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MAT001 – MODULES 3 & 4

FRACTIONS & MIXED NUMBERS

1 of 18

P. 1 of 105

Section 3.1/4.1

Understanding Fractions

Modules 3 & 4Fractions &

Mixed Numbers

P. 2 of 105

Representing Part of a Whole

A fraction represents parts of a whole.

The whole is the circle on the left.

14

numerator

denominator

The fraction represents the

shaded part of the circle. 1 out of

4 pieces is shaded. is read “one-

fourth.”

1

4

1

4

P. 3 of 105


The fraction represents the

portion of the box that is shaded.

We can also think of a fraction as a division problem.

55 8

8 and

55 8

8

58

P. 4 of 105

CQ-3/4-01. Write a fraction that represents

the portion of the box that is shaded.

1.

2.

3.

4.

10

7

11

6

7

4

11

7

P. 5 of 105


Example:

Use a fraction to describe the situation.

1. 11 out of 15 of the math students received an

“A” for the semester.

1115

2. Patty needed three-fourths of a yard of material

to make the doll.34

P. 6 of 105

CQ-3/4-02. Use a fraction to describe

the situation.

3 out of 25 students don’t have the

computer at home.

1.

2.

3.

4.

25

22

3

25

25

3

28

25



2 of 18

P. 7 of 105

Equivalent Fractions

and

Simplifying Fractions

Section 3.2/4.2

P. 8 of 105

Prime Numbers

A prime number is a whole number greater than 1

that cannot be evenly divided except by 1 and itself.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

The first 10 prime numbers

A composite number is a whole number greater

than 1 that can be evenly divided by whole numbers

other than 1 and itself.

24 = 2 12 24 = 3 8 24 = 4 6

P. 9 of 105

Divisibility Tests

1. A number is divisible by 2 if the last digit is 0, 2, 4,

6, or 8.

2. A number is divisible by 3 if the sum of the digits

is divisible by 3.

3. A number is divisible by 5 if the last digit is 0 or 5.

Example:

The number 450 is divisible by 2. (It ends in 0.)

The number 450 is divisible by 3. (4 + 5 + 0 = 9 and 9 is

divisible by 3.)

The number 450 is divisible by 5. (It ends in 0.)

P. 10 of 105

Prime Factorization

Example:

Write the number 24 as a product of primes.

Write 24 as the product of any two factors.24 = 4 6

If the factors are not prime, they must

be factored.2 2 2 3

When all of the factors are prime, the

number has been completely factored.24 = 2 2 2 3

Instead of writing 2 2 2 3, we can also write 23 3.

P. 11 of 105

The Fundamental Theorem of Arithmetic

24 = 4 6

2 2 2 3

The order of the prime factors is not important

because multiplication is commutative.

The Fundamental Theorem of Arithmetic

Every composite number can be written in exactly

one way as a product of primes.

24 = 3 8

2 4

2 2

24 = 2 2 2 3 24 = 3 2 2 2

P. 12 of 105

CQ-3/4-03. Write 54 as a product of primes,

i.e., write the prime factorization of 54.

1.

2.

3.

4.

96

932

3332

272



3 of 18

P. 13 of 105

Equivalent Fractions

Equivalent fractions can be written in more than one

way. The value of the fractions is the same.

is shaded.36

is shaded.12

Equivalent fractions

P. 14 of 105

Equal Fractions

If two fractions are equivalent, their diagonal products or

cross products will be equal.

1 8 8

2 4 8

Equality Test for Fractions

For any two fractions where a, b, and c are whole

numbers and b 0, d 0, if then a d = b c.,a cb d

1 4?

2 8=

The products are equal,

1 4therefore .

2 8

P. 15 of 105

CQ-3/4-04. Are the following fractions

equivalent?

1. Yes

2. No

63

32

9

4and

P. 16 of 105

Simplest Form (Lowest Term)

1 3 32 3 6

Any nonzero number divided by itself is equal to 1.

A fraction is in simplest form when the numerator

and denominator have no common factors (other

than 1).

