LESSON 2: Adding and Subtracting Fractions – A Quick Review
NUMERATOR
= Numerator
DENOMINATOR
Denominator
In a fraction, the denominator (bottom) tells us how many equal parts the whole is divided into, and the
numerator (top) tells us how many of those parts we have. Suppose you had a pizza that was evenly
sliced into eight parts. Then five of those parts were taken. You end up with three parts remaining.
3
The fraction of the pizza remaining is three parts out of eight (three-eighths):
8
Using whole numbers we say: 8 slices
– 5 slices
= 3 slices
Each slice is 1/8 pizza, so:
8(1/8ths) – 5(1/8ths) = 3(1/8ths) pizza
8
5
3
With fractions we say:
pizza pizza =
pizza
8
8
8
Note: Subtracting fractions with the same denominator is like subtracting numbers with the same units.
Now, suppose one slice was returned and the remaining pizza was to be placed in the refrigerator.
Using whole numbers we say: 3 slices + 1 slice
= 4 slices
Each slice is 1/8 pizza, so:
3(1/8ths) + 1(1/8ths) = 4(1/8ths) pizza
3
1
4
With fractions we say:
pizza +
pizza =
pizza
8
8
8
Note: Adding fractions with the same denominator is like adding numbers with the same units.
When asked how much of the pizza is in the refrigerator we could say 4 pieces, 4/8ths, or ½ pizza.
The answer of ½ pizza is better than 4/8 pizza because ½ is easier for most people to visualize. For
now, we will not concern ourselves with simplifying a result. That is covered at the end of this lesson.
8 5 1
8 5 1
4
- + =
=
8 8 8
8
8
We could summarize as follows:
The above example demonstrates that dealing with “Like Fractions” makes counting them easy.
Like fractions are fractions with the same denominator. You can easily add and subtract like fractions
- simply add or subtract the numerators and write the result over the common denominator.
More examples:
5
4
9
+
=
14 14 14
Now you try it:
7
3
2
5
13 5
73 25
+
= --- ;
+
= --- ;
= --- ;
= --12 12
6
6
16 16
80 80
;
7
4 11
14 8
6
+
=
;
=
9
9
9
15 15 15
;
41 19
22
=
48 48
48
Answers: 10/12 7/6 8/16 48/80
Now consider problems such as the following:
1
2
+
4
3
= ?
or
7
1
12
3
= ?
The first step in adding or subtracting fractions