CHAPTER 3: FRACTIONS - CCBC Faculty Webfaculty.ccbcmd.edu/~lwalte19/Math081C3S1Text.pdf137 CHAPTER 3: FRACTIONS CHAPTER 3 CONTENTS 3.1 Introduction to Fractions 3.2 Multiplication

137

CHAPTER 3: FRACTIONS

CHAPTER 3 CONTENTS

3.1 Introduction to Fractions

3.2 Multiplication of Fractions and of Mixed Numbers

3.3 Division of Fractions and of Mixed Numbers

3.4 Addition of Fractions and of Mixed Numbers

3.5 Subtraction of Fractions and of Mixed Numbers

3.6 Order of Operations

3.7 U.S. Measurement Conversions

3.8 Application Problems

Image from Microsoft Office Clip Art

CCBC Math 081 Introduction to Fractions Section 3.1

Third Edition 23 pages

138

3.1 Introduction to Fractions

Before we begin, we will review some ideas that will be used in our studies of fractions.

Prime Number: A prime number is a number greater than 1 that is only divisible by 1 and

itself.

The number 3 is a prime number; it is only divisible by the number 3 and 1. The number 4 is not

a prime number because it is divisible by 1, 2 and 4.

Prime Factorization: The prime factorization of a number is the product of all of the prime

numbers that equals the number.

Example 1: Write the prime factorization of the numbers 32 and 40.

One way to determine the prime factorization of a number is to create a factor tree. We will

create factor trees for both 32 and 40 below. To create a factor tree, split each number into two

factors that multiply to equal the number above. Continue to split the factors into factors until

each factor is prime. A prime number is only divisible by 1 and itself. The prime factors in

these factor trees have a box around them. The product of all of the prime fractors at the bottom

of the factor tree is the prime factorization.

32 4 8

2 2 2 4

2 2

40 4 10

2 2 2 5

The prime factorization for 32 is: The prime factorization for 40 is:

32 2 2 2 2 2 40 2 2 2 5

Practice 1: Write the prime factorization of the numbers 36 and 24.

Answer: 36 2 2 3 3 24 2 2 2 3

Watch it: http://youtu.be/aXxzz99urEU

Greatest Common Factor: The Greatest Common Factor (GCF) of two numbers is the greatest

number that divides both numbers evenly. One way to determine the GCF is to use the prime

factorization. The product of all of the prime factors that are common in both numbers will give

the GCF.

http://youtu.be/aXxzz99urEU



139

Example 2: Determine the Greatest Common Factor of 32 and 40. Denoted as GCF(32, 40).

The product of the prime factors the two numbers have in common equals the GCF.

GCF(32,40) = 2 2 2 8

Therefore, 8 is the largest number that divides 32 and 40.

Practice 2: Determine the Greatest Common Factor of 36 and 24. Denoted as GCF(36, 24).

Answer: 12

Watch it: http://youtu.be/102S-l4-9pQ

Least Common Multiple: The Least Common Multiple (LCM) of two numbers is the smallest

number that both numbers divide evenly. One way to determine the LCM of two numbers is to

multiply the two numbers together and divide that product by the GCF.

Example 3: Determine the Least Common Multiple of the numbers 32 and 40. Denoted

LCM(32, 40).

LCM(32,40) 32 40 GCF(32,40)

32 40 8

1280 8

160

Therefore, 160 is the smallest number that both 32 and 40 divide evenly.

FORMULA FOR DETERMINING THE LEAST COMMON MULTIPLE

LCM( , ) GCF( , )a b a b a b

http://youtu.be/102S-l4-9pQ



140

Practice 3: Determine the Least Common Multiple of the numbers 36 and 24. Denoted

LCM(36, 24).

Answer: 72

Watch it: http://youtu.be/JBKzWeZvsBM

We will be using GCF, LCM, and prime factorization as we work with fractions. You may need

to refer back to this explanation as you work through this chapter.

Fraction: Fractions are used in mathematics for many purposes. Most commonly fractions

express the number of equal parts out of a whole (or out of a part), a division problem, a

remainder, or a ratio (a comparison of two numbers).

If a pizza is cut into eight equal pieces and you take three pieces you have the fraction3

8, which

is read “three eighths” or “3 divided by 8” or “3 out of 8.”