2 431

2 43bb

1 4 42 4 8

1 50 502 50 100

are equivalent fractions.1 3 4 50

, , , and 2 6 8 100

We can multiply both the numerator and the

denominator by a non-zero number.

P. 17 of 105

Common Factors

When a fraction is not in simplest form, it can be reduced.

25 5 540 5 8

5 is the common factor.

To reduce a fraction, find a common factor in the

numerator and the denominator and divide it out.

25 540 8

A fraction is called simplified, reduced, or in lowest

terms if the numerator and the denominator only

have 1 as a common factor.

P. 18 of 105

CQ-3/4-05. Simplify to lowest

terms.

1.

2.

3.

4.

8

1

30

4

60

8

15

2

300

40



4 of 18

P. 19 of 105


terms.

1.

2.

3.

4.

49

21

21

9

11

7

7

3

147

63

P. 20 of 105

Converting Between

Improper Fractions

and

Mixed Numbers

Section 3.3/4.3

P. 21 of 105

Proper and Improper Fractions

If the value of a fraction is less than 1 (the numerator is

less than the denominator), the fraction is proper.

If the value of a fraction is greater than or equal to 1 (the

numerator is greater than or equal to the denominator),

the fraction is improper.

3 5 1 27, , ,

4 9 2 40

9 5 18 27, , ,

5 2 13 27

The numerator is

greater than or equal

to the denominator.

P. 22 of 105

CQ-3/4-07. What is the name of a fraction

with a numerator that is greater than or

equal to its denominator?

1. Improper

2. Mixed

3. Proper

4. Unmixed

P. 23 of 105

CQ-3/4-08. The fraction is best

described to by which word.18

6

1. Proper

2. Improper

3. Simplified

4. Factored

P. 24 of 105

Mixed Numbers

A mixed number is the sum of

a whole number greater than zero and a proper fraction.

3 5 12 , 1 , 22

4 9 2

32

4



5 of 18

P. 25 of 105

Mixed Numbers to Improper Fractions

Changing a Mixed Number to an Improper Fraction

1. Multiply the whole number by the denominator of the fraction.

2. Add the numerator of the fraction to the product found in step 1.

3. Write the sum found in step 2 over the denominator of the

fraction.

Example: Change into an improper fraction.2

43

4 3 23

Multiply the whole number

by the denominator.

12 23

143

Add the numerator

to the product.

Write the sum over

the denominator.

P. 26 of 105

CQ-3/4-09. Write as an improper

fraction.

1.

2.

3.

4.

3

10

3

13

3

15

3

17

3

25

P. 27 of 105

Improper Fractions to Mixed Numbers

Changing an Improper Fraction to a Mixed Number

1. Divide the numerator by the denominator.

2. Write the quotient followed by the fraction with the

remainder over the denominator.remainder

quotientdenominator

Example: Change into a mixed number.214

4 21

5 R1 21 15

4 4

denominator

quotient

remainder

P. 28 of 105

CQ-3/4-10. Write as a mixed

number.

1.

2.

3.

4.

7

47

38

7

36

7

342

7

45

P. 29 of 105

Reducing a Mixed Number

Example: Reduce the improper fraction 125

.15

125 5 5 515 3 5

common factors

253

1

1

Example: Reduce the mixed number fraction 11

4 .66

11 11 14 4

66 11 6

1

1

14

6

3

18

P. 30 of 105


terms. Write the result as a mixed

number if possible.

1.

2.

3.

4.

22

26

11

21

11

13

44

81

44

52



6 of 18

P. 31 of 105

Converting

Fractions to Decimals

&

Decimals to Fractions

Section 3.4/4.4

P. 32 of 105

Equivalent Fractions and Decimals

A number can be expressed in two equivalent forms: as a

fraction and as a decimal.

Fraction

three and one-fourth

13

4

Decimal

three and twenty-five hundredths

3.25

Same quantity,difference appearance

10.5

2

10.25

4

10.2

5

10.1

10

Common equivalent fractions and decimals

P. 33 of 105

Fractions & Equivalent Decimals

Note the relationship between fractions and their equivalent numbers’ decimal form.

