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Section 1.5 Reducing Fractions
Integers help describe whole units. Rational numbers help when describing a piece of a whole unit.
1.5.1 The Idea of a Rational Number
If your thermometer only had integers you could measure a temperature
of 98 or 99 but you couldn’t measure a temperature of 98.6. To measure the
temperature 98.6 we divide the distance between 98 and 99 into 10 parts of equal
length and then use 6 of the 10 parts or
6
6
parts. The number
is an example of a rational number.
10
10
Definition – Rational Number
English: A number that can be written as the quotient of two integers.
Example: 3 , 0.1 , 15%
5
Note: Rational numbers with a denominator of 0 are undefined.
Fractions are a common type of rational number. In this section we’ll begin reviewing operations
with fractions. Notice the integers are a subset of fractions since every integer can be written with a
denominator of 1. For instance the integer 3 can be written as the fraction
3
.
1
1.5.2 Prime Factorization
Many procedures with fractions rely on prime factoring. To discuss prime factoring we need a little
vocabulary. A prime number is a natural number, greater than 1, which only has factors 1 and itself. The
first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. A composite number is a natural number, greater
than 1, which is not prime. The first few composite numbers are 4, 6, 8, 9, 10, 12, and 14. You have prime
factored a number when the number is written as the product of prime factors. We consider 2  2  3 , the
prime factorization of 12 since the factors 2 and 3 are prime. We do not consider 2  6 a prime
factorization of 12 because 6 is composite.
To help organize a prime factorization it’s common to use the commutative property to write the
prime factors from smallest to largest.
Practice 1.5.2 Prime Factorization
Prime factor any composite factors and use the commutative property to write the
final prime factors from smallest to largest.
a) 3  6  2
3 23 2

Prime factored 6.
2 233

Used the commutative property to reorder the factors.
Homework 1.5 Prime factor any composite factors and use the commutative property to write the
final prime factors from smallest to largest.
1)
5  5  15
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2)
Scott Storla
4362
3)
14  6  3
1.5 Reducing Fractions
4)
15  6  8
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1.5.3 Factor Rules
1
It’s easy to look at 6 and realize it has prime factors 2 and 3. It’s harder to look at a number like 48
and do the same. Here are some helpful rules for prime factoring a number.
Factor Rules
2 is a factor if the number is even. Even numbers end in 0,2,4,6 or 8.
3 is a factor if the sum of the number’s digits is a multiple of 3.
5 is a factor if the number ends in 5 or 0.
Practice 1.5.3 Factor Rules
Decide if the following numbers have 2, 3, or 5 as factors.
a) 12345
2 is not a factor

Since the number does not end in 0,2,4,6 or 8 the number is not even.
3 is a factor

Since the sum of the digits is 15, 1 2  3  4  5  15 , and 15 is a multiple
of 3, so is 12345.
5 is a factor

12345 ends in 5 so it has a factor of 5.
2 is a factor

Since the number is even, it ends in 0, two is a factor.
3 is a factor

The sum of the digits is 3, 1 2  0  3 , so three is a factor.
5 is a factor

120 has a factor of five since it ends in 0.
b) 120
Homework 1.5 Decide if the following numbers have 2, 3, or 5 as factors.
5)
6)
18
7)
540
385
8)
119
1.5.4 Factor Trees.
Sometimes a prime factorization will “pop into your head.” A factor tree can help if nothing “pops”.
For instance say I wanted to prime factor 20. I might start by using the factors 5 and 4
Then I’ll factor 4.
Next I’ll circle 5 since 5 is prime.
and circle all prime factors
Since there are no composite factors left the prime factorization of 20 is 2  2  5 .
again.
Instead of starting with 5 and 4 what if I had started with 2 and 10?
1
These are also known as divisibility rules.
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Scott Storla
1.5 Reducing Fractions
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Factored 20 into 2 and 10.
Factored
Circled all factors that are prime.
the composite factor.
and circled all prime factors again.
Again the prime factorization is 2  2  5 . This idea, that every natural number greater than 1 has a unique
prime factorization, is often called the Fundamental Theorem of Arithmetic.
The Fundamental Theorem of Arithmetic
Every natural number, greater than 1, has a unique prime factorization.
Example: 20  5  4  2  10  2  2  5
Here’s another example. When I see 120 I think of 12 times 10
so that’s how I’ll start. The prime factorization for 120 is 2  2  2  3  5 .
Sometimes a composite number doesn’t have a factor of 2, 3 or
5. For instance 119 has prime factorization 7  17 . Here’s a procedure
that will help you find all prime factorizations.
Procedure – To Prime Factor a Natural Number
1. Build a factor tree using the factor rules for 2,3,and 5.
2. After step 1 look for other prime factors by dividing uncircled factors by the
prime numbers 7 up to the square root of the number.
3. Write your prime factors in order from smallest to largest.
Homework 1.5 Prime factor the following numbers.
9)
24
11)
10) 42
80
13) 102
12) 91
14) 429
1.5.5 Reducing Fractions Using the Multiplicative Identity
A fraction is reduced when the numerator and denominator have no common factors (other than
1). If the numerator and denominator do have a common factor we can reorder and regroup the factors to
reduce using the multiplicative identity. For instance to reduce
regroup the factors,
2 3
3
 , notice that the first factor is really a factor of one, 1 and use the multiplicative
2 5
5
identity to simplify the product to
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23
23
I could reorder the factors,
,
52
25
Scott Storla
3
.
5
Here’s some practice.
1.5 Reducing Fractions
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Practice 1.5.5 Reducing Fractions Using the Multiplicative Identity
Reduce using the multiplicative Identity
a)
12
18
223
233

