1
Fractions
Adding & Subtracting
A fraction is a noninteger quantity expressed in terms of a numerator and a denominator.
1. FRACTION DEFINITIONS
1) Proper fraction: numerator is less than the denominator.
1 2
,
2 3
2) Improper fraction: numerator is greater than or equal to denominator.
5 9
,
3 9
3) Mixed number: whole number and a fraction.
1 4
2 ,5
3 9
4) Equivalent fractions: fractions that represent the same number.
1 2 3
, ,
3 6 9
5) Reciprocal: the multiplicative inverse of a number. For a fraction,
1
and10 it's obtained by "turning the fraction over".
10
5
6 1
and
and10
6
5 10
Any whole number can be made into a fraction by placing it over 1.
Example:
10 =
10
1
2 a. COMMON DENOMINATORS
The bottom number of a commonly written fraction is called the denominator.
When adding fractions with the same denominators,
you add the numerators to arrive at the sum. The denominators remain the same
and only the numerators are added.
Example
3
2
5
+
=
7
7
7
Since the denominators are both 7, you add the numerators 3 + 2 = 5 and the sum of the
fraction is
5
. If the numerator is greater than the denominator, the fraction is called an
7
improper fraction and can be simplified to a mixed number. Mixed numbers contain
both an integer and a fraction.
2
Example
4
7 11
+
=
9
9
9
This can also be written as 1
2
. This is calculated by removing whole numbers from
9
the improper fraction.
11
9 2
2
can be written as
+
or 1
9
9 9
9
If the denominators of the fractions you are adding are different, an operation must be
performed before the fractions can be added. The denominators must be manipulated
so that they are the same.
Example
To add
3
4
and , you have to find an equivalent denominator for both fractions. Before
5
7
you can add or subtract fractions with different denominators, you must first find
equivalent fractions with the same denominator. Multiply the denominator and use the
product as the common denominator. Find the smallest multiple of both numbers.
5 x 7 = 35
( 3)( 7 )
( 5)( 7 )
+
( 4 )( 5)
( 5)( 7 )
=
21 20
+
35 35
41
=
35
=
Multiply the numerator and denominator by the same number.
Perform the multiplication and add the numerators.
This is an improper fraction and should be simplified
by removing the whole numbers to get
35
35
=1
6
35
+
6
35
=1+
6
35
Answer
b. LOWEST COMMON DENOMINATOR
Follow these steps to add fractions with different denominators.
1
1
+
2
4
1
2
as
so it has the same denominator as the
The first step is to express
2
4
2
1
second fraction. The addition is now
+
4
4
2
1
3
Add the fractions, now that they have the same denominator
+
=
4
4
4
Example We need to add
3
c. SIMPLIFICATION
If the numerator and denominator can both be divided by the same factor,
the fraction can be simplified.
Example
( 3)( 3)
9
can be written
15
( 3)( 5 )
If both the numerator and denominator are divided by the common factor, then the
fraction becomes
3
5
3. ADDING TWO POSITIVE FRACTIONS
When adding two positive fractions, the sum will be positive and larger than each
individual fraction. If you were to eat
1
3
of a pizza on Monday and
of the pizza
5
5
on Tuesday, by Wednesday you will have eaten more pizza than on either day.
Example
1 3 4
+
=
of the pizza.
5 5 5
4. ADDING MORE THAN TWO POSITIVE FRACTIONS
When adding more than two positive fractions, the sum will be positive
and larger than each individual fraction. The order does not matter when
adding two more than two positive fractions.
Example
3
6
9
2
20
2
3
9
6
20
+
+
+
=
or
+
+
+
=
33 33 33 33 33
33 33 33 33 33
5. ADDING TWO NEGATIVE FRACTIONS
When adding two negative fractions, the sum will be negative and smaller
than either fraction.
Example (-
4
3
7
) + (- ) =( - )
10
10
10
6. ADDING MORE THAN TWO NEGATIVE FRACTIONS
4
When adding more than two negative fractions, the sum will be negative
and smaller than each of the numbers.
3
8
7
4
22
) + () + () + () =()
47
47
47
47
47
Example (-
7. SUBTRACTING TWO NEGATIVE FRACTIONS
When subtracting two negative fractions, the sign following the subtraction sign
(-) must be changed.
Example (-
8
5
8 5
3
1
) – (- ) = - +
= - which can be simplified to –
9
8
9 9
9
3
The negative (-) sign in front of the
5
was changed to a positive (+) sign.
