Download Unit 3 Section 2(2012)

UNIT 3: Prealgebra in a Technical World
3.2 Factors and Simplifying Fractions
SWBAT 1. Find all the factors of a number.
2. Use divisibility rules to find factors.
3. Find the prime factors of a number.
4. Simplify fractions by factoring.
Any fraction can be written in an infinite number of ways, for example:
We could keep naming fractions that equal
1
2
1
2
28
97
= 56 = 194 = …
for the rest of the quarter and beyond.
We say and write fractions in their simplest form so that we do not have to think so
hard!
In this section we study strategies for factoring numbers, and
we use these strategies to simplify fractions.
Factors and Finding ALL the Factors
DEFINITION: The factors of a number are all the integers that divide evenly into
the original number. For example, 3 and 7 are factors of 21, because they both
divide evenly into 21.
Most often, though, we are only concerned with the positive integers that divide evenly
into our number.
EXAMPLE 1: Amir purchased two dozen strawberry plants at the farmers' market. He brought
them home and looked up how far apart to plant them. His garden book recommended a
spacing of 1 foot apart. If Amir plants his strawberry plants in a rectangle, how long and wide
can he make his strawberry bed?
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SECTION 3.2 Factors and Simplifying Fractions
Think it through: Understand: We are asked to find lengths and widths of rectangles that have
areas of 24 ft2.
Plan: Sketch the different arrays of 24 strawberries.
Solve: Amir quickly sketched the four rectangular beds he could make.
Check: Amir could create strawberry beds that are 1 ft by 24 ft, 2 ft by 12 ft,
3 ft by 8 ft, or 4 ft by 6 ft. These lengths and widths: 1, 2, 3, 4, 6, 8, 12, and
24 are all of the factors of 24.
RULE: To find all the factors of a number, we can find all the possible integer
lengths and widths of rectangles that have an area equal to the given number.
 Check Point 1
Rachel has 30 tiles. How long and how wide could she make a countertop using all of these
tiles? (What are all the factors of 30?) _______________________________________________
An easier way to find all of the factors of a number is to create a two column list with
the smallest factor (we can call this the width) on the left and the larger, missing factor of the
number on the right (we can call this the length). For instance, look at the following example.
UNIT 3: Prealgebra in a Technical World
EXAMPLE 2: Find all the factors of 72 by listing the smallest factors in order on the left. For
each, find the missing factor that produces the product 72. Continue until all factors of 72 are
listed. We can think of this as finding widths of rectangles and the lengths that go with them.
Using this double listing, we have found all the factors of 72.
ANSWER: The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
RULE: To find all the factors of a number:
1. Start with 1, and list the original number beside it.
2. Count until you reach the next factor. Then write
this factor and the factor that goes with it.
3. Continue counting and writing pairs of factors.
4. When you arrive at a factor that is already written
on the right, you have found all of the factors.
5. Write the list.
All the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
Example:
All the factors of 30
1
30
2
15
3
10
5
6
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SECTION 3.2 Factors and Simplifying Fractions
Using this method, you have found the smallest factors of your number on the left. If
there were another, larger factor that would divide evenly into your number, it would have had
the other factor that went with it on the left.
 Check Point 2
Find all the factors of 80 by creating a two column list like the one in the example. The last row
has been finished for you.
Widths Lengths
8
10
To simplify fractions, we often need to find factors quickly. We have rules to help us do this.
Divisibility Rules
We can determine quickly whether a number is divisible by 2, 3, 4, 5, 6, 9, or 10 using
divisibility rules. The rules allow us to check for factors without actually dividing. If you learn
the rules, you can skip lots of division!
For instance, what do we know about numbers that are divisible by 2? All even
numbers are divisible by 2, and even numbers end in 2, 4, 6, 8, or 0.
Numbers divisible by 5 end in 5 or 0. Numbers divisible by 10, or any power of 10, end in 0.
RULE: If a number ends in one of the even digits: 2, 4, 6, 8, or 0, it is divisible by 2.
If a number ends in 0 or 5, it is divisible by 5.
If a number ends in 0, it is divisible by 10.
UNIT 3: Prealgebra in a Technical World
The number 4 has two factors of 2. If a number is divisible by 4, it must be even;
however, not all even numbers are divisible by 4; for instance, 18 cannot be divided by 4.
Because 100 is divisible by 4, the pattern of 4, 8, 12, 16, etc., repeats at every hundred.
