Transcript Columbus State Community College Chapter 4 Section 2 Writing Fractions in Lowest Terms Ch 4 Sec 2: Slide #1

Columbus State Community College
Chapter 4 Section 2
Writing Fractions in Lowest Terms
Ch 4 Sec 2: Slide #1
Writing Fractions in Lowest Terms
1. Identify fractions written in lowest terms.
2. Write fractions in lowest terms using common factors.
3. Write a number as a product of prime factors.
4. Write a fraction in lowest terms using prime factorization.
5. Write a fraction with variables in lowest terms.
Ch 4 Sec 2: Slide #2
Note on Factors
NOTE
Recall that factors are numbers being multiplied to give a product.
For example,
1 • 5 = 5, so 1 and 5 are factors of 5.
5 • 7 = 35, so 5 and 7 are factors of 35.
5 is a factor of both 5 and 35, so 5 is a common factor of those
numbers.
Ch 4 Sec 2: Slide #3
Writing a Fraction in Lowest Terms
Writing a Fraction in Lowest Terms
A fraction is written in lowest terms when the numerator and
denominator have no common factors other than 1.
1
5
2
11
Examples are
,
,
, and
.
9
8
7
34
When you work with fractions, always write the final answer in
lowest terms.
Ch 4 Sec 2: Slide #4
Identifying Fractions Written in Lowest Terms
EXAMPLE 1
Identifying Fractions Written in Lowest Terms
Are the following fractions in lowest terms?
(a)
9  The factors of 9 are 1, 3, and 9.
14  The factors of 14 are 1, 2, 7, and 14.
The numerator and denominator have no common factor other
than 1, so the fraction is in lowest terms.
(b)
10  The factors of 10 are 1, 2, 5, and 10.
25  The factors of 25 are 1, 5, and 25.
The numerator and denominator have a common factor of 5,
so the fraction is not in lowest terms.
Ch 4 Sec 2: Slide #5
Using Common Factors to Write Fractions in Lowest Terms
EXAMPLE 2
Using Common Factors – Lowest Terms
Write each fraction in lowest terms.
(a)
27
36
The largest common factor of 27 and 36 is 9. Divide both
numerator and denominator by 9.
27
36
=
27 ÷ 9
36 ÷ 9
=
3
4
Ch 4 Sec 2: Slide #6
Using Common Factors to Write Fractions in Lowest Terms
EXAMPLE 2
Using Common Factors – Lowest Terms
Write each fraction in lowest terms.
(b)
40
55
The largest common factor of 40 and 55 is 5. Divide both
numerator and denominator by 5.
40
55
=
40 ÷ 5
55 ÷ 5
=
8
11
Ch 4 Sec 2: Slide #7
Using Common Factors to Write Fractions in Lowest Terms
EXAMPLE 2
Using Common Factors – Lowest Terms
Write each fraction in lowest terms.
(c)
– 32
72
The largest common factor of 32 and 72 is 8. Divide both
numerator and denominator by 8.
– 32
72
=
– 32 ÷ 8
72 ÷ 8
=
– 4
9
Keep the negative sign
Ch 4 Sec 2: Slide #8
Using Common Factors to Write Fractions in Lowest Terms
EXAMPLE 2
Using Common Factors – Lowest Terms
Write each fraction in lowest terms.
(d)
60
80
Suppose we made an error and thought that 10 was the largest
common factor of 60 and 80.
60
80
=
60 ÷ 10
80 ÷ 10
=
6 ÷ 2
8 ÷ 2
=
3
4
Not in lowest terms
Ch 4 Sec 2: Slide #9
Dividing by a Common Factor – Fractions in Lowest Terms
Dividing by a Common Factor – Fractions in Lowest Terms
Step 1
Step 2
Step 3
Find the largest number that will divide evenly into both
the numerator and denominator. This number is a
common factor.
Divide both numerator and denominator by the common
factor.
Check to see if the new numerator and denominator have
any common factors (besides 1). If they do, repeat Steps
1 and 2. If the only common factor is 1, the fraction is in
lowest terms.
Ch 4 Sec 2: Slide #10
Prime Numbers
Prime Numbers
A prime number is a whole number that has exactly two different
factors, itself and 1.
