Download 4.1 Introduction to Fractions For example, is a proper fraction where

Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
4.1 Introduction to Fractions
Fraction
a
A number of the form , where a and b are integers, and b is not zero is called a
b
fraction.
a is called the numerator.
b is called the denominator.
Proper Fraction
A proper fraction is a fraction whose numerator is less than the denominator.
A positive proper fraction is somewhere in between 0 and 1.
A negative proper fraction is somewhere in between 0 and 1 .
2
is a proper fraction where the numerator is 2 and the denominator is 3.
3
2
Example 1: Graph
on a number line.
3
For example,
Solution:
Since
2
is a positive fraction, it is located to the right of zero on the number line.
3
The denominator describes how many equal parts in the whole.
Since
2
is a positive, proper fraction, the whole is thought of as the distance between 0 and 1.
3
The denominator is 3 so we break the distance between 0 and 1 up into three equal parts.
Notice that we use two tick marks to divide the distance into three equal parts.
The numerator is 2 so we plot a closed circle on the second tick mark (counting from zero).
Write
2
under the closed circle to label it properly.
3
-4
-3
-2
-1
0
2
1
3
2
3
4
1
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
3
You Try It 1: Graph
on a number line.
8
-4
-3
Example 2: Graph 
-2
-1
0
1
2
3
4
3
on a number line.
4
Solution:
Since 
3
is a negative fraction, it is located to the left of zero on the number line.
4
The denominator describes how many equal parts in the whole.
Since 
3
is a negative, proper fraction, the whole is thought of as the distance between 0 and 1 .
4
The denominator is 4 so we break the distance between 0 and 1 up into four equal parts.
Notice that we use three tick marks to divide the distance into four equal parts.
The numerator is 3 so we plot a closed circle on the third tick mark (counting from zero).
Write 
-4
3
under the closed circle to label it properly.
4
-3
You Try It 2: Graph 
-4
-3
-2
-1 
3
4
0
1
2
3
4
0
1
2
3
4
4
on a number line.
5
-2
-1
2
2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Improper Fraction
An improper fraction is a fraction whose numerator is greater than the denominator.
A positive improper fraction is greater than 1.
A negative improper fraction is less than 1 .
For example,
9
is an improper fraction where the numerator is 9 and the denominator is 4.
4
Example 3: Graph
9
on a number line.
4
Solution:
9
is a positive fraction, it is located to the right of zero on the number line.
4
9
To further pinpoint its location, notice that
is an improper fraction.
4
9
This means that
is located to the right of 1.
4
Since
The denominator describes how many equal parts in the whole.
Since
9
is a positive fraction, the whole is thought of as the distance between 0 and 1.
4
The denominator is 4 so we break the distance between 0 and 1 up into four equal parts.
Notice that we use three tick marks to divide the distance into four equal parts.
We extend the tick marks in the same manner beyond 1 since
9
is improper.
4
The numerator is 9 so we plot a closed circle on the ninth tick mark (counting from zero).
Notice that the ninth tick mark is located to the right of 2.
Write
9
under the closed circle to label it properly.
4
-4
-3
-2
-1
0
1
2
9
4
3
4
3
2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
You Try It 3: Graph
-4
11
on a number line.
8
-3
Example 4: Graph 
-2
-1
0
1
2
3
4
5
on a number line.
3
Solution:
5
is a negative fraction, it is located to the left of zero on the number line.
3
5
To further pinpoint its location, notice that  is an improper fraction.
3
5
This means that  is located to the left of 1 .
3
Since 
The denominator describes how many equal parts in the whole.
Since 
5
is a negative fraction, the whole is thought of as the distance between 0 and.
3
The denominator is 3 so we break the distance between 0 and 1 up into three equal parts.
Notice that we use two tick marks to divide the distance into three equal parts.
We extend the tick marks in the same manner beyond 1 since 
5
is improper.
3
The numerator is 5 so we plot a closed circle on the fifth tick mark (counting from zero).
Notice that the fifth tick mark is located to the left of 1.
Write 
5
under the closed circle to label it properly.
3
-4
-3
-2 
5
3
-1
0
1
2
3
4
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
You Try It 4: Graph 
-4
-3
13
on a number line.
4
-2
Example 5: Graph 
-1
0
1
2
3
4
4
2
and  on the same number line.
8
4
Solution:

4
is a negative, proper fraction so it is located in between 0 and 1 . Since the denominator is 8, we
8
break
the distance between 0 and 1 up into eight equal parts. The numerator is 4 so we plot a closed circle on
the fourth tick mark (counting from zero) and write 

