole number as a fraction, multiply across the top and across the bottom, and reduce if

necessary.

E.g.: Simplify 8

69

8 8 6 8 6 48 48 3 16 16 5

9 9 1 9 1 9 9 3 3 3

E.g.: Simplify 7

525

7 5 7 5 7 35 35 5 7 25 1

25 1 25 1 25 25 25 5 5 5

Do #’s 13, 15, p. 145 text in your PB.

Multiplying Mixed Numbers

To multiply mixed numbers, change each mixed number to an improper fraction, multiply across the top

and across the bottom, and reduce if necessary.

E.g.: Simplify 3 5

35 6

3

3 5 3 5 3 5 18 5 183

5 6 5 6 5 6

1

51

51

6

3 1 33

1 1 1

https://www.youtube.com/watch?v=qmfXyR7Z6Lk

17

E.g.: Simplify 1 4

5 44 7

3

1 4 5 4 1 4 7 4 21 32 21 32 215 4

4 7 4 7 4 7 4 7

8

321

41

7

3 8 2424

1 1 1

E.g.: Simplify 1 3

3 52 8

1 3 3 2 1 5 8 3 7 43 7 43 301 133 5 18

2 8 2 8 2 8 2 8 16 16

Do #’s 21, 23, 25, 29, 31, 33, 35, p. 145 text in your PB.

Do #’s 47, 49, 51, 53, p. 146 text in your PB.

18

Module 2, Section 5: Dividing Fractions & Mixed Numbers

Outcomes:

Divide proper fractions.

Divide improper fractions.

Divide proper & improper fractions.

Divide a whole number by a fraction.

Divide mixed numbers.

Dividing Two Proper or Improper Fractions

Dividing fractions is very similar to multiplying fraction but requires one extra step. Compare the two

columns in the following table.

Find 1

2of 4.

Divide 4 into 2 equal pieces.

This can be written as 1

42 or

14

2 This can be written as 4 2 or

24

1

Note that you are really doing the same thing in each column as taking half of 4 is the same as dividing 4

by 2. This means that 2 1

4 41 2

.

When dividing fractions, multiply the first fraction by the reciprocal of the second fraction. Don’t forget

to reduce, if necessary.

E.g.: Simplify 3 9

13 26

1

3 9 3 26 3

13 26 13 29

2

261

133

9

1 2 2

1 3 3

E.g.: Simplify 5 5

8 8

5 5 5 8 5 8 401

8 8 8 5 5 8 40

19

E.g.: Simplify 11 1

12 5

11 1 11 5 11 5 55 74

12 5 12 1 12 1 12 12

E.g.: Simplify 2 34

17 6

2 34 2 6 2 6 12 12 2 6

17 6 17 34 17 34 578 578 2 289

Do #’s 3, 5, 7, 9, 11, 13, p. 153 text in your PB.

Dividing a Whole Number and a Fraction

Write the whole number as a fraction, multiply the first fraction by the reciprocal of the second fraction,

and reduce if necessary.

E.g.: Simplify 3

617

3 3 6 3 1 3 1 3 3 3 16

17 17 1 17 6 17 6 102 102 3 34

E.g.: Simplify 7

1415

7 14 7 14 7 14 7 98 814 6

15 1 15 1 15 1 15 15 15


Dividing Mixed Numbers

To divide mixed numbers, change each mixed number to an improper fraction, multiply the first fraction

by the reciprocal of the second fraction, and reduce if necessary.

E.g.: Simplify 3 5

35 6

3 5 3 5 3 5 18 5 18 6 18 6 108 83 4

5 6 5 6 5 6 5 5 5 5 25 25

20

E.g.: Simplify

41

51

62

41

4 1 1 5 4 6 2 1 9 13 9 2 9 2 185 1 61 5 2 5 2 5 2 5 13 5 13 65

62

Do #’s 31, 33, 45, 49, pp. 153-154 text in your PB.

Solving Division Equations with Variables

E.g.: Find the value of x if 3 8

4 15x

3 8

4 15

4 8

3 15

x

x

Since 2 4 8 and 5 3 15 , we know that 2

5x


Do #’s 69, 73, 75, pp. 154-155 text in your PB.

Complete “How Am I Doing?”, p. 156 & pp. 157-158

21

Module 2, Section 6: The Least Common Denominator (LCM) & Equivalent Fractions

Outcomes:

Find the least common multiple (LCM) of two numbers.

Use the LCM to find the common denominator for two or more fractions.

Create equivalent fractions for a given LCM.

nDef : The least common multiple (LCM) of two or more numbers is the smallest (least) number that

the given numbers will divide into evenly.

E.g.: The LCM of 2 & 3 is 6 as 6 is the smallest number that 2 divides into evenly and that 3 divides into

evenly.

E.g.: The LCM of 10 and 30 is 30.



Finding the Least Common Multiple

An easy way is to list the multiples of each number until there is a number in each list that is the same.

E.g.: Find the LCM of 14 and 21

The multiples of 14 are 14, 28, 42, 56, 70, …

The multiples of 21 are 21, 42, 63, …

42 is the smallest number in both lists so 42 is the LCM of 14 and 21.