Here are the prime numbers smaller than 50.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Ch 4 Sec 2: Slide #11
Composite Numbers
Composite Numbers
A number with a factor other than itself or 1 is called a composite
number.
Ch 4 Sec 2: Slide #12
Zero and One
CAUTION
A prime number has only two different factors, itself and 1.
The number 1 is not a prime number because it does not
have two different factors; the only factor of 1 is 1.
Also, 0 is not a prime number. Therefore, 0 and 1 are neither
prime nor composite numbers.
Ch 4 Sec 2: Slide #13
Finding Prime Numbers
EXAMPLE 3
Finding Prime Numbers
Label each number as prime or composite or neither.
Composite
Prime
Neither
numbers
numbers
0
6
7
9
12
15
17
20
29
35
42
43
Ch 4 Sec 2: Slide #14
Prime and Odd Numbers
CAUTION
All prime numbers are odd numbers except the number 2. Be
careful though, because not all odd numbers are prime numbers.
For example, 21, 25, and 27 are odd numbers but they are not
prime numbers.
Ch 4 Sec 2: Slide #15
Prime Factorization
Prime Factorization
A prime factorization of a number is a factorization in which every
factor is a prime factor.
Examples
Prime factorization of 60
Prime factorization of 126
60 = 2 • 2 • 3 • 5
126 = 2 • 3 • 3 • 7
All Prime Numbers
Ch 4 Sec 2: Slide #16
Methods for Finding the Prime Factorization
We will discuss two methods for finding the prime
factorization of a number.
1. The Division Method
2. The Factor Tree Method
Ch 4 Sec 2: Slide #17
Factoring Using the Division Method
Let’s say we want to use the division method to find the prime
factorization of 30.
2
30
3
15
5
5
1
30 = 2 • 3 • 5
You’re done!
Ch 4 Sec 2: Slide #18
Prime Factorization – the Order of Factors
NOTE
You may write the factors in any order because multiplication is
commutative. So you could write the factorization of 30 as
5 • 3 • 2. We will show the factors from smallest to largest in our
examples.
Ch 4 Sec 2: Slide #19
Factoring Using the Division Method
EXAMPLE 4
Factoring Using the Division Method
(a) Find the prime factorization of 84.
2
84
2
42
3
21
7
7
1
84 = 2 • 2 • 3 • 7
You’re done!
Ch 4 Sec 2: Slide #20
Factoring Using the Division Method
EXAMPLE 4
(b)
Factorizing Using the Division Method
Find the prime factorization of 150.
2
150
3
75
5
25
5
5
1
150 = 2 • 3 • 5 • 5
You’re done!
Ch 4 Sec 2: Slide #21
Factoring Using the Division Method
CAUTION
When you’re using the division method of factoring, the last
quotient is 1. Do not list 1 as a prime factor because 1 is not a
prime number.
Ch 4 Sec 2: Slide #22
Factoring Using the Factor Tree Method
Let’s say we want to use the factor tree method to find the prime
factorization of 120.
120
120
10
2
12
5
4
2
6
3
2
120 = 2 • 2 • 2 • 3 • 5
2
20
3
4
2
5
2
120 = 2 • 2 • 2 • 3 • 5
Ch 4 Sec 2: Slide #23
Factoring Using the Factor Tree Method
EXAMPLE 5
Factoring Using the Factor Tree Method
(a) Find the prime factorization of 270.
270
9
3
30
3
6
2
5
3
270 = 2 • 3 • 3 • 3 • 5
Ch 4 Sec 2: Slide #24
Factoring Using the Factor Tree Method
EXAMPLE 5
(b)
Factoring Using the Factor Tree Method
Find the prime factorization of 108.
108
2
54
9
3
6
3
2
3
108 = 2 • 2 • 3 • 3 • 3
Ch 4 Sec 2: Slide #25
Divisibility Tests
NOTE
Here is a reminder about the quick way to see whether a number is
divisible by 2, 3, or 5; in other words, there is no remainder when
you do the division.
A number is divisible by 2 if the ones digit is 0, 2, 4, 6, or 8. For
example, 68, 994, and 560 are all divisible by 2.
