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Fractions: Simplification,
Multiplication & Division
Lesson 1e
Fractions
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3 12
, ,
5 4
Numerals such as
and
called fractions.
Fractions have two parts:
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
A Numerator &
A Denominator
a
b
2
8
are
Numerator
Denominator
Since you cannot divide by zero, the denominator of a
fraction can never equal zero. However, a numerator can
equal zero.
Fractions
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Any fraction with the same nonzero
numerator and denominator equals 1.
a
1
a
22
1
22
223
1
223
Fractions
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Any fraction with a denominator of 1
names the same number as its
numerator.
22
 22
a
1
1
a
0
0
1
Product Rule for Fractions
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Multiply the numerators.
Multiply the denominators
“•” is a symbol for multiplication (X).
a c a  c ac
 

b d b  d bd
2 5 2  5 10
 

3 7 3  7 21
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Examples: Multiplying Fractions
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a
c aa cc
aac
c
Multiply the numerators.
   
b d bb dd bbd
d
Multiply the denominators
“•” is a symbol for multiplication (X).
Multiply:
Click here to check your answers!
3 1
1. 
4 5
2 4
2. 
3 5
3 7
3. 
5 8
1 3
4. 
2 8
5 1
5. 
6 3
9 3
6.

10 7
ac

bd
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Examples: Multiplying Fractions
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
a
c aa cc
aac
c
Multiply the numerators.
   
b d bb dd bbd
d
Multiply the denominators
“•” is a symbol for multiplication (X).
ac

bd
Multiply:
3 1
3
1.  
4 5 20
2 4
8
2.  
3 5 15
1 3
3
4.  
2 8 16
5 1 5
5.  
6 3 18
3 7 21
3.  
5 8 40
9 3 27
6.
 
10 7 70
Writing Fractions in Lowest
Terms
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3
6
2
4
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A fraction is in lowest terms when the
numerator and denominator have no
common prime factors.
In each of the fractions below, what
common prime factor do each the
numerator and denominator share?
The common factor
is 3
The common factor
is 2
5
10
9
12
The common factor
is 5
The common factor
is 3
Writing Fractions in Lowest
Terms: Example 1
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The first step in writing fractions in lowest
terms is to write the numerator and
denominator as a product of prime factors.
7
1
1
Write in lowest terms:


Solution:
 Write the prime factorization.
42
 Write as a product of two fractions.
 Any number multiplied by one is
equal to itself.
6
6
7
7

42 2  3  7
7 1
 
7 23
1
1
 1

6
6
Writing Fractions in Lowest
Terms: Example 2
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12
2
Write in lowest terms:

30
5
Solution:
 Write the prime factorization.
2

5
12 2  2  3

30 2  3  5
1
 Divide the numerator and the
denominator by common factors.
 Write in lowest terms.
1
223

235
1
2

5
1
2

5
Practice
Next
Click here to check your answers!
Write the following in lowest terms:
5
1.
20
9
2.
15
16
3.
36
Practice
Next
Write the following in lowest terms:
5
5
1.

20 2  2  5
1
9
33
2.

15
35
1
16 2  2  2  2
3.

36 2  2  3  3
1 1
5

225
33

35
1
2222

2233
1

4
3

5
4

9
1
1 1
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More on Multiplying Fractions
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Sometimes after multiplying fractions your
answer needs to be reduced to lowest terms.
For example:
3 4 33 4 12
12
 

8 5 88 5 40
40
1
12 3  4 3


40 28  5 10
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More on Multiplying Fractions
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
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To reduce fractions, you can often use a
shortcut when multiplying.
If possible reduce by dividing a denominator
and a numerator by the same number(a
common factor).
Then multiply. For Example:
1
5 and 10 have
a common factor
of 5.
7
5 7


8 102 16
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Examples: Multiplying Fractions
Multiply and then write your answers in lowest terms.
Click here to check your answers!
2 14
1. 
7 15
6 5
2. 
11 12
3 5
3.

10 6
6 14
4. 
7 15
7
5.  36
8
5 3 4
6.
 
12 20 7
Next
Examples: Multiplying Fractions
Multiply and then write your answers in lowest terms.
2
2 14 4
1. 

7 15 15
1
2
2
1
5
6 14 4
4. 

7 15 5
1
6 5
5
2. 

11 12 22
2
9
1
1
2
2
3 5
1
3.
 
10 6
4
1 1
7 36 63
1 15 3 4 1
5. 
  31 6.
  
8 1
2
2 412 204 7 28
2
1
Multiplying Mixed Numbers
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In order to multiply mixed numbers you
must change all mixed numbers to
fractions. For Example:
3 11
?
2 
4
2• 4 = 8 4
Multiply
the 4 & 5
Now add the 8 to
the numerator 3.
Now add the 20 to
the numerator 1.
Multiply
the 2 & 4
8 + 3 = 11
1 21
?
4 
4
4 • 5 = 20 5
20 + 1 = 21
Multiplying Mixed Numbers
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Now that we can change mixed numbers into
fractions, we can multiply mixed numbers.
First, we change the mixed number to a fraction.
 Next, we multiply the fractions as we have learned
previously.
Example 1:
Example 2:
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2
3 11

4 4
3 1
2   ?
4 6
11 1 11
 
4 6 24
1 1
2 5 
3 6
1 7
2 
3 3
?
1 31
5 
6 6
7 31 217
1
 
 12
3 6
18
18
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Examples: Multiplying Mixed Numbers
Multiply and then write your answers in lowest terms.
Click here to check your answers!
1 4
1. 9 
2 7
2 5
2. 1 
5 8
1
4. 5  4
3
3 1
5. 3  1
5 6
1 3
3. 3 
7 4
4 4
6. 2  3
5 7
Next
Examples: Multiplying Mixed Numbers
Multiply and then write your answers in lowest terms.
2
1 4 19 4
1. 9   
2 7 12 7
38
3

5
7
7
16 4
1
4. 5  4  
3 1
3
64
1

 21
3
3
1
2 5 7 5
2. 1   
5 8 15 8
7

8
3
3 1 18 7
5. 3  1  
5 6
5 61
21
1

4
5
5
11
1 3 22 3
3. 3  

7 4 7 42
33

14
2
5
4 4 14 25
6. 2  3  
5 7 1 5 71
10

 10
1
Dividing Fractions
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Now that you can multiply fractions, you can
learn to divide fractions.
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First, we change any mixed numbers to fractions.
The second step is to write the reciprocal of the fraction that
follows the division symbol (That fraction is called the divisor).
Lastly, we change the division to multiplication, and multiply the
fractions as we have learned previously.
Example 1:
3 2
11 22
2   ? 
4 5
4 55
7
11  52 55

6
 
8
8
4 25
Example 2:
4
4 8
8  ? 
5
5 1
1
1
4  18

 
5 812 10
Next
Examples: Dividing Fractions
Multiply and then write your answers in lowest terms.
Click here to check your answers!
5 2
1. 
6 3
10 5
2.

17 8
7
3. 10 
10
Next
Examples: Dividing Fractions
Multiply and then write your answers in lowest terms.
1
5 2
5 3
1. 
 
6 3 26 2
5

4
2
10 5 10 8
2.
  
17 8 17 51
16

17
7 10 10
3. 10   
10 1 7
100
2

 14
7
7
BACK
Definition: Reciprocal
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Two numbers whose product is 1 are
reciprocals.
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The reciprocal of  is .
  X  = 1
The reciprocal of 7 is .
 7 X  = 1
The reciprocal of 2 is 3/7
2 =    X 3/7 = 1
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