4
to label it properly.
8
2
is a negative, proper fraction so it is located in between 0 and 1 . Since the denominator is 4, we
4
break
the distance between 0 and 1 up into four equal parts. The numerator is 2 so we plot a closed circle on
the second tick mark (counting from zero) and write 
-4
-3
You Try It 5: Graph
-4
-3
-2
-1 
4
8

2
4
0
2
to label it properly.
4
1
2
3
4
1
2
3
4
6
12
and
on the same number line.
4
8
-2
-1
0
5
2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
In Example 5 and You Try It 5, we see that even though some fractions look different, they still represent the
same place on the number line.
Equivalent Fractions
Two fractions are equivalent if they represent the same number.
The numerator and denominator of equivalent fractions share a common factor (other
than 1).
We can find equivalent fractions by multiplying or dividing both the numerator and the denominator by the
same nonzero number.
Finding Equivalent Fractions
If a, b, and c are numbers (and b and c are not zero), then:
a ac
a a c
and 

b bc
b bc
It is important to note the “whatever you do to the denominator, you must do to the numerator” rule really
comes from the multiplication and division properties of 1:
Multiplying any number by 1 results in the same number:
Dividing any number by 1 results in the same number:
a a
a c ac
 1   
b b
b c bc
a a
a c a c
 1   
b b
b c bc
When we find an equivalent fraction with a larger denominator than that of the original, we are building a
fraction to higher terms.
Building a Fraction to Higher Terms
1) Figure out what number you need to multiply the original denominator by to turn it
into the denominator specified in the directions.
2) Multiply the numerator by the same number.
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Example 6: Find a fraction equivalent to the given fraction with the specified denominator.
a)
2
with denominator 24
3
b) 
6
with denominator 15
5
Solution:
a) The original denominator is 3. Multiplying 3 by 8 gives us 24, the new denominator. To
write an equivalent fraction, multiply both the numerator and denominator by 8.
2 2  8 16


3 3  8 24
b) The original denominator is 5. Multiplying 5 by 3 gives us 15, the new denominator. To
write an equivalent fraction, multiply both the numerator and denominator by 3.

6
63
18


5
53
15
You Try It 6: Find a fraction equivalent to the given fraction with the specified denominator.
a) 
1
with denominator 21
7
b)
8
with denominator 33
3
When we find an equivalent fraction with a smaller denominator than that of the original, we are reducing a
fraction to lower terms.
Reducing a Fraction to Lower Terms
1) Figure out what number you need to divide the denominator by to turn it into the
denominator specified in the directions.
2) Divide the numerator by the same number.
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2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Example 7: Find a fraction equivalent to the given fraction with the specified denominator.
a)
36
with denominator 7
21
b) 
12
with denominator 6
24
Solution:
a) The original denominator is 21. Dividing 21 by 3 gives us 7, the new denominator. To write
an equivalent fraction, divide both the numerator and denominator by 3.
36 36  3 12


21 21  3
7
b) The original denominator is 24. Dividing 24 by 4 gives us 6, the new denominator. To write
an equivalent fraction, divide both the numerator and denominator by 4.

12
12  4
3


24
24  4
6
You Try It 7: Find a fraction equivalent to the given fraction with the specified denominator.
a)
4
with denominator 10
20
b) 
28
with denominator 4
16
Does it Matter Where the Negative Sign Goes?
For negative fractions, the rules for dividing signed integers explain the three possible placements of the
negative sign in the property below.
Fractions and Negative Signs
If a and b are integers and b is not zero, then the following is true:

a a
a


b
b
b
note: It is most common to see the negative sign to be placed at the fraction bar or in the numerator.
Ask your instructor if they have a particular preference.
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2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
For example, 
below:
3 3
3
since the quotient of two integers with different signs is negative. We show this


4
4
4
3
3
3
3
  3  4   and
 3   4   
4
4
4
4
Also note that when a fraction has a negative sign in the numerator and the denominator, the following holds:
Fractions and Negative Signs
If a and b are integers and b is not zero, then the following is true:
a a

b b
For example,
3 3
 since the quotient of two integers having the same sign is positive. We show this below:
4 4
3
3
  3   4  
4
4
Example 8: Find a fraction equivalent to the given fraction with the specified denominator.
3
with denominator 8x
2
a) 4x with denominator 9
b)
a) The original denominator is 1, since 4 x 
4x
. Multiplying 1 by 9 gives us 9 , the new
1
Solution:
denominator. To write an equivalent fraction, multiply both the numerator and denominator
by 9 .
4x 
4 x 4 x  9 36 x