Equivalent Fractions: Changing the numerator and denominator of the fraction without

changing its value forms an equivalent fraction. There are two ways to accomplish this: by

expanding the fraction or by reducing the fraction.

PARTS OF A FRACTION

Fractions have two numbers that are separated by a fraction bar.

Numerator This number indicates how many equal parts we have.

Denominator This number indicates how many equal parts make up a whole.

http://youtu.be/JBKzWeZvsBM



141

You can multiply the numerator and denominator by the same number and produce an equivalent

fraction, and you can divide the numerator and denominator by the same number and produce an

equivalent fraction.

A fraction whose numerator and denominator are the same number has the value 1 unless the

number is 0. For instance, the fraction, 8

8, means there are 8 pieces that make a whole and we

have 8 pieces. Therefore, we have 1 whole.

Note: The fraction 0

0 is undefined.

Reducing Fractions to Lowest Terms

In some cases, fractions may be reducible (also called simplifying fractions); however, the

numerator and the denominator of the fraction must have a common factor, which will evenly

divide both numbers without a remainder. For example, if an egg carton that holds 12 eggs has

EQUIVALENT FRACTIONS

a a c

b b c

and

a a c

b b c

for b, 0c

A FRACTION THAT EQUALS 1

1a

a when a is not 0.



142

exactly 6 eggs in the container, you might say you have 6

12 of a carton of eggs. However, you

might just as easily remark that you have 1

2 of a carton of eggs. Both observations are correct

since 6

12 is equivalent to

1

2. In fact,

1

2 is

6

12 reduced to its lowest terms.

Let’s look at this more closely and determine how to do this mathematically so that we can

reduce other fractions that we are not as familiar with.

Example 4: Simplify 6

12

First, determine the prime factorization for the numbers 6 and 12. We will show how to do this

again by factor trees as follows.

6

2 3

12

4 3

2 2

The prime factorization for 6 is: The prime factorization for 12 is:

6 2 3 12 2 2 3

6 2 3

12 2 2 3

Rewrite the fraction using the prime factorization.

2

2

3

2

3 Divide out a 2 in the numerator and denominator.

3

2 3

1

1

Note: 2

12

3

2 3 Divide out a 3 in the numerator and denominator.

1

12

Note: 3

13

1

2

Note: If all factors in the numerator or denominator are divided out, the number that will

remains is a 1.



143

Practice 4: Simplify 8

28. Answer:

2

7

Watch it: http://youtu.be/QiIP5PwDaqM


12

Another way to simplify fractions is to divide the numerator and the denominator by the Greatest

Common Factor (GCF). First determine the GCF for 6 and 12 is 6. To simplify the fraction

divide both the numerator and denominator by 6.

6 6 6 1

12 12 6 2

Since you cannot simplify1

2any further, that is there are no common factors for 1 and 2 other

than 1, the fraction is reduced to its lowest terms.

Sometimes at first we do not choose the largest common factor to divide into both the numerator

and denominator. In that case, just keep reducing until you cannot reduce any further. This

problem could also be simplified in the following way.

6 6 2

12 12 2

First, divide both the numerator and denominator by 2.

3

6 Simplify.

3 3

6 3

Then, divide both the numerator and denominator by 3.

1

2

http://youtu.be/QiIP5PwDaqM



144


28. Answer:

2

7

Watch it: http://youtu.be/CM6a7Lc1q6A


42

We will first do this problem by using the prime factorization of the numerator and denominator.

28 2 2 7

42 2 3 7

Determine the prime factorization of the numerator and

denominator.

2

72

2 73 Divide out common factors in the numerator and

denominator.

2

3

We can also simplify this fraction by diving the numerator and denominator by common factors.

28 28 2 14

42 42 2 21

Since both 28 and 42 are even, they can be divided by 2.

without a remainder. So, divide 28 by 2 and 42 by 2

14 14 7 2

21 21 7 3

Both 14 and 21 can be divided by 7 without a remainder; so

the fraction is not fully simplified. Divide both numbers by

7.

28

42=

2

3 The fraction is now fully simplified.

Note that you could have gotten the answer in one step by dividing the original numerator and

denominator by 14 (the greatest common factor for 28 and 42).

28 28 14 2

42 42 14 3

Divide the numerator and denominator by the GCF.

