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Math 300
Basic College Mathematics
Chapter 2
Multiplying and Dividing Fractions
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
Page 1 of 25
2.1 Introduction of Fractions and Mixed Numbers
The following are some examples of
fractions:
1
2
8
5
11
23
This way of writing number names is called
fraction notation. The top number is
called the numerator and the bottom
number is called the denominator.
Example: Identify the numerator and the
denominator.
3
1. 4
9
2. 2
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
Page 2 of 25
2.1 Introduction of Fractions and Mixed Numbers
(cont)
The number 1 in Fraction Notation
Any number divided by itself is equal to 1.
Example: Simplify.
5
1
5
23
1
23
9
1
9
The number 0 in Fraction Notation
Zero divided by any number is equal to 0.
Example: Simplify.
0
0
1
0
0
9
0
0
23
Excluding Division by 0
Any number divided by zero is “undefined” or “not defined”.
Example: Simplify.
1
 undefined
0
9
 undefined
0
23
 undefined
0
Division by 1
Any number divided by 1 is equal to that number.
Example: Simplify.
5
5
1
9
9
1
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
23
 23
1
Page 3 of 25
2.1 Introduction of Fractions and Mixed Numbers
(cont)
When you are given one picture and have to
determine which part is shaded or indicated, you
can write a fraction for the picture.
To write the correct fraction for the picture, you
must understand the denominator of the fraction is
the number of equal parts in which the picture is
divided into and the numerator is the number of
parts that are shaded on the picture.
Example: What part is shaded?
The picture is divided into 4 equal parts.
There is only 1 shaded area.
1
Therefore, the fraction is: 4
Writing Fraction from Real-Life Data
Example: Write each fraction.
1. In a family with 11 children, there are 4 boys and
7 girls. What fraction of the children is girls?
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.1 Introduction of Fractions and Mixed Numbers
(cont)
Identifying Proper Fractions, Improper Fractions, and
Mixed Numbers
 A proper fraction is a fraction whose numerator is
less than its denominator. Proper fractions are less
than 1.
 An improper fraction is a fraction whose numerator
is greater than or equal to its denominator. Improper
fractions are greater than or equal to 1.
 A mixed number contains a whole number and a
fraction. Mixed numbers are greater than 1.
Examples: Identify each number as a proper fraction,
improper fraction, or mixed number.
5
1.
8
1
5
2.
4
7
3.
7
4.
14
13
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.1 Introduction of Fractions and Mixed Numbers
(cont)
There are some cases of when you will see
more than one picture that is the same
picture. This represents fractions that are
greater than 1.
In that case, the write the correct fraction,
you would identify how many equal parts
is in each picture for your denominator
and the numerator will consist of how
many shaded areas there are altogether.
Example: What part is shaded?
In the picture above, there are 3 equal
parts in each.
There are 7 shaded areas.
7
Therefore, the fraction is: 3
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.1 Introduction of Fractions and Mixed Numbers
(cont)
Writing a Mixed Number as an Improper Fraction
 To write a mixed number as an improper fraction, you
would multiply the denominator times the whole
number add the numerator and write it over the
original denominator.
Example: Write each as an improper fraction.
1. 6
2. 9
1
2
1
10
Writing an Improper Fraction as a Mixed Number or a
Whole Number
 To write an improper fraction as a mixed number or a
whole number, you would divide the denominator into
the numerator.
 You answer would be written as follows:
remainder
Whole number
or just a whole number
divisor
Example: Write a mixed number or a whole number.
3.
4.
17
4
84
6
5.
467
100
6.
7672
85
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.2 Factors and Prime Factorization
Factors
 A number is a factor of another number if it divides
evenly into it.
Example: Find all the factors of 15.
1, 3, 5, 15
Think: 1  15 = 15
3  5 = 15
factor  factor = multiple
Because all four numbers will divide evenly into 15.
Prime, Composite, & Neither
 A prime number is a natural number that has
exactly two different factors, 1 and itself.
Example: 2 is a prime number, because 1  2 = 2
4 is not a prime number, because
1  4 = 4, 2  2 = 4
 A composite number is any natural number, other
than 1, that is not prime.
Example: 6 is composite, because
1  6 = 6 and 3  2 = 6
 The natural number 1 is neither prime nor
composite.
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.2 Factors and Prime Factorization (cont)
 Prime factorization is to express a number as a product of
prime numbers only!!
 There are 2 ways to do prime factorization.
1.) Prime factor using division
2.) Prime factor tree
Way # 1: Prime factoring using division
Example: Find the prime factorization of 72
 First, make sure you know the list of primes
(at least the first 11 prime numbers)
Then write the number that you are prime factoring like this:
72
 Put the first prime number that you are dividing by in front
like this:
2
72
(Note: you may start dividing by any prime number that will
divide evenly into that number.)
 Continue dividing until you reach one at the top like this:
1
2
2
2
4
3 12
3 36
2 72
 Once you reach one at the top, STOP DIVIDING!!!
 Your prime factorization is the numbers that appear down the
left side.
 Write your answer like this: 72 = 2  2  2  3  3
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.2 Factors and Prime Factorization (cont)
Way #2:
Prime factoring using the factor tree
Example: Find the prime factorization of 72
 Begin with the number that has to be factor.
72
 Choose two factors of that number. (Two factors
that when you multiply them together, you get the
number in which your factoring.)
72
89
 For each number that is not prime, factor it by
choosing 2 more numbers that are factors of that
number.
 For each number that is prime, bring down to the
next row.
72
8  9
2433
22233
 Your end result should be ALL PRIME NUMBERS!!!
The prime factorization of 72 is 2  2  2  3  3
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.2 Factors and Prime Factorization (cont)
Rules for Divisibility by 2, 3, 4 or 5
 If a number is even (the last digit is 0, 2, 4, 6, 8),
then it will divide by 2!!
Example: 4786 is divisible by 2, because the last digit
is 6.
 If you add up the digits in a number and the
result divides evenly by 3, then the number will
divide evenly by 3!!
Example: 201 is divisible by 3, because 2 + 0 + 1 = 3
 If the last two digits of a number is divisible by
4, then the number is will divide evenly by 4.
Example: 1712 is divisible by 4, because the last two
digits (12) is divisible by 4.
 If a number ends in 0 or 5, then it will divide by
5.
Example: 790 is divisible by 5, because the number
ends in 0.
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.2 Factors and Prime Factorization (cont)
Rules for Divisibility by 6, 9, or 10
 A number is divisible by 6 if it is both an even
number and divisible by 3.
Example: 810 is divisible by 6, because it is an even #
and
8 + 1 + 0 = 9, well 9 is divisible by 3.
 A number is divisible by 9 if the sum of their digits
is divisible by 9.
Example: 29, 223 is divisible by 9, because
2 + 9 + 2 + 2 + 3 = 18.
 A number is divisible by 10 if the number had a
zero in the ones place.
Example: 300 is divisible by 10, because there is a zero
in the ones place.
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.3 Simplest Form of a Fraction
 To simplify a fraction means to write a fraction in
simplest or lowest terms. A fraction is considered in
simplest or lowest terms when the numerator and
denominator have no common factors other than 1.
Example: Simplify.
4
1.
8
2.
100
20
42
3.
48
9
4. 50
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.3 Simplest Form of a Fraction (cont)
A Test for Equality
 A test for equality is how we compare fractions.
 The signs that you would use is either equal to  or
not equal   to.
 You would use cross products or you may find
common denominators to determine the equality.
Example: Determine whether each pair of fractions is
equivalent.
3
2
1.
and
9
6
2
14
2.
3 and 20
Solving Problems by Writing Fractions in Simplest
Form
Example: Write each fraction in simplest form.
3. Two thousand baseball caps were sold one year at the
U.S. Open Golf Tournament. What fractional part of this
total does 200 caps represent?
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
Page 14 of 25
2.4 Multiplying Fractions
Multiplying a Fraction and a Whole Number
To multiply a fraction and a whole number, you would
multiply the numerator (top number) by the whole number
and keep the denominator (bottom number) the same.
Example: Multiply.
1
2