For instance 328 is divisible by 4 because 28 is divisible by 4. Because 76 is divisible by 4, 576,
876, and 435,678,001,976 are all divisible by 4.
RULE: If the last two digits of a number are divisible by 4, then the number is
divisible by 4.
If we forget the rule for 4, we can still divide by 2—twice, but it is always more efficient
to divide less often!
Example 3: For each number, check the boxes that apply.
Number
a. 70
Divisible
by 2

b. 142

c.

1,234,910
Divisible
by 4
d. 435,678,890,876,988


e. 778,128,235,124,280


Divisible
by 5

Divisible
by 10





Divisible
by 5
Divisible
by 10
 Check Point 3
For each number, check the boxes that apply.
Number
a. 740
b. 144
c.
1,234,912
d. 435,678,890,876,985
e. 778,128,235,124,281
Divisible
by 2
Divisible
by 4
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SECTION 3.2 Factors and Simplifying Fractions
Probably the most interesting rules for divisibility are the rules for 3 and 9. But since 3
and 9 do not divide evenly into powers of ten, the reasoning here is different.
Every power of ten is just one more than a number divisible by 9. For instance,
10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1, and 10,000 = 9,999 + 1. The digit 1 and the powers of
ten, 10, 100, 1000 and 10,000, are not divisible by 9, but 9, 99, 999 and 9,999 are each divisible
by 9. We can use the distributive property to look at this pattern of divisibility. Complete the
table:
Thousands
Hundreds
Tens
1000 = 1 ∙ 𝟗𝟗𝟗 + 1
100 =
10 = 1 ∙ 9 + 1
1
2000 =
200 = 2 ∙ 99 +2
20 =
2
3000 =
300 = 3 ∙ 99 + 3
30 =
3
4000 = 4 ∙ 𝟗𝟗𝟗 + 4
400 =
40 = 4 ∙ 9 + 4
4
5000 =
500 = 5 ∙ 99 + 5
50 =
5
6000 = 6 ∙ 999 + 6
600 = 6 ∙ 99 + 6
60 = 6 ∙ 9 + 6
6
7000 = 7 ∙ 999 + 7
700 =
70 = 7 ∙ 9 + 7
7
8000 =
800 = 8 ∙ 99 + 8
80 =
8
9000 = 9 ∙ 999 + 9
900 = 9 ∙ 99 + 9
90 = 9 ∙ 9 + 9
9
The table rewrites each digit times a power of ten as the sum of the same digit times a
multiple of 9 and the original digit. Using this idea we can investigate any number.
UNIT 3: Prealgebra in a Technical World
We had to choose a number to investigate, so we chose 567. To determine if 567 was
divisible by 9, we did this:
Since 5 + 6 + 7 = 18, which is divisible by 9, then the original number, 567, is divisible by 9. In
fact, 567 = 9 ∙ 63.
RULE: If a number is divisible by 3, the sum of its digits is divisible by 3.
If a number is divisible by 9, the sum of its digits is divisible by 9.
The year this book was written is 2009. 2009 is not divisible by 3 because the sum of
the digits, 11, is not divisible by 3. On the other hand, the enrollment at RCC this year is
reported at the Web site StateUniversity.com as 4,341 students1. The number 4,341 is divisible
by 3 because 4 + 3 + 4 + 1 = 12. We also know that 4,341 is not divisible by 9 because the
sum of the digits, 12, is not divisible by 9.
1
*http://www.stateuniversity.com/universities/OR/Rogue_Community_College.html
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SECTION 3.2 Factors and Simplifying Fractions
 Check Point 4
For each number, check the boxes that apply.
Number
Divisible Divisible Divisible Divisible
by 2
by 3
by 4
by 9
a. 745
b. 288
c.
1,234,912
d. 435,678
e. 778,128
We have one more rule to consider. We know that 6 = 2 ∙ 3, and because 2 and 3 do
not share factors, we have a rule for 6.
RULE: If a number is divisible by both 2 and 3, then it is divisible by 6.
Just because a number is divisible by two factors, it does not always follow that it
is divisible by the product of those factors. For example, 2 is a factor of 12 and
4 is a factor of 12, and while 2 ∙ 4 = 8, the number 8 is not a factor of 12.
Because 4 has a factor of 2, a number divisible by 4 is not always divisible by 8.