Prime factored the numerator and denominator.
2 3 2
 
2 3 3

Reordered and regrouped factors.
2
3

Rewrote factors of 1.
2
3

Multiplied.
1 1
b) 56  42
1 2  2  2  7
237

Prime factored the numerator and denominator. It’s useful to think of a
factor of 1 in the numerator.
2 7 1 2  2
 
2 7
3

Reordered and regrouped the factors.
1 2  2
4

3
3

Rewrote as factors of 1 and multiplied. Whether the factor of 1 is
written in the numerator or in front of the fraction the value of the
fraction is the same.
2 23
22335

Prime factored the numerator and denominator.
2 2 3
1
  
2 2 3 35

Reordered and regrouped the factors. Made sure to leave a factor of 1
in the numerator.
1
1

3  5 15

Rewrote as factors of 1 and multiplied.
1 1
c)
12
180
1 1 1
Homework 1.5 Reduce using the multiplicative identity.
15)
52
36
16) 16  20
17)
42
36
18) 63  84
19)
56
120
20)
105
126
1.5.6 Reducing Fractions
Instead of rewriting fractions, and using the multiplicative identity, people often “cancel” the
common factors in the numerator and denominator. Although this does work, please be aware that you are
actually identifying, and reducing, common factors of 1. Here’s a helpful procedure.
Procedure – Reducing Fractions
1. Prime factor the numerator and denominator.
2. Reduce all common factors to a factor of 1.
3. Multiply the remaining factors in the numerator or denominator together.
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Scott Storla
1.5 Reducing Fractions
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Practice 1.5.6 Reducing Fractions
Reduce using prime factorization.
36
30
a)
1 2  2  3  3
235

Prime factored the numerator and denominator. It’s useful to think of a
factor of 1 in the numerator.
1 2  2  3  3
2  3 5

Reduced common factors of 1. Usually we don’t actually move them, we
just reduce them or “cancel” them.
6
5

Multiplied the factors left in the numerator and denominator.
b) 18  54
233
2333

Prime factored the numerator and denominator.
233
1

2  3  3 3 3

Reduced and multiplied. Remembered to include a factor of 1 in the
numerator since factors are reduced to a factor of 1, not 0.
Homework 1.5 Reduce using prime factorization.
21)
28
26
22) 16  20
27)
143
165
28) 
Homework 1.5
1) 3  5  5  5
42
36
23)
221
153
29) 
2 2 2 233
2)
24) 84  294
3)
2 2337
2 and 3 are factors.
6)
2,3,and 5 are factors.
8)
All three are not factors.
9)
2 2 23
13) 2  3  17
17)
7
6
18)
3
4
19) 
22) 
4
5
23)
7
6
24)
2
7
25)
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29) 
315
225
2 2 2 2335
7)
5 is a factor.
11) 2  2  2  2  5
14) 3  11 13
4
5
13
9
4)
10) 2  3  7
16) 
28)
26)
72
792
5)
12) 7  13
105
126
25)
7
15
5
6
15)
13
9
20)
5
6
21)
14
13
26)
7
5
27)
13
15
1
11
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1.5 Reducing Fractions
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