A number is divisible by 3 if the sum of the digits is divisible by 3.
For example, 435 is divisible by 3 because 4 + 3 + 5 = 12 and 12 is
divisible by 3.
A number is divisible by 5 if it has a 0 or 5 in the ones place. For
example, 95, 820, and 17,225 are all divisible by 5.
Ch 4 Sec 2: Slide #26
Using Prime Factorization – Fractions in Lowest Terms
EXAMPLE 6
Using Prime Factorization to Write Fractions in
Lowest Terms
24
(a) Write
in lowest terms.
84
24 can be written as 2 • 2 • 2 • 3  Prime factors
84 can be written as 2 • 2 • 3 • 7  Prime factors
24
84
=
1 1
1
2•2•2•3
2•2•3•7
1 1 1
=
2
7
Ch 4 Sec 2: Slide #27
Using Prime Factorization – Fractions in Lowest Terms
EXAMPLE 6
Using Prime Factorization to Write Fractions in
Lowest Terms
54
(b) Write
in lowest terms.
63
54 can be written as 2 • 3 • 3 • 3  Prime factors
63 can be written as 3 • 3 • 7
54
63
=
 Prime factors
1 1
2•3•3•3
3•3•7
1 1
=
6
7
Ch 4 Sec 2: Slide #28
Using Prime Factorization – Fractions in Lowest Terms
EXAMPLE 6
Using Prime Factorization to Write Fractions in
Lowest Terms
14
(c) Write
in lowest terms.
70
14 can be written as 2 • 7
 Prime factors
70 can be written as 2 • 5 • 7  Prime factors
14
70
=
1 1
2•7
2•5•7
1
1
=
1
5
Ch 4 Sec 2: Slide #29
1 in the Numerator
CAUTION
In Example 6(c), all factors of the numerator divided out.
But 1 • 1 is still 1, so the final answer is
14
70
=
1 1
2•7
2•5•7
1
1
=
1
(not 5).
5
1
5
Ch 4 Sec 2: Slide #30
Using Prime Factorization – Fractions in Lowest Terms
Using Prime Factorization to Write a Fraction in Lowest Terms
Step 1 Write the prime factorization of both numerator and
denominator.
Step 2 Use slashes to show where you are dividing the numerator
and denominator by any common factors.
Step 3 Multiply the remaining factors in the numerator and in the
denominator.
Ch 4 Sec 2: Slide #31
Writing Fractions with Variables in Lowest Terms
EXAMPLE 7 Writing Fractions with Variables in Lowest Terms
8
(a) Write
in lowest terms.
4n
8 can be written as 2 • 2 • 2  Prime factors
4n can be written as 2 • 2 • n  4n means 4 • n = 2 • 2 • n
8
4n
=
1 1
2•2•2
2•2•n
1 1
=
2
n
Ch 4 Sec 2: Slide #32
Writing Fractions with Variables in Lowest Terms
EXAMPLE 7 Writing Fractions with Variables in Lowest Terms
6ab
(b) Write
in lowest terms.
9abc
6ab can be written as 2 • 3 • a • b
 6ab = 2 • 3 • a • b
9abc can be written as 3 • 3 • a • b • c  9abc = 3 • 3 • a • b • c
6ab
9abc
=
1 1 1
2•3•a•b
3•3•a•b•c
1
1 1
=
2
3c
Ch 4 Sec 2: Slide #33
Writing Fractions with Variables in Lowest Terms
EXAMPLE 7 Writing Fractions with Variables in Lowest Terms
(c) Write
30 m2 n5
42 m3 n2
in lowest terms.
Alternative Method
Reduce
Now
Do
By
how
you
take
the
have
many?
care
coefficients
more
of the
m’svariables.
n’s
on
on
using
top
topor
any
orbottom?
bottom?
method you choose.
30 m2 n5
42 m3 n2
=
5 n3
7m
The 1 is optional as the exponent on the m.
Ch 4 Sec 2: Slide #34
Writing Fractions in Lowest Terms
Chapter 4 Section 2 – Completed
Written by John T. Wallace
Ch 4 Sec 2: Slide #35