 We do not include a rule for 8. It is much simpler to divide by 4 and then divide by 2.
 Similarly the rule for 7 is not as easy to remember. It is easier to simply divide by 7.
 If you would like to research the divisibility rules for 7, 11 and 13, you can find these on
 the Web.
We use these rules often when finding the prime factorization of a number. We factor
numbers into primes when it is the most efficient way to simplify fractions and find equivalent
fractions.
UNIT 3: Prealgebra in a Technical World
Prime Factors
DEFINITION: A prime number has exactly two factors: one and itself.
Finding the prime factors of a number can be useful in solving problems and simplifying
answers. (7, 13 and 19 are primes.)
The remaining counting numbers are called composites. Composite numbers have at
least three factors. (8, 9, 10 and 12 are composites.)
EXAMPLE 4: Determine which numbers are prime and which are composite.
a. 2 is prime because its only factors are 1 and 2.
b. 3 is prime because 1 and 3 are its only factors.
c. 4 is composite because it has the factor 2.
d. 5 is prime because 1 and 5 are its only factors.
 Check Point 5
Determine which numbers are prime and which are composite.
a. 6 is ___________________
b. 7 is ___________________
c. 8 is ___________________
d. 9 is ___________________
The Sieve of Eratosthenes (Era-tos-the-knees) is a “net” for catching primes. This
method for finding the primes was developed by the Greek mathematician Eratosthenes who
lived around 200 BCE.
To create a small Eratosthenes’ sieve and catch the prime numbers less than 100, follow the
directions here. When finished, you will have your own list of primes.
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SECTION 3.2 Factors and Simplifying Fractions
1. On the hundred table that follows, cross out 1 because it is neither prime nor composite.
2. Circle 2. This is the first prime number. Any other number that has 2 as a factor cannot be
prime, so cross out all of these numbers. We have started this step for you. Complete it!
3. The next number, three, is un-circled, and 3 is prime. Circle the 3 and cross out every third
number after 3 since these numbers have 3 as a factor.
4. The next number, 4, is crossed out since it has a factor of two. Thus 4 cannot be prime.
5. Five is not crossed out. Circle 5 and cross out all numbers that have a factor of 5.
6. Seven is not crossed out. Circle 7 and cross out all numbers that have a factor of 7.
7. Now that you have checked off all numbers that have factors of primes less than ten, the
remaining numbers on the hundred table are all primes. Circle them all.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
This last step is the “mathemagic” of Eratosthenes’ sieve. It is not magic at all, but it is
math! All of the composite numbers less than 100 must have at least one prime factor less
than 10. So all of the composite numbers less than 100 have been crossed out when we
checked these primes.
As you might guess here, the prime factors of a number are important. Every composite
number has a prime factorization, which is expressing a number as a product of prime factors.
UNIT 3: Prealgebra in a Technical World
For example the prime factorization of 12 = 2 ∙ 2 ∙ 3. We can choose to use exponents when
applicable, 12 = 22 ∙ 3.
To find the prime factorization of a number, we use our divisibility rules and a factor
tree. A factor tree is an algorithm, a step-by-step process, which produces the prime factors of
any number.
RULE: To find the prime factorization of a number by using a factor tree, follow
these steps:
1. Write the number to be factored into primes.
2. Directly below the number, write two factors that multiply to this number
and draw branches down to these numbers. (Do not write 1 and the
number—this will not help)!
3. For each of these two factors, if the factor is a composite, repeat Step 2. If
the factor is prime, circle it.
4. For new factors as they are written down, repeat Step 3.
5. When all of the branches of the tree end in circled factors, the factor tree is
complete. The circled primes, when multiplied together, are the prime
factorization of your number.
EXAMPLE 5: Write the prime factorization of 96.
Think it through: Use a factor tree to find all prime factors
Write the number to be factored first.
96 = 24 ∙ 4.
24 = 12 ∙ 2 and 4 = 2 ∙ 2. The 2s are prime.
12 = 4 ∙ 3. The 3 is prime
4 = 2 ∙ 2. The 2s are prime.
ANSWER: 𝟗𝟔 = 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟑 = 𝟐𝟓 ∙ 𝟑
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SECTION 3.2 Factors and Simplifying Fractions
EXAMPLE 6: Write the prime factorization of 168.
Think it through: Use a factor tree to find all the prime factors.
Write the number to be factored first.