E.g.: Find the LCM of 7 and 8

The multiples of 7 are 7, 14, 21, 28, 35,42, 49, 56, 63, 70, …

The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, …

56 is the smallest number in both lists so 56 is the LCM of 7 and 8.

22

E.g.: Find the LCM of 5, 12, & 20

The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, …

The multiples of 12 are 12, 24, 36, 48, 60, 72, …

The multiples of 20 are 20, 40, 60, 80, …

60 is the smallest number in both lists so 60 is the LCM of 5, 12, and 20.

Do #’s 1, 3, 5, 7, 9, p. 166 text in your PB.

Finding the Least Common Denominator (LCD) of Two or More Fractions

We can use the LCM to find the LCD of two or more fractions.

E.g.: Find the LCD of 5 7

and 4 12

.

We need to find the LCM of 4 and 12.

The multiples of 4 are 4, 8, 12, 16, 20, …


12 is the smallest number in both lists so 12 is the LCD of 4 and 12.

E.g.: Find the LCD of 7 11

and 24 30

.

We need to find the LCM of 24 and 30.


The multiples of 30 are 30, 60, 90, 120, 150, 180, 210, …

120 is the smallest number in both lists so 120 is the LCD of 7 11

and 24 30

.

E.g.: Find the LCD of 11 3 2

, and 16 20 5

.

We need to find the LCM of 16, 20 and 5.



The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, …

23

80 is the smallest number in both lists so 80 is the LCD of 11 3 2

, and 16 20 5

.

Do #’s 13, 15, 17, 19, 23, 27, 29, 37, 39, p. 166 text in your PB.

Finding Equivalent Fractions

We will create equivalent fractions by taking a given fraction and multiplying it by 1. However, the 1

will usually be written as a whole fraction. For example, the 1 may be written as 2 3 4 5

or or or 2 3 4 5

E.g.: Write three fractions that are equivalent to 1

2.

1 2 1 2 2 1 3 1 3 3 1 4 1 4 4 or or

2 2 2 2 4 2 3 2 3 6 2 4 2 4 8

E.g.: Write three fractions that are equivalent to 5

8.

5 6 5 6 30 5 7 5 7 35 5 8 5 8 40 or or

8 6 8 6 48 8 7 8 7 56 8 8 8 8 64

E.g.: Make an equivalent fraction with the given denominator.

1 ?

5 35

1 7 1 7 7

5 7 5 7 35

So 1

5 is equivalent to

7

35.

E.g.: Make an equivalent fraction with the given denominator.

3 ?

25 125

3 5 3 5 15

25 5 25 5 125

So 3

25 is equivalent to

15

125.


24

E.g.: Create equivalent fractions for the given LCD. 9 3

LCD 20, Fractions are and 10 4

9 2 9 2 18 3 5 3 5 15

10 2 10 2 20 4 5 4 5 20

E.g.: Create equivalent fractions for the given LCD. 9 3

LCD 120, Fractions are and 5 8

9 24 9 24 216 3 15 3 15 45

5 24 5 24 120 8 15 8 15 120


E.g.: Create equivalent fractions with the LCD of 7 35

and 9 54

.

The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 73, …


The LCD is 54.

7 6 7 6 42 35 1 35 1 35

9 6 9 6 54 54 1 54 1 54

E.g.: Create equivalent fractions with the LCD of 3 5 13

, and 8 14 16

.

The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, …

The multiples of 14 are 14, 28, 42, 56, 70, 84, 98, 112, 126…


The LCD is 112.

3 14 3 14 42 5 8 5 8 40 13 7 13 7 91

8 14 8 14 112 14 8 14 8 112 16 7 16 7 112


25

Module 2, Section 7: Adding & Subtracting Fractions

Outcomes:

Add & subtract fractions with the same denominator.

Add & subtract fractions with the different denominators.

Adding Fractions with the Same Denominator

E.g.: Simplify 1 2

5 5 .

1 2

5 5 can be represented by the picture below. Since the shaded portions of each circle are the same

shape and size, we can write:

1 2 3

5 5 5

E.g.: Simplify 3 2

8 8 .

3 2



3 2 5

8 8 8


26

Subtracting Fractions with the Same Denominator

The method for subtracting is very similar to that for adding.

E.g.: Simplify 4 1

5 5 .

4 1



4 1 3

5 5 5

E.g.: Simplify 3 2

4 4 .

3 2



3 2 1

4 4 4

Do #’s 5,7, p. 174 text in your PB.

27

Adding Fractions with Different Denominators

E.g.: Simplify 1 1

4 2 .

1 1

4 2 can be represented by the picture below. However, this time the pieces are not the same shape and

size so we cannot add the fractions as they are now.

We have to subdivide the second diagram so the shaded pieces become the same size and the same

shape.

So the equation 1 1

4 2 becomes

1 2

4 4 which has the same denominator.

So

1 1 1 2 3

4 2 4 4 4

E.g.: Simplify 2 7

5 8 .

The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 …


The LCD is 40.

2 8 16 7 5 35

5 8 40 8 5 40

So 2 7 16 35 51 11

15 8 40 40 40 40

28

E.g.: Simplify 5 7

8 12 .