1
1 9
9
b) The original denominator is 2. Multiplying 2 by 4x gives us 8x , the new denominator. To
write an equivalent fraction, divide both the numerator and denominator by 4x .
3 3  4 x 12 x


2
2  4x
8x
9
2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
You Try It 8: Find a fraction equivalent to the given fraction with the specified denominator.
a) 5 with denominator 3n
b)
5
with denominator 32n
8n
Lowest Terms
A fraction is in lowest terms when the numerator and denominator have no common
factor other than 1.
5 7 1 12
9
are some examples of fractions in lowest terms.
,  , ,  , and
8
3 2 13
4
Example 9: Are the following fractions in lowest terms? If not, find at least one common factor of the
numerator and denominator (other than 1).
a)
27
12
b)
4
15
Solution:
a) Factors of the numerator, 27, are 1, 3, 9, and 27.
Factors of the denominator, 12, are 1, 2, 3, 4, 6, and 12.
Since the numerator and denominator have a common factor of 3, the fraction
27
is not in
12
lowest terms.
b) Factors of the numerator, 4, are 1, 2, and 4.
Factors of the denominator, 15, are 1, 3, 5, and 15.
Since the only common factor of the numerator and denominator is 1, the fraction
4
is in
15
lowest terms.
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
You Try It 9: Are the following fractions in lowest terms? If not, find at least one common factor of the
numerator and denominator (other than 1).
a) 
3
16
b)
12
21
c)
23
6
d) 
15
40
Writing a Fraction in Lowest Terms
The following directions all mean the same thing:
“Reduce to lowest terms.”, “Write in lowest terms.”, “Write in simplest form.”, and “Simplify.”
We will show you two methods to write a fraction in lowest terms:
Writing a Fraction in Lowest Terms: Dividing By a Common Factor
This method works best when the numerator and denominator are small numbers.
1) Find the greatest number that will divide evenly into both the numerator and the
denominator. (This number is called the greatest common factor.)
2) Divide both the numerator and denominator by this number.
3) Check to see that the numerator and denominator do not have any common factors
(other than 1). If they do, repeat steps 2 and 3.
Example 10: Divide by a common factor to reduce the following fractions to lowest terms.
a) 
18
30
b)
90
72
Solution:
a) Factors of the numerator, 18, are 1, 2, 3, 6, 9, and 18.
Factors of the denominator, 30, are 1, 2, 3, 5, 6, 10, 15, and 30.
The greatest common factor numerator and denominator is 6.
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Divide both the numerator and denominator by 6: 
18  6
3

30  6
5
Check! The only common factor 3 and 5 share is 1 so 
3
is in lowest terms.
5
b) Notice there is a negative sign in both the numerator and denominator. Will the reduced
fraction be negative or positive?______________
Factors of the numerator, 90, are _________________________________.
Factors of the denominator, 72, are _______________________________.
The greatest common factor numerator and denominator is _____.
Divide both the numerator and denominator by _____:
Check! Is the only common factor of the numerator and denominator 1?
What if we made a mistake and thought 9 was the greatest common factor of 90 and 72?
Dividing the numerator and denominator by 9 gives us
90  9 10

72  9
8
10
is not in lowest terms because 10 and 8 still have 2 as a common factor!
8
10  2 5
We still need to divide the numerator and denominator by 2:

82
4
5
is in lowest terms since 5 and 4 only have a common factor as 1.
4
But
If we had divided the numerator and denominator by 18, the greatest common factor or 90 and
72, we would get
90  18 5
 (same as the answer above)
72  18 4
Comparing the paths we took above and some more of the possible ways to reduce
90  9 10  2 5


72  9
82
4
90  18 5

72  18 4
90  2 45  3 15  3 5



72  2 36  3 12  3 4
90
:
72
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
90  3 30  2 15  3 5