168 = 21 ∙ 8.
21 = 3 ∙ 7 and 8 = 2 ∙ 4.
4 = 2 ∙ 2 , and each branch ends is a prime number.
ANSWER: 𝟏𝟔𝟖 = 𝟐 ∙ 𝟐 ∙ 𝟐 ∙ 𝟑 ∙ 𝟕 = 𝟐𝟑 ∙ 𝟑 ∙ 𝟕
The remarkable idea about factor trees and primes is that, at each step, you
get to choose the multiplication to use. It does not matter what choices you
make; in the end, we all have the same prime factorization!
 Check Point 6
Use a factor tree to find the prime factorization of these numbers. Write the prime
factorization as a product of its primes.
a. 60
b. 42
c. 81
We now have the tools we need to efficiently simplify fractions.
Simplifying Fractions
UNIT 3: Prealgebra in a Technical World
All of the previous circles have the same amount of shaded area. The only difference
among the three is how many equal size pieces have been cut in each circle. The simplest
diagram is the one with the fewest pieces cut. A simplified fraction is a fraction whose
numerator and denominator share no common factors other than one.
4
From the diagram above, we see that 12 =
2
6
=
1
3
. Only
1
3
is simplified. For the
2
4
fraction 6 , the numerator and denominator share a common factor of 2. For 12 the numerator
and denominator share a common factor of 4.
We have two methods for simplifying fractions that do not require us to draw circles!
METHOD 1: To simplify fractions divide common factors from numerator and
denominator until the numerator and denominator no longer share any common
factors.
EXAMPLE 7: Simplify
36
54
Think it through: Check divisibility rules for 36 and 54. The rule for 9 works!
36 ÷ 9
4
3 + 6 = 9 and 5 + 4 = 9, so divide both by 9.
=
54 ÷9
6
Because 4 and 6 share a factor of 2, divide both by 2:
4÷2
6÷2
=
2
3
The numerator 2 and the denominator 3 share no factors.
ANSWER:
𝟐
is the simplified fraction.
𝟑
EXAMPLE 8: Simplify
30
75
Think it through: The numerator ends in 5 and the denominator in 10, so divide both by 𝟓.
𝟑𝟎 ÷ 𝟓
𝟕𝟓 ÷𝟓
=
𝟔
𝟏𝟓
Since 6 and 15 are divisible by 3, divide both by 3:
6÷3
15 ÷ 3
The numerator 2 and the denominator 5 share no factors.
ANSWER:
𝟐
𝟓
is the simplified fraction.
=
2
5
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SECTION 3.2 Factors and Simplifying Fractions
Joshua thought through the same problem using prime factorization. He simplified
by writing the prime factorization first:
30
75
𝟐∙𝟑∙𝟓
2
= 𝟑∙𝟓∙𝟓 = 5 .
30
75
Using the prime factorization,
Joshua saved steps in simplifying, and he knows he has factored out all possible common
factors. Our second method is the one Joshua used.
METHOD 2: To simplify fractions write the prime factorization of both the
numerator and denominator. Then divide away all common prime factors. This
method is best used when numerators and denominators are so large that common
factors are hard to recognize.
Example 9: Simplify:
224
256
Think it through: Because both numbers are so large, find the prime factorizations of both
numbers. Use a tree for each:
224 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 7
256 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
224
𝟐∙𝟐∙𝟐∙𝟐∙𝟐∙𝟕
7
= 𝟐∙𝟐∙𝟐∙𝟐∙𝟐∙𝟐∙𝟐∙𝟐 = 8
256
ANSWER:
224
7
=8
256
 Check Point 7
Simplify. (HINT: Use prime factorization.)
a.
35
65
b.
18
48
c.
96
432
UNIT 3: Prealgebra in a Technical World
We finish this section by stating two properties we rely upon when we simplify fractions.
PROPERTIES: For all 𝑎, 𝑏 and 𝑐, with 𝑏 and 𝑐 not equal to 0,
𝑎𝑐
𝑏𝑐
For example:
28
70
÷
÷
𝑐
𝑐
14
14
=
=
𝑎
𝑏
2
5
and
and
𝑐
𝑐
14
14
=1
=1
Not only do we use factors to simplify fractions; many, many mathematical procedures
and applications, with and without fractions, depend on finding and using factors of a number.
You may spend quite a bit of time learning to think in terms of factors, but this will help you
with almost all of your future math work.