30.3

10

490.049

1000

one zero

one decimal place

three zeros

three decimal places

P. 34 of 105

Changing from Fractions to Decimals

Example:

1. Write the fraction as a decimal.61

100

610.61

100

three zeros three decimal places

2. Write the fraction as a decimal.27

51000

275 5.027

1000

P. 35 of 105

Changing from Decimals to Fractions

Example:

1. Write the decimal 0.371 as a fraction.

3710.371

1000

Simplify when possible.

2. Write the decimal 4.0038 as a fraction.38

4.0038 410000

194

5000

P. 36 of 105

CQ-3/4-12. Write 0.155 in fractional

form and reduce, if possible.

2000

31.

2.

3.

4.

200

31

2000

31

20

3



7 of 18

P. 37 of 105

Converting a Fraction to a Decimal

Converting a Fraction to an Equivalent Decimal

Divide the denominator into the numerator until

a) the remainder becomes zero, or

b) the remainder repeats itself, or

c) the desire number of decimal places is achieved.

10.5

2

10.25

4

10.2

5

10.1

10

Common equivalent fractions and decimals

P. 38 of 105

Terminating and Repeating Decimals

30.375

8

10.333 0. 3

3

130.59090 0.5 90

22

10.0625

16

Terminating decimals(The remainder is zero when converting

the fraction into a decimal.)

Repeating decimals(When converting, the remainder is a

digit or group of digits that repeats.)

repeating digit repeating group of digits

P. 39 of 105

Converting a Fraction to a Decimal

Example:

Write as an equivalent decimal. 5

18

5 18 5

18

repeating

remainders

0.277

.00036140126

5 0.27

18

140

P. 40 of 105

CQ-3/4-13. Write as a decimal.

1. 1.

2. 2.

3. 3.

4. 4.

8

5

6.0

6.1

625.0

5125.0

P. 41 of 105

CQ-3/4-14. Write as a decimal.

1. 1.

2. 2.

3. 3.

4. 4.

38.0

2.1

6.5

1.1

6

5

P. 42 of 105

Ordering Fractions and Decimals

Example:

Fill in the blank with one of the symbols <, =, or >.

3 ____ 0.7

4

Change the fraction into a decimal

for easier comparison.

0.75 ___ 0.7>



8 of 18

P. 43 of 105

CQ-3/4-15. Select the correct statement.

1.1.

2.2.

3.3.

4. 4.

14.37

22

14.37

22

14.37

22

37

22

P. 44 of 105

Multiplying

Fractions

and

Mixed Numbers

Section 3.5/4.5

P. 45 of 105

Multiplying Fractions

Multiplication of fractions is used when we want to

take a fractional part of something.

1 5 52 9 18

12

59

1 5 of

2 9yields 5 out of 18

squares.

P. 46 of 105

Multiplying Fractions

To multiply two fractions, we

multiply the numerators and multiply the denominators.

3 2 67 5 35

In general, for all positive whole numbers a, b, c, and d,

a c a cb d b d

(when b and d are not 0).

3557

623

P. 47 of 105

Multiplying Proper or Improper Fractions

Example: Multiply12 3

.17 24

12 317 24

Simplify the

fraction.

12 317 24

12 317 24

36408

3

34

To make multiplying easier, the fractions may be

simplified before multiplying.

3 4 317 2 3 4

3

17 2

3

34

1 1

1 1

3 4 317 2 3 4

P. 48 of 105

CQ-3/4-16. Multiply:

1.

2.

3.

4.

56

15

35

24

15

8

24

35

8

5

7

3



9 of 18

P. 49 of 105

CQ-3/4-17. Multiply:

1.

2.

3.

4.

429

143

3

1

507

121

39

13

11

13

39

11

P. 50 of 105

CQ-3/4-18. Simplify:

1.

2.

3.

4.

9

7

72

52

14

10

49

25

2

7

5

P. 51 of 105

Multiplying Mixed Numbers

Example: Multiply4 4

2 1 .5 7

To multiply mixed numbers,

first change each mixed number into an

improper fraction.

14 115 7

22 2 or 4

5 5

2

1

4 42 1

5 7

=

P. 52 of 105


1.

2.

3.

4.