Unless directions indicate otherwise, you should simplify a fraction to its lowest terms.

http://youtu.be/CM6a7Lc1q6A



145


36 Answer:

2

3

Watch it: http://youtu.be/Y5GixJsD-es

Expanding Fractions

At times, especially when adding or subtracting fractions, we need to rewrite fractions in an

equivalent form. To do this we will multiply the numerator and the denominator by the same

number (usually a positive integer).

1

2

2

1

2

2

2

4

3

1

2

3

3

6

4

1

2

4

4

8

Example 7: Rewrite 3

5 by multiplying both the numerator and the denominator by 4.

3 3 4 12

5 5 4 20

Practice 7: Rewrite 5

7 by multiplying both the numerator and the denominator by 3.

Watch it: http://youtu.be/_PDJ9HwYeh0 Answer: 15

21

EXPANDING FRACTIONS

a a c

b b c

for b, 0c

http://youtu.be/Y5GixJsD-es

http://youtu.be/_PDJ9HwYeh0



146


5 by multiplying both the numerator and denominator by 2.

3 3 2 6

5 5 2 10


7 by multiplying both the numerator and denominator by 4.

Watch it: http://youtu.be/tlLTgE6xylw Answer: 20

28

As long as you multiply both the numerator and the denominator by the same number (except 0),

you will obtain a fraction that is equal in value to the original fraction.

There are three types of numbers that involve fractions: proper fractions, improper fractions and

mixed numbers.

Proper fraction: A proper fraction is any fraction whose numerator is smaller than its

denominator. Some examples of proper fractions include:

1 16 4 2, , ,

3 17 5 11

Improper fraction: A fraction whose numerator is equal to or greater than the denominator is

an improper fraction. Some examples of improper fractions include:

4 17 7 53 7, , , ,

4 17 6 19 1

Mixed number: When a number consists of both a whole number and a fraction number it is

called a mixed number or a mixed fraction. The fraction part of a mixed number is a proper

fraction. Some examples of mixed numbers include:

3 1 5 35 , 1 , 2 , 23

4 7 6 19

PARTS OF A DIVISION STATEMENT

Dividend Divisor Quotient

8 2 4

Dividend

Quotient

Divisor

84

2

Quotient

Divisor Dividend

42 8

http://youtu.be/tlLTgE6xylw



147

Example 9: Rewrite the division problem 5 3 as a fraction.

5 is the dividend and 3 is the divisor, so as a fraction we have:

5

3

Practice 9: Rewrite the division problem 6 5 as a fraction.

Watch it: http://youtu.be/NcuJE2zLGeA Answer: 6

5

Example 10: Rewrite the fraction 5

3 as you would perform long division.

3 5

In previous discussions regarding division problems, when there was a remainder you simply

wrote the remainder as a whole number; however, we often use fractions to express that

remainder.

3

Whole number part

Denominator 5

3

Nume

2

rator

1

12

3 (See, it’s as easy as 1, 2, 3)

What does 5

3mean? Picture a whole candy bar that has 3 pieces. You have 5 of these pieces.

1 2

3

Practice 10: Rewrite the fraction 6

5 as you would perform long division.

Watch it: http://youtu.be/uH6f2zagJjI Answer: 5 6

http://youtu.be/NcuJE2zLGeA

http://youtu.be/uH6f2zagJjI



148


5 as a mixed number.

Divide the denominator into the numerator.

5

Whole number part

Denominator 17

15

Nume

2

rat r

3

o

32

5

Therefore, 17

5 53

2 .


8 as a mixed number. Answer:

52

8

Watch it: http://youtu.be/YIL3CA-FBtA


6 as a mixed number.


6

Whole number part

Denominator 27

24

Nume

3

rat r

4

o

43

6

This answer is not finished. Look at the fraction on the mixed number. It can be simplified by

dividing 3 into the 3 in the numerator and the 6 in the denominator. This gives the final answer:

33 3 1

4 4 46 6 23

Therefore, 27 1

46 2 .

Always simplify fractions (when possible) before finishing a problem.

http://youtu.be/YIL3CA-FBtA



149

The previous problem could have been simplified first and then the long division performed as

we will show below.

27 3 3 3

6 2 3

Write prime factorization of numerator and denominator.

33 3

2 3

Divide out a 3 in the numerator and denominator.