9
8. SUBTRACTING TWO POSITIVE FRACTIONS
When you subtract two positive fractions, the sign following the subtraction sign
(-) must be changed.
Example (+
5
1
5
1
4
) – (+ ) =
–
=
7
7
7
7
7
The positive (+) sign in front of the
1
was changed to a negative (-) sign.
7
9. SUBTRACTING FRACTIONS WITH MIXED SIGNS
When you subtract two fractions with mixed signs, the sign following the
subtraction sign (-) must be changed.
Example (-
9
6
9
6
15
) – (+
)=–
–
= –
17
17
17 17
17
Fractions Multiplying
1. MULTIPLYING FRACTIONS
When multiplying fractions, look for common factors in the numerators. If there is a
common factor in the numerators and denominators, dividing both by the factor will
simplify the question.
5
3 4 1
x x
1 9 2
Example
= 3 x (2 x 2) x 1
1 x (3 x 3) x 2
=
2
3
In this example there is one factor of 2 and one factor of 3 common to both numerator
and denominator. Dividing the numerator and denominator by
(2 x 3) produced
2
as the product. If there are negative fractions, use the negative sign
3
with a factor in either the numerator or denominator.
2. MULTIPLYING TWO POSITIVE FRACTIONS
When multiplying two positive fractions, the product is always positive and larger than
either of the factors. To arrive at the product, multiply the numerator by the numerator,
and the denominator by the denominator.
Example (+
5
8
40
5
8 40
) x (+ ) = (+
) or
x
=
11
9
99
11 9 99
3. MULTIPLYING MORE THAN TWO POSITIVE FRACTIONS
When you multiply more than two fractions, the product is always positive and larger
than each of the factors. To arrive at the product, multiply the numerator by the
numerator and the denominator by the denominator.
Example
2 × 3 × 3 × 5 15
5
=
=
which is simplified to lowest terms.
2 × 3 × 3 × 11 33 11
Divide the numerator and denominator by 2 x 3 x3 or 18
4. MULTIPLYING TWO NEGATIVE FRACTIONS
When multiplying two negative fractions, the product is always positive and larger than
either of the factors. To arrive at the product, multiply the numerator by the numerator,
and the denominator by the denominator.
6
Example ( -
7
3
21
) x ( – ) = (+
)
8
5
40
5. MULTIPLYING MORE THAN TWO NEGATIVE FRACTIONS
When multiplying more than two negative fractions, the product is negative if there is an
odd number of factors, and positive if there is an even number of factors. To arrive at
the product, multiply the numerator by the numerator and the denominator by the
denominator.
Example (-
3
2
5
30
5
)x
x – =
=–
4
3
6
72
12
There are three factors and 3 is an odd number
-
30 6
5
÷
=72 6
12
Divide the numerator and denominator by 2 x 3 or by 6.
6. MULTIPLYING FRACTIONS WITH MIXED SIGNS
When multiplying a series of fractions, the product is negative if the number of negative
factors is odd, and positive if the number of negative factors is even.
Example (-
3
5
2
5 150
) x (+ ) x ( – ) x
=
5
3
3
2
90
There are two negative factors (-
3 2
- ) and two is an even number.
5 3
150
is an improper fraction, which can be written as
90
( 2 × 2 × 3 × 5) = 1
60
which can be simplified as 1
1
90
( 2 × 3 × 3 × 5)
90 60
+
or as
90 90
2
3
Divide the numerator and denominator by 2 x 3 x 5 or 30.
Fractions Dividing
1. DIVIDING TWO POSITIVE FRACTIONS
To divide one fraction by another, invert the second fraction and multiply. When dividing
one positive fraction by another positive fraction, the quotient is positive.
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Example
2 3
3
4
to get
and then multiply.
÷ , invert
3 4
4
3
2 4 8
× =
3 3 9
5 3
÷
6 4
is
5 4 20
10
, which is equivalent to
.
× =
6 3 18
9
 4
+ 
 7
2. DIVIDING TWO NEGATIVE FRACTIONS
When you divide one negative fraction by another negative fraction, the quotient is
positive.
 6  3
 7  2
 6   2  12
, which is equivalent to
 7   3  21
Example  −  ÷  −  is  −  ×  −  =
 4
+ 
 7
3. DIVIDING FRACTIONS WITH MIXED SIGNS
When you divide one fraction by another, if one is positive and the other is negative, the
quotient is negative.
Example
8  4
8  7   56   14 
 2
÷  −  is ×  −  =  −  =  −  , which is equivalent to  −4 
3  7
3  4   12   3 
 3