1.
3
Multiplying a Fraction by a Fraction
To multiply a fraction by a fraction, you would multiply the
numerator times numerator and denominator times
denominator.
Example: Multiply.
1 1
2. 6  4
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.4 Multiplying Fractions (cont)
Multiplying and Simplifying
 This section is where you would put both multiplying
and simplifying together in one problem.
 There are 2 ways you can approach multiplying and
simplifying:
1.
You may multiply the fractions first, then simplify
your answer
Or
2.
You may simplify first by using cancelling correctly,
then multiply your answer.
Example: Multiply and simplify.
3 1
1. 8  3
2.
16 5

15 4
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.4 Multiplying Fractions (cont)
Multiplying Fractions and Mixed Numbers
 To multiply with mixed numbers, first write the mixed
number as a fraction and then multiply.
Example: Multiply.
1 8
2 15
1. 2 
3.
6
3
7
*5
10 10
Solving Problems by Multiplying Fractions
 To solve real-life problems that involve multiplying
fractions, use the steps for problem-solving steps from
Chapter 1. A new key word that implies multiplication
is used…that key word is “of.”
Example: Solve. Write answer in simplest form.
2
3. A recipe calls for
of a cup of flour. How much flour
3
1
should be used if only
of the recipe is being made?
2
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
Page 17 of 25
2.5 Dividing Fractions
 To find a reciprocal of a fraction, interchange the
numerator and the denominator. (In other words, flip
the fraction over.)
 To find a reciprocal of a whole number, first place the
whole number over 1 to make it a fraction then flip the
fraction over.
 Zero has no reciprocal.
Example: Find the reciprocal.
2
1. 5
2. 9
1
3. 5
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.5 Dividing Fractions (cont)
Division of Fractions
 Steps for dividing fractions:
1. Keep the first fraction the same.
2. Change the division symbol  to a multiplication
symbol  .
3. Take the reciprocal of the second fraction.
4. Multiply and simplify.
Example: Divide and simplify.
1.
2 3