5

218

3

133

8

59

24

5

33

3

26

P. 53 of 105


1.

2.

3.

4.

42

19

9

15

3

119

42

118

9

13

14

36

P. 54 of 105

Dividing

Fractions

and

Mixed Numbers

Section 3.6/4.6



10 of 18

P. 55 of 105

Dividing Fractions

A cup of lemonade that is full must be divided

into -cup servings. How many cups of lemonade

will there be?34

34

14

How many

1 3's are in ?

4 4

14

14

14

14

three1 3

There are 's in .4 4

There will be three -cup servings in the cup.14

P. 56 of 105

Dividing Proper or Improper Fractions

When fractions are divided, we invert the second

fraction and multiply. [Multiplied by the reciprocal.]

3 14 4

3 44 1

1

1

Rules for Division of Fractions

a c a db d b c

(when b, c, and d are not 0).

To divide two fractions, we invert the second fraction and multiply.

31

3

P. 57 of 105

CQ-3/4-21. What do we call two

numbers which have a product of one?

1. Opposites

2. Factors

3. Quotients

4. Reciprocals

P. 58 of 105

CQ-3/4-22. Find the reciprocal of 8.

1.

2.

3.

4.

8

1

8.0

8

0

P. 59 of 105


1.

2.

3.

4.

25

6

8

3

6

14

3

23

10

3

5

4

P. 60 of 105


1.

2.

3.

4.

35

36

7

20

35

15

20

7

7

12

10

6



11 of 18

P. 61 of 105

Dividing Mixed Numbers

Example: Divide4 7

2 1 .5 10

To divide mixed numbers,

first change each mixed number into an improper

fraction.

4 7 14 172 1

5 10 5 10

14 105 17

2

1

28 11 or 1

17 17

P. 62 of 105


1.

2.

3.

4.

3

110

49

441

7

18

7

12

3

12

7

34

P. 63 of 105


1.

2.

3.

4.

8

4

19

36

2

140

4

1218

P. 64 of 105


1.

2.

3.

4.

8

12

4

32

2

12

8

112

64

312

P. 65 of 105

The Least Common

Denominator

and

Creating Equivalent

Fractions

Section 3.7/4.7

P. 66 of 105

A multiples of a number are the products of that

number and the numbers 1, 2, 3, 4, 5, …

The multiples of 3 are 3, 6, 9, 12, 15, …

3 13 2

3 3

The least common multiple, or LCM, of two

natural numbers is the smallest number that is

a multiple of both.

Least Common Multiple (LCM)



12 of 18

P. 67 of 105

Least Common Multiple (LCM)

Example: Find the LCM of 4 and 6.

The multiples of 4 are 4, 8, 12, 16, 20, 24 …

The multiples of 6 are 6, 12, 18, 24, 30, 36 …

The first number that appears on both lists is the

LCM.

12 is the least common multiple (LCM) of 4 and 6.

P. 68 of 105

Least Common Denominator (LCD)

A least common denominator (LCD) of two or more

fractions is the smallest number that can be divided

evenly by each of the fractions’ denominators.

7 3 and

12 4

Since 4 can be divided into 12, the LCD of

is 12.7 3

and 12 4

P. 69 of 105

Least Common Denominator (LCD)

3 4 and .

4 5

20 is also the smallest number that can be divided

by 4 and 5 without a remainder.

Example: Find the LCD for

4 5 = 20

The LCD of is 20.3 4

and 4 5

P. 70 of 105

Finding the Least Common Denominator(A more systematic/longer way)

Three-Step Procedure for Finding the LCD1. Write each denominator as the product of prime factors.

2. List all the prime factors that appear in either product.

3. Form a product of those prime factors, using each factor the greatest number of times it appears in any one denominator.

Example: Find the LCD for5 7

and .12 30

2 2 3 2 3 5Product of primes

Prime factors in either product: 2 2 3 5

The LCD is 60.

P. 71 of 105

CQ-3/4-28. Find the LCD for

1.

2.

3.

4.