9

2 Simplify, and then complete long division using the

simplified fraction.


Whole number part

Denominator 9

8

Numerat

4

1

o

2

r

41

2

As we saw before, 27

6 24

1 .


8 as a mixed number. Answer:

16

2

Watch it: http://youtu.be/KYepikF1DvU

Integers as Fractions

Integers such as 0 or 7 or -12 or 38 sometimes need to be written as fractions. Since fractions

represent division and dividing by 1 does not change the value of the number, whole numbers

can be written as a fraction “over” 1. That is, divided by 1.

HOW TO WRITE AN INTEGER AS A FRACTION

For any integer, a , 1

aa .

http://youtu.be/KYepikF1DvU



150

Example 13: Write 0, 7, –12, and 38 as fractions.

0 =0

1 7 =

7

1

1212

1

38 =

38

1

Practice 13: Write 0, 3, –23, and 5 as fractions. Answer: 0 3 23 5

, , ,1 1 1 1

Watch it: http://youtu.be/g5-JTiVxRKs

Mixed Numbers

When you have a mixed number, you may need to rewrite it as an improper fraction in order to

work with it. In this section you will be introduced to an algorithm (procedure) for changing a

mixed number to an improper fraction. In section 3.5, you will learn why this procedure works.

To change a mixed number into an improper fraction, the numerator of the improper fraction will

be determined by multiplying the whole number times the denominator of the mixed number and

then adding this product to the numerator of the mixed number. The denominator of the

improper fraction will be the same as the mixed number.

Note:

b bA A

c c


8 as an improper fraction.

Multiply the denominator with the whole number, and add that product to the numerator. Then

place that resulting sum in the numerator, and keep the same denominator.

5 (8 1) 5 8 5 13

18 8 8 8

HOW TO CHANGE A MIXED NUMBER TO AN IMPROPER FRACTION

( )b A c bA

c c

( )b b A c bA A

c c c

http://youtu.be/g5-JTiVxRKs



151


34

as an improper fraction. Answer: 15

4

Watch it: http://youtu.be/UrXadoqagVM


9 as an improper fraction.

4 (9 3) 4 27 4 31

39 9 9 9


27


7

Watch it: http://youtu.be/YJrl5nlu-K8


25

as an improper fraction.

The number 3

25

means 3

25

. We will first determine the improper fraction in the

parentheses and then that number will be negative.

3 (5 2) 3 10 3 13 13

25 5 5 5 5

Note: Disregard the negative sign until the mixed number has been turned into an improper

fraction.


43


3

Watch it: http://youtu.be/oYFuIXgLLNI

Mixed numbers are sometimes needed when finding the mean and median value of a set of data.

Recall that to compute an arithmetic mean (an average), add all of the values and divide that sum

by the number of values. If the answer is not a whole number we can write the result as a mixed

number or fraction.

http://youtu.be/UrXadoqagVM

http://youtu.be/YJrl5nlu-K8

http://youtu.be/oYFuIXgLLNI



152

Example 17: Find the mean of 75, 45, 80, 94, 65. Add all of the values and divide by the number of values.

75 + 45 + 80 + 94 + 65

5=

359

5 This is the answer as an improper fraction.

359

35

09

5

7

5

1

4

5

359

571

4

Since 359 does not divide evenly by 5, we write

the mean as a mixed number by performing long

division.

Practice 17: Determine the mean of 85, 76, 55, 98. Answer: 1

782

Watch it: http://youtu.be/lxJkvDjfQ0o

Also recall that to find the median of a set of data values, arrange the values in numerical order

and locate the middle. If there are an odd number of data values, then the median is the number

in the middle of the list. If there is an even number of data values, then there are two data values

in the middle of the list: compute the mean (the average) of these two middle values to

determine the median. Again, if it is not a whole number we can write the result as a mixed

number or a fraction.

Example 18: Find the median of 20, –49, 31, –56, 67, –97.

We begin by listing the values in ascending order: –97, –56, –49, 20, 31, 67.

Note that there is an even number (six) of data values. The two middle data

values are -49 and 20. We find the mean (average) of these:

49 20 29

2 2

This is the answer as an improper fraction.

29

2

0

1

2

9

4

8

1

129

2 214

Since -29 does not divide evenly by 2, we can

write the result as a mixed number by performing

long division.