3 4
12
4
2.
7
3. 24 
3
8
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.5 Dividing Fractions (cont)
 To divide with a mixed number, first write the mixed
number as a fraction and then follow the steps for
dividing a fraction.
Example: Divide.
1.
5
 10
6
2. 2
5
6
4
6
7
2
0
3.
3
4.
5.
18  2
1
4
4
1
5 2
5
2
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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2.5 Dividing Fractions (cont)
 To solve real-life problems that involve dividing
fractions, we continue to use our problem-solving
steps from Chapter 1.
Example: Write each answer in simplest form.
1. If there are 13
1
grams of fat in 4 ounces of lean
3
hamburger meat, how many grams of fat are in an ounce?
2. A heart attack patient in rehabilitation walked on a
treadmill 12
3
4
miles over 4 days. How many miles is this
per day?
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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Math 300 Chapter 2
Glossary
Fraction – number used to express equal parts of a whole.
Numerator - Top number in a fraction, it is the number of parts of the whole
being considered.
Denominator – Bottom number in a fraction, it is the number of equal parts in
the whole.
Proper fraction - a fraction with the numerator less than the denominator.
Improper fraction – a fraction with the numerator greater than or equal to the
denominator.
Mixed Number - A number and a fraction together.
Natural number – Natural numbers are whole numbers starting with 1.
Prime number – a natural number that has exactly two different factors, 1 and
the number itself.
Composite number – any natural number other than 1 that is not a prime
number.
Prime factorization – factorization of a number such that all the factors are
prime number.
Equivalent fractions – two fractions that represent the same portion of a
whole. They are equal.
Simplest form of a fraction - a fraction is in its simplest form when the
numerator and denominator have no common factors other than 1.
Reciprocal - Two numbers are reciprocals of each other when their product is
1.
Cross product – used to determine if two fractions are equivalent. The cross
product of a/b and c/d is (a)(d) and (b)(c). If those products are equal, then the
fractions are equal.
Properties
Multiplication of fractions:
a/b * c/d = a * c / b * d
Division of fractions:
a/b divided by c/d is a/b * d/c
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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Math 300 Chapter 2
Glossary (cont)
Hints
The mixed number 2 ¼ represents 2 and ¼
Convert a mixed number to an improper fraction
1. Multiply the denominator of the fraction by the whole number
2. Add the numerator of the fraction to the product from step 1 above
3. Write the sum from 3 above as the numerator of the improper fraction
with the denominator from the original fraction.
Writing an improper fraction as a mixed number
1. Divide the denominator into the numerator
2. The whole number part of the mixed number is the quotient. The
fractional part of the mixed number is the remainder from step 1 above
over the original denominator.
When converting an improper fraction to a mixed number, if the remainder is
0, then the answer is just a natural number without the fraction.
Prime factorization
1. Factor tree
2. Division
Multiplication is commutative so 2x2x3 = 2x3x2 = 3x2x2 = 22x3
Divisibility tests – a whole number is divisible by:

2 if the last digit is even (0, 23, 4, 6, 8)

3 if the sum of the digits is divisible by 3. For example, 255 = 2+5+5=12
and 12 is divisible by 3

5 if the last digit is a 0 or a 5
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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Math 300 Chapter 2
Glossary (cont)
Hints (continued)

4 if the last two digits are divisible by 4. For example, 324 is divisible by
4 since 24 is divisible by 4

6 if it is divisible by 2 and 3

9 if the sum of the digits is divisible by 9
Factor Tree
36
12
4
2
3
3
2
2x2x3x3=22 x 32
Any nonzero number when divided by itself is 1. 12/12 = 1
When putting a fraction into simplest form, it is helpful to write the prime
factorizations of the numerator and denominator and then removing any
common factors.
When simplifying a fraction, remember if all of the prime factors can be
removed there is still 1 left.
To determine if two fractions are equal, just find the cross product.
It is not always necessary to do a complete prime factorization to simply a
fraction. For example, if both the numerator and denominator end in 0, you
can divide both by 10 (not just 2 and then 5).
Multiplication of mixed number is done by converting the mixed numbers into
improper fractions and multiplying.
The reciprocal of a fraction if done by interchanging the numerator and
denominator. The product of the reciprocals is 1.
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
Page 24 of 25
Math 300 Chapter 2
Glossary (cont)
Hints (continued)
To divide two fractions, simply multiply the first fraction time the reciprocal of
the second fraction.
Dividing fractions and mixed number is done by converting all mixed number s
to improper fractions and then divide by multiplying the first fraction by the
reciprocal of the second.
When dividing fractions, do not remove common factors until after the
multiplication is complete.
Math 300 M-G 4e Chapter 2; Rev: Mar 2011
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