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SECTION 3.2 Factors and Simplifying Fractions
UNIT 3: Prealgebra in a Technical World
3.2 Exercise Set
Name _______________________________
Skills
Use the divisibility rules for the following. Justify your answer by stating why or why not.
1. Is 69 divisible by 3?
2. Is 108 divisible by 3?
3. Is 1063 divisible by 9?
4. Is 1364 divisible by 4?
5. Is 1845 divisible by 4?
6. Is 1338 divisible by 6?
For problems 7 and 8, circle the prime numbers.
7.
19, 36, 5, 62, 91, 12, 51, 21
8.
60, 53, 3, 85, 73, 35, 75, 97
For problems 9 and 10, circle the composite numbers.
9.
48, 23, 31, 42, 11, 57, 63, 7
10.
73, 49, 89, 74, 61, 24, 64, 34
Find all of the factors of the following numbers by making a two column list. For some you will
need to use your own paper. Make sure to label and attach your work.
11.
77
12.
126
13.
192
14.
50
15.
300
16.
90
17.
150
18.
24
19.
72
20.
96
21.
60
22.
240
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SECTION 3.2 Factors and Simplifying Fractions
Use a factor tree to find all of the prime factors. For some you will need to use your own paper.
Make sure to label and attach your work.
23.
77
24.
126
25.
192
26.
50
27.
300
28.
90
29.
150
30.
24
31.
72
32.
96
33.
60
34.
240
Simplify the following fractions if possible. If the fraction is already simplified state “simplified.”
35. 10
20
36.
8
12
37. 12
60
38.
10
24
39.
7
77
40.
17
51
41. 21
56
42. 15
65
43.
33
44
44. 24
32
45.
45
81
46. 18
48
47. 12
45
48.
21
56
49. 16
64
50.
48
64
51. 14
56
52. 24
80
53.
30
75
54. 27
51
UNIT 3: Prealgebra in a Technical World
Applications UPS
55. Hector has volunteered to set up folding chairs in the gym for a middle school band
concert. He finds that he has 216 chairs.
a. How many different ways can he arrange the chairs so that he has the same number of
chairs in a row? (Hint: Make a list and you will discover several ways.)
b. If the most efficient way to arrange the chairs is as close to a square as possible, which
way would be best?
56. Larry has 6 boxes of ceramic tile left over from a project. Each box has 20 pieces of tile.
He thinks that the leftover tile would look good on his patio. How many possible
rectangular patterns could he make with the leftover tile?
57. A friend has given Elisha 90 one-foot-square paving blocks that he had left over from a
patio he built. Elisha has decided to build a small patio for himself with these leftover
blocks. How many ways can he arrange the blocks to make a rectangular patio?
45
of the way through his math
58. Sam has completed 45 of 60 math problems, so he is
60
homework. What is the simplified form of this fraction?
59. Nora’s Internet friend found out she was taking prealgebra and sent her a math puzzle.
The puzzle asks, “What two numbers multiply to 32 and add to 12?” Nora decides to use a
factor table from this chapter. What are the two numbers?
60. Now that Nora (from the previous problem), knows how to solve factor puzzle problems,
she sends a problem back to her Internet friend, “What two numbers multiply to 120 and
add to 23?” Find these two numbers.
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SECTION 3.2 Factors and Simplifying Fractions
Review and Extend
Draw a diagram and use factor tables for problems 61 and 62.
61. A contractor is picking up molding and flooring for a rectangular room she is renovating.
She has jotted down the perimeter of the room, 40 feet, and the area of the room,
96 feet. What are the length and width of this room?
62. Carlo is planting grape vines for a small vineyard. He has acquired 245 plants and wants to
plant them in a rectangle to maximize irrigation.
a. If he keeps exactly the same number of plants in each row, what are the possible
numbers of rows he can plant?
b. Carlo will plant the grapes 6 feet apart in rows that are also 6 feet apart. He will fence
his vineyard so that he has a 6-foot walkway around each side of his plot of grapes.
What is the length and width of his plot?
What is the area that he will need to irrigate?
How much fencing will he need?
For problems 63 to 66, name the indicated measures on each scale. (Hint: not all are fractions!)
63.
From a micrometer:
64. From an insulin syringe:
?
?
65.
A different syringe:
?
3
3
66. A tape measure with different units:
12
13
?
4
4