A negative divided by a positive is a negative. Therefore our answer is negative.

Practice 18: Determine the median of 3, -9, 2, -5, 6, -6. Answer: 1

12

Watch it: http://youtu.be/Rr3QHvccn8Y

http://youtu.be/lxJkvDjfQ0o

http://youtu.be/Rr3QHvccn8Y



153

Order of Fractions

When fractions have the same denominator they can be placed in order from least to greatest by

comparing their numerators. The following fractions all have the denominator 7. Since they

have a common denominator they are arranged from least to greatest according to their

numerators.

1

7,2

7,3

7,4

7,5

7,6

7,7

7

When fractions do not have a common denominator, we will need to first determine the common

denominator and then make all of the fractions into equivalent fractions with the common

denominator. The Least Common Denominator will be the Least Common Multiple of all the

denominators.

Example 19: Order the following fractions from least to greatest. 1 3 5 2 1

, , , ,3 6 6 3 6

To put fractions in order we must first write all of the fractions with a common denominator.

The Least Common Multiple of the denominators 3 and 6 is the smallest number they divide.

Multiples of 3: 3, 6, 9, 12…

Multiples of 6: 6, 12, 18…

The LCM(3, 6) = 6

Now, write an equivalent fractions for each fraction with the denominator of 6.

1 1 2

3 3

2

2 6

Multiply the numerator and denominator by 2.

3

6 This fraction already has a denominator of 6.

5


2 2 4

3 3

2

2 6


1


2 3 5 4 1, , , ,

6 6 6 6 6

These are the equivalent fraction for 1 3 5 2 1

, , , ,3 6 6 3 6

with a

common denominator of 6.

1 2 3 4 5, , , ,

6 6 6 6 6

Put the fractions in order from least to greatest from the

smallest numerator to the largest numerator.

1 1 3 2 5, , , ,

6 3 6 3 6 Then write the fractions in their original form



154

Practice 19: Order the following fractions from least to greatest. 1 3 5 2 1

, , , ,4 4 8 4 8

Watch it: http://youtu.be/AsBhn5LUIU4 Answer: 1 1 2 5 3

, , , ,8 4 4 8 4

Example 20: Determine the median of the following numbers. 1 3 3 2 5

, , , ,2 8 4 4 8

To determine the median, we must first put the numbers in order. To put fractions in order we

must first write all of the fractions with a common denominator. The Least Common Multiple of

the denominators 2, 4 and 8 is the smallest number they all divide.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16…

Multiples of 4: 4, 8, 12, 16, 20, 24…

Multiples of 8: 8, 16, 24…

The LCM(2, 4, 8) = 8

Now, write an equivalent fractions for each fraction with the denominator of 8.

1 1 4

2 2

4

4 8


3


3 3 6

4 4

2

2 8


2 2 4

4 4

2

2 8


5


The equivalent fraction for

1 3 3 2 5, , , ,

2 8 4 4 8with a common denominator of 8 are:

4 3 6 4 5

, , , ,8 8 8 8 8

Put the fractions in order from least to greatest from the smallest numerator to the largest

numerator.

3 4 4 5 6

, , , ,8 8 8 8 8

3 4 4 5 6

, , , ,8 8 8 8 8

Determine the middle number.

Therefore, 4 1

8 2 is the median value.

http://youtu.be/AsBhn5LUIU4



155

Practice 20: Determine the median of the following numbers. 1 3 4 2 1

, , , ,3 4 12 4 4

Watch it: http://youtu.be/VXknaa77Iys Answer: 1

3

Example 21: Determine the mode of the following numbers. 1 3 3 2 5

, , , ,2 8 4 4 8

To determine the mode we must first look at all of the fractions in order from least to greatest. In

Example 20, we have already put this data set in order from least to greatest.

3 4 4 5 6

, , , ,8 8 8 8 8

The mode is the data value that is repeated the most often. As you can see the data value 4

8is

repeated the most often. The fraction 4

8is equal to

1

2. Therefore,

1

2 is the mode.

Practice 21: Determine the mode of the following numbers. 1 3 4 2 1

, , , ,3 4 12 4 4

Answer: http://youtu.be/P6zxxx0khg8 Watch it: 1

3

Zero and Fractions

Earlier in your studies, you discovered that the number 0 has some special qualities. When you

are working with fractions, there are a couple of things that you need to keep in mind.