72  3 24  2 12  3 4
90  6 15  3 5


72  6 12  3 4
Notice that using the greatest common factor resulted in less steps but the division became
slightly more difficult while some paths took more steps, using smaller common factors, made
it quicker to mentally calculate the division.
So which is better? Using the greatest common factor or a smaller common factor?
This is up to you! Some fractions you will find that using the greatest common factor is
easier; some fractions you will find it easier to start with using a smaller common factor.
All paths lead to the same correct answer, you just have to make sure that you keep dividing
by common factors until the greatest common factor of the numerator and denominator is 1.
You Try It 10: Divide by a common factor to reduce the following fractions to lowest terms.
a) 
15
20
b)
20
58
c) 
18
16
d)
74
62
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2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Sometimes, students have a hard time with the previous method due to all of the mental calculations that have to
be made. If you find yourself in this situation, the following method may work better for you.
Writing a Fraction in Lowest Terms: Using Prime Factorization
1) Write the prime factorization of the numerator and denominator.
2) Divide out common factors that the numerator and denominator share.
(Use the concept
same number
 1 and use a slash to show you are “cancelling the
same number
common factors.)
3) Multiply the remaining factors in the numerator and write the result as your new
numerator.
4) Multiply the remaining factors in the denominator and write the result as your new
denominator.
You might be wondering what prime factorization means. Before we get into that, let’s review what prime and
composite numbers are.
Recall that the set of natural numbers is 1, 2, 3, 4, 5, 6, 7, ...
Prime Number
A prime number is a natural number greater than one that has exactly two factors, itself
and 1.
Composite Number
A composite number is a natural number greater than one with three or more factors.
(It has at least one factor other than itself or 1.)
2 is a prime number since it can only be divided evenly by 2 and 1. 17 is a prime number since it can only be
divided by 17 and 1.
9 is a composite number since it can be divided evenly by 3 as well as 9 and 1. 24 is a composite number since
it can be divided evenly by 2, 3, 4, 6, 8, and 12 as well as 24 and 1.
1 is neither prime nor composite since it is not a natural number greater than 1. 0 is neither prime nor
composite since it is not a natural number.
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
It is useful to be able to recognize prime numbers when finding the prime factorization of a number. For
reference, here are the prime numbers smaller than 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 , 61, 67, 71, 73, 79, 83, 89, and 97
Caution: 2 is the only even prime number. All prime numbers are odd (except for 2) but not all odd numbers
are prime.
Prime Factorization of a Number
The prime factorization of a number is when you rewrite a composite number as a
product of only prime factors.
There are many techniques to find the prime factorization of a number. We encourage you to find a technique
that works for you. We will present the Factor Tree Method as it tends to be the most user-friendly.
Finding the Prime Factorization: The Factor Tree Method
1) Write the composite number down with enough space below it to create a factor
tree.
2) Find two factors of the composite number (besides 1 and the number itself) and
write them side by side below the composite number. Draw a branch from the
original composite number to each factor.
3) Decide whether each factor is prime or composite. If a factor is prime, circle it to
signal the end of the branch. If a factor is composite, repeat step 2 by finding
factors and drawing new branches.
4) Repeat steps 2 and 3 until no uncircled factors remain.
5) The prime factorization of the original composite number is a product of all the
circled factors. Write them out in order from least to greatest.
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Example 11: Write the prime factorization of each number.
a) 42
b) 180
Solution:
a)
42
6
2
7
The prime factorization of 42 is 2  3  7 .
3
What if we start with different factors?
Here are three different ways to show the prime factorization of 42:
42
6
2
42
7
2
3
42
21
3
3
14
2
7
7
Notice that all three trees lead to the same prime factorization: 42  2  3  7 So, it does not matter what factors
you start with (as long as they are correct factors).
b)
180
10
2
18
5
9
2
3
3
The prime factorization of 180 is 2  2  3  3  5 .
180
Again, we could have chosen different factors to get the following tree:
2
90
2
45
5
9
3
3
Which still leads us to the same prime factorization, 180  2  2  3  3  5 .
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2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Why can’t we use exponents to write the repeated factors in our prime factorization?
It is true that the prime factorization of 180, 2  2  3  3  5 , is the same as 22  32  5 . If the directions only ask you to
write the prime factorization, it would be perfectly fine to write the answer using exponents.
However, our main purpose for prime factorization in this chapter is to use it to write fractions in lowest terms.
If the prime factorizations involve exponents, it may be confusing to see which factors divide out, and lead to
unnecessary errors. For this reason, we promote writing out all of the factors in the prime factorizations.
You Try It 11: Write the prime factorization of each number.
a) 32
b) 90
c) 88
d) 150
Example 12: Use prime factorization to reduce the following fractions to lowest terms.
a) 
18
30
b)
90
72
Solution:
a) The prime factorization of the numerator, 18, is 2  3  3 .
The prime factorization of the denominator, 30, is 2  3  5 .
1
1
So,  18   2  3  3
30
2  35
1
Don’t forget to rewrite the negative sign!
1
Divide out common factors in the numerator and denominator.
2  2  1 and 3  3  1
Multiply the remaining factors in the numerator: 11 3  3
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Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Multiply the remaining factors in the denominator: 11 5  5
1 1  3