36

56

32

72

.18

11 and

8

3

P. 72 of 105

Building Fraction Property

Building Fraction Property

For whole numbers a, b, and c where b 0, c 0,

1 .a a a c a cb b b c b c

Example:

Build to an equivalent fraction with a LCD of 20.34

3 ?4 20

cc

3 5 154 5 20

3 15and are

4 20

equivalent fractions.



13 of 18

P. 73 of 105

Creating Equivalent Fractions

Fractions with unlike denominators cannot be added.

To change the denominators and make them the same,

1) find the LCD and

2) build up the addends into equivalent fractions that

have the LCD as the denominator.

3 4 +

4 5The LCD is 20.

3 ?4 20

cc

4 ?5 20

cc

55 , 1

5c

44 , 1

4c

The building

fraction property

P. 74 of 105

CQ-3/4-29. Find the equivalent fraction

for the given fraction to have 48 as its

new denominator.

1.

2.

3.

4.

8

3

48

9

48

15

8

18

48

18

P. 75 of 105

Adding and Subtracting

Fractions

Section 3.8/4.8

P. 76 of 105

Fractions with Common Denominators

Fractions must have common denominators before

they can be added or subtracted.

24

+ =

14

34

2 1 34 4 4

P. 77 of 105


1.

2.

3.

4.

9

8

9

4

18

12

3

4

9

12

3

11

P. 78 of 105

Fractions with Different Denominators

If fractions have different denominators, find the

LCD and build up each fraction so that its

denominator is the LCD.

3 18 6

Example:

Add .

LCD = 24

3 3 98 3 24

1 4 46 4 24

9 4 1324 24 24



14 of 18

P. 79 of 105


1.

2.

3.

4.

10

5

21

19

2

1

21

4

7

4

3

1

P. 80 of 105


1.

2.

3.

4. 1

7

34

7

12

7

31

7

34

P. 81 of 105

Fractions with Different Denominators

5 712 30

Example:

Subtract .

LCD = 60

5 5 2512 5 60

7 2 1430 2 60

25 14 1160 60 60

P. 82 of 105


1.

2.

3.

4.

1

6

11

36

5

36

57

12

7

18

13

P. 83 of 105


1.

2.

3.

4.

40

5

8

5

40

19

3

5

5

2

8

7

P. 84 of 105

Comparing Two Fractions

When comparing two fractions, find the LCD

and build up each fraction so that its denominator

is the LCD. Then, the fraction with the larger

numerator is the larger fraction.

Example:

Compare .6

5

8

7 and

24

20

4

4

6

5

6

5

24

21

3

3

8

7

8

7

6

5

8

7,,

24

20

24

21 soBeacuse



15 of 18

P. 85 of 105

CQ-3/4-35. Compare

1.

2.

3.

4.

.5

4

7

4 and

5

4

7

4

5

4

7

4

5

4

7

4

35

16

5

4

7

4

P. 86 of 105

Adding and Subtracting

Mixed Numbers

&

the Order of Operations

Section 3.9/4.9

P. 87 of 105

Adding Mixed Numbers

When adding mixed numbers, it is best to add the

fractions together and then add the whole

numbers together.

1 29 2 .

6 3

Example:

Add

LCD = 61

9 6

42

6

2 2 42 2

3 2 6

19

6

22

3

5 6

11

Add the

fractions

first.

Add the whole

numbers.

P. 88 of 105


1.

2.

3.

4.

21

135

10

35

21

26

10

36

3

12

7

23

P. 89 of 105


1.

2.

3.

4.

36

5121

5

3120

216

65120

5

3121

18

1382

12

538

P. 90 of 105

Subtracting Mixed Numbers

Subtracting mixed numbers is like adding mixed

numbers.

2 1 9 7 .

3 8

Example:

Add

LCD = 24

169

24

37

24

1 3 37 7

8 3 24

29

3

17

8

13 24

2

Subtract the

fractions

first.

Subtract the whole

numbers.

2 8 169 9

3 8 24



16 of 18

P. 91 of 105

Subtracting Mixed Numbers

1 7 5 3 .

12 18

Example:

Add

LCD = 36

394

36

143

36

7 2 143 3

18 2 36

35

36

143

36

251

36

1 3 35 5

12 3 36

We borrow 1 from 5 to obtain

3 3 39 395 4 1 4 4 .