The LCD is 24.

5 3 15 7 2 14

8 3 24 12 2 24

So 5 7 15 14 29 5

18 12 24 24 24 24


8 6 20 .

The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120 …

The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 64, 68, 72, 78, 84, 90, 96, 102, 108, 114,

120…

The multiples of 20 are 20, 40, 60, 80, 100, 120, 140,…

The LCD is 120.

7 15 105 5 20 100 7 6 42

8 15 120 6 20 120 20 6 120

So 7 5 7 105 100 42 247 7

28 6 20 120 120 120 120 120

Do #’s 9, 11, 13, 17, 19, 21, 23, 43, 47, pp. 174-175 text in your PB.

29

Subtracting Fractions with Different Denominators

E.g.: Simplify 7 2

8 5 .


The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 …

The LCD is 40.

7 5 35 2 8 16

8 5 40 5 8 40

So 7 2 35 16 19

8 5 40 40 40

E.g.: Simplify 5 7

8 12 .



The LCD is 24.

5 3 15 7 2 14

8 3 24 12 2 24

So 5 7 15 14 1

8 12 24 24 24

E.g.: Solve 1 7

8 16x for x.

First write each fraction with a common denominator (Find the LCD).



The LCD is 16.

1 2 2

8 2 16

So 1 7

8 16x can be written as

2 7

16 16x .

30

Since 5 2 7

16 16 16 ,

5

16x



31

Module 2, Section 8: Adding & Subtracting Mixed Numbers & the Order of Operations

Outcomes:

Add & subtract fractions mixed numbers.

Simplify expressions containing fractions using the order of operations.

Adding Mixed Numbers

E.g.: Simplify 5 3

2 416 16

.

Since each fraction has the same denominator, we can add the whole numbers and add the fractions.

52

16

34

16

8 16 6

16 2

E.g.: Simplify 7 3

3 48 8 .

Since each fraction has the same denominator, we can add the whole numbers and add the fractions.

73

8

34

8

10 8 2 2 2 2 17 7 7 1 8 8 8

8 8 8 8 8 8 4


32

E.g.: Simplify 1 1

20 34 8 .

Since the fractions have different denominators, we must find a common denominator for the fractions

and then we can add the whole numbers and add the fractions. The LCD is 8.

1 2 2

4 2 8

220

8

13

8

323

8

E.g.: Simplify 2 3

6 75 4 .



2 4 8 3 5 15;

5 4 20 4 5 20

86

20

157

20

23 20 3 3 3 313 13 13 1 14 14

20 20 20 20 20 20

E.g.: Simplify 3

8 24

.

We can add the whole numbers and add the fractions.

8

32

4

310

4


33

Subtracting Mixed Numbers

E.g.: Simplify 5 3

7 416 16

.

Since each fraction has the same denominator, we can subtract the whole numbers and subtract the

fractions.

57

16

34

16

2 13 3

16 8

E.g.: Simplify 3 7

6 38 8 .

Since each fraction has the same denominator, it seems that we can subtract the whole numbers and

subtract the fractions. However, we can’t take 7

8 away from

3

8 so we have to borrow 1 from the 6 and

add it to the 3

8.

3 3 3 8 3 116 6 5 1 5 5

8 8 8 8 8 8

115

8

74

8

4 11 1

8 2

34

E.g.: Simplify 4

8 35

.

First rewrite 8 with a whole number part and a fractional part.

5 58 7 1 7 7

5 5

57

5

43

5

14

5


E.g.: Simplify 1 1

20 34 8 .


and then we can subtract the whole numbers and subtract the fractions. The LCD is 8.

1 2 2

4 2 8

220

8

13

8

117

8

E.g.: Simplify 2 3

9 65 4 .



2 4 8 3 5 15;

5 4 20 4 5 20

So 2 3

9 65 4 is the same as

8 159 6

20 20 .

35

We can’t take 15

20 away from

8

20 so we have to borrow from the 9.

So 8 8 20 8 28 28

9 8 1 8 8 820 20 20 20 20 20

288

20

156

20

132

20

Do #’s 15, 17, 23, 33, 35, 37, 39, p. 182 text in your PB.


Order of Operations with Fractions

The order of operations is applied the same way to fractions as it is to

whole numbers.


5 15 13

3 1 10 3 1 10 3 10 3 39 10

5 15 13 5 15 13 5 195 5 39 195

117 10 107

195 195 195

E.g.: Simplify 1 5 5 7

7 6 13 6

1 5 5 7 5 5 7 5 5 6 5 30 5 60 65 231

7 6 3 6 42 3 6 42 3 7 42 21 42 42 42 42

E.g.: Simplify

24 3 1

3 5 10

2 2 2

4 3 1 4 6 1 4 5 4 25 4 100 400 25 15 5

3 5 10 3 10 10 3 10 3 100 3 25 75 75 3


36

Module 2, Section 9: Solving Problems Involving Fractions

Outcome:

Solve word problems involving fractions.

37

Do #’s 1, 3, 5, 9, 15, pp. 193-194 text in your PB.

AM1101 Math Essentials Class Notes Module Two Module 2 ...

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