1 1  5

3
5
Notice that this is the same answer from example 10!
b) The prime factorization of the numerator, 90, is 2  3  3  5 .
The prime factorization of the denominator, 72, is 2  2  2  3  3 .
1
1
1
So, 90   2  3  3  5
72  2  2  2  3  3
1
1
Don’t forget to divide out the negative signs!
1
  
Divide out common factors in the numerator and denominator.
2  2  1 , 3  3  1 , and 3  3  1
Multiply the remaining factors in the numerator: 111 5  5
Multiply the remaining factors in the denominator: 1 2  2 11  4
1 1 1  5

1  2  2 1 1

5
4
Notice that this is the same answer from example 10!
You Try It 12: Use prime factorization to reduce the following fractions to lowest terms.
a) 
15
20
b)
20
58
c) 
18
16
d)
74
62
18
2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Sometimes fractions have variables in the numerator or denominator. To reduce fractions like these to lowest
terms just write the variable factors in expanded form and use the concepts of prime factorization to divide out
common factors.
Example 13: Reduce the following fractions to lowest terms.
a)
6
30a
54 xy
48 x
b)
5
c) 20 x 2
32 x
2
d)  a 4b
 ab c
Solution:
a) The prime factorization of the numerator, 6, is 2  3 .
The prime factorization of the denominator, 30a, is 2  3  5  a .
1
1
So, 6  2  3
30 a 2  3  5  a
1
1
Divide out common factors in the numerator and denominator.
2  2  1 and 3  3  1
Multiply the remaining factors in the numerator: 11 
Multiply the remaining factors in the denominator: 11 5  a  5a
1 1

1 1  5  a

1
5a
b) The prime factorization of the numerator, 54xy, is 2  3  3  3  x  y .
The prime factorization of the denominator, 48x, is 2  2  2  2  3  x .
1
1
1
So, 54 xy  2  3  3  3  x  y Don’t forget to rewrite the negative sign!
48 x  2  2  2  2  3  x
1
1
1
Divide out common factors in the numerator and denominator.
2  2  1 , 3  3  1 , and x  x  1
Multiply the remaining factors in the numerator:
111 3  3 1 y  9 y
Multiply the remaining factors in the denominator:
1 2  2  2  2 11  16
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2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
1 1 1  3  3 1  y

1 2  2  2  2 11

9y
16

9y
16
c) The prime factorization of 20 is 2  2  5 and x 5 in expanded form is x  x  x  x  x .
Rewrite 20x5 as 2  2  5  x  x  x  x  x .
The prime factorization of 32 is 2  2  2  2  2 and x 2 in expanded form is x  x .
Rewrite 32x2 as 2  2  2  2  2  x  x .
1
1
1
1
5
So, 20 x2  2  2  5  x  x  x  x  x
32 x
2 222 2 x x
1
1
1
1
Divide out common factors in the numerator and denominator.
2  2  1 , 2  2  1 , x  x  1 , and x  x  1
Multiply the remaining factors in the numerator:
11 5 11 x  x  x  5x3
Multiply the remaining factors in the denominator:
1 1  2  2  2 1 1  8

1 1  5 1 1  x  x  x
1 1  2  2  2 1 1

5 x3
8
d) a 2 b in expanded form is a  a  b . Rewrite a2b as a  a  b .
ab4c in expanded form is a  b  b  b  b  c . Rewrite ab4c as a  b  b  b  b  c .
1
1
2
 aa b
So,  a 4b 
Don’t forget to divide out the negative signs!
 ab c  a  b  b  b  b  c
1
1
  
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2015 Carreon
Math 40
Prealgebra
Section 4.1 – Introduction to Fractions
Divide out common factors in the numerator and denominator.
a  a  1 and b  b  1
Multiply the remaining factors in the numerator: 1 a 1  a
Multiply the remaining factors in the denominator:
11 b  b  b  c  b3c

1  a 1
1 1  b  b  b  c

a
b3c
You Try It 13: Reduce the following fractions to lowest terms.
a)
32 x
4
b)
12ab
54abc
c)
55a 4
22 a
2
d) x y2
xy
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2015 Carreon