36 36 36 36

3 14We cannot subtract

36 36

so we need to borrow.

P. 92 of 105


1.

2.

3.

4.

3

26

8

16

8

15

4

16

2

16

8

512

P. 93 of 105


1.

2.

3.

4.

4

111

4

110

4

311

4

310

4

1718

P. 94 of 105


1.

2.

3.

4.

15

1126

15

825

20

1126

10

126

15

1387

5

2113

P. 95 of 105

Order of Operations

Order of Operations

1. Perform operations inside any parentheses.

2. Simplify any expressions with exponents.

3. Multiply or divide from left to right.

4. Add or subtract from left to right.

Do first

Do last

Example: Evaluate2

5 2.

8 5

5 48 25

1 12 5

1

10

Exponents

Multiplication1 1

2 5

P. 96 of 105

Order of Operations

Example: Evaluate3 1 5

.4 4 3

3 1 34 4 5

3 1 54 4 3

3 34 20

Express division

as multiplication.

Multiply.

15 320 20

Rewrite fractions

using the LCD.

18 920 10

Add and simplify.



17 of 18

P. 97 of 105


1.

2.

3.

4.

135

41

135

121

135

49

72

11

2

3

1

3

8

5

3

P. 98 of 105


1.

2.

3.

4.

85

93

5

41

85

462

85

24

8

12

5

36

P. 99 of 105

Solving Applied Problems

Involving Fractions

Section 3.10/4.10

P. 100 of 105

Problem Solving Steps1. Understand the problem.

a) Read the problem carefully.

2. Draw a picture if it is helpful.

a) Fill in the Mathematics Blueprint so that you have the facts and a method of proceeding in this situation.

3. Solve and state the answer.

a) Perform the calculations.

b) State the answer and include the unit of measure.

4. Check.

a) Estimate the answer.

b) Compare the exact answer with the estimate to see if

your answer is reasonable.

P. 101 of 105

Mathematics Blueprint

Mathematics Blueprint for Problem Solving

Key Points to Remember

How Do I Proceed?

What Am I Asked to Do?

Gather the Facts

The Mathematical Blueprint is simply a sheet of paper with four columns. Each column tells you something to do.

P. 102 of 105

All fractions must

use the LCD = 24.


Example #1:

A carpenter is using an 8-foot length of wood for a frame. He needs to cut a

notch in the wood that is feet from one end and feet from the other

end. How long does the notch need to be?

74

8

213



How Do I Proceed?


Gather the Facts

The board is 8 ft.

long. There is a

cut from each

side of the board.

Add the fractions

and subtract the

total from 8.

Find the length of

the notch.

Example continues.



18 of 18

P. 103 of 105


Example #1:

A carpenter is using an 8-foot length of wood for a frame. He needs to cut a

notch in the wood that is feet from one end and feet from the other

end. How long does the notch need to be?

74

8

213

2 71 43 8

16 211 424 24

37 135 6

24 24

This is the part of the

board that is not

notched.

13 24 138 6 7 6

24 24 24

11124

The length of the

11notch is 1 feet.

24

P. 104 of 105

“of” means to

multiply.




How Do I Proceed?


Gather the Facts

The total income

is $450.There are

three deductions

to be subtracted

from the 450.

Multiply each

fraction by 450,

then subtract the

three products

from 450.

Find how much

money Patty has

after the

deductions.

Example continues.

Example #2:

Patty earns $450 per week. She has of her income withheld for federal

taxes, of her income withheld for state taxes, and of her income

withheld for medical coverage. How much money per week is left for Patty

after those three deductions?

115

15 1

25

P. 105 of 105


1450

5

Example #2:

Patty earns $450 per week. She has of her income withheld for federal

taxes, of her income withheld for state taxes, and of her income

withheld for medical coverage. How much money per week is left for Patty

after those three deductions?

115

15 1

25

45090

5

1450

15

45030

15

1450

25

45018

25

The total of the deductions is 138.

450 – 138 = 312

Patty has $312 left

after deductions.`

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