When the numerator is 0 (and the denominator is not 0), the value of that fraction is 0. Thus,

0

6= 0. To illustrate why this is true, let’s look closely at this equation.


6

We know that this is a division problem of 0 6 , which has a quotient of 0. We can express this

as 0

6= 0 .

On the other hand, if the denominator is 0, you have the case of division by 0. As you learned

earlier in this book, you cannot divide by 0: the answer is “undefined.” Fractions give us a

numeric way to demonstrate why you cannot divide by 0.

http://youtu.be/VXknaa77Iys

http://youtu.be/P6zxxx0khg8



156


12 Answer: 0

Watch it: http://youtu.be/8LfUW4AghoE


0

70

0 because multiplying the denominator, 0, by the whole number quotient, 7, does not

produce, 7: 0 7 7 .

Similarly, 7

00 because multiplying the denominator, 0, by the whole number quotient, 0, does

not produce, 7: 0 0 7 .

As you can see, division by 0 is not defined; we say that division by 0 is undefined.


0 Answer: undefined

Watch it: http://youtu.be/KsS3Vkkv5yA

Watch All: http://youtu.be/56TSha9Row0

RULES FOR FRACTIONS WITH ZERO

If 0 is divided by any number (except 0),

the answer is 0.

00

n

If any number is divided by 0,

the answer is undefined.

0

nundefined

http://youtu.be/8LfUW4AghoE

http://youtu.be/KsS3Vkkv5yA

http://youtu.be/56TSha9Row0



157

3.1 Introduction Exercises

1. The fraction 3

4 is equivalent to ___ ÷ ___

2. The fraction 6

5 is equivalent to ___ ÷ ___

3. Write 35 19 as a fraction. 4. Write 9 8 as a fraction.

5. Write 15 as a fraction. 6. Write 3 4 as a fraction.

7. The numerator of 7

10 is ___ and the denominator is ___.

8. The numerator of

100

13 is ___ and the denominator is ___.

9. Simplify 4

12 by dividing 4 and 12 by 4.

10. Simplify 36


11. Simplify 55


12. Simplify 6

8 13. Simplify

15

60

14. Simplify 20

64 15. Simplify

12

66

16. Simplify 16

56 17. Simplify

48

98

18. Convert 51

2 to an improper fraction. 19. Convert 3

4

7 to an improper fraction.

20. Convert 1

23

to an improper fraction.

21. Arrange from smallest to largest: 2 1 1 3 1

, , , ,3 12 3 4 2

22. Arrange from smallest to largest: 2 1 5 3

, , ,9 2 72 8



158

23. Write 14

3 as a mixed number in simplest form.

24. Write 78

8 as a mixed number in simplest form.

25. Find the mean of 15, 12, 3, 11, 7, 9.

26. Find the mean of –31, 20, –10, 17, –5.

27. Find the mean of 18, 21, 8, 11.

28. Find the median of 1 3 2

, ,2 4 3

29. Find the median of 1 10 4 3 7

, , , ,11 11 11 11 11

30. Find the mode of 3 5 2 1 5 10 1 4

, , , , , , ,8 7 6 2 15 14 10 12



159

3.1 Introduction Exercises Answers

1. 3 4 16. 2

7

2. 6 5 17. 24

49

3. 35

19 18.

11

2

4. 8

9 19.

25

7

5. 15

1 20.

7

3

6. 3

4 21.

1 1 1 2 3, , , ,

12 3 2 3 4

7. numerator: 7, denominator: 10 22. 5 2 3 1

, , ,72 9 8 2

8. numerator: 100; denominator: 13 23. 42

3

9. 1

3 24. 9

3

4

10. 6

7 25.

57

6= 9

1

2

11. 5

7 26.

9 41

5 5

12. 3

4 27.

29 114

2 2

13. 1

4 28.

2

3

14. 5

16 29.

4

11

15. 2

11 30.

1

3

Documents

CHAPTER 3: FRACTIONS - CCBC Faculty Webfaculty.ccbcmd.edu/~lwalte19/Math081C3S1Text.pdf137 CHAPTER 3: FRACTIONS CHAPTER 3 CONTENTS 3.1 Introduction to Fractions 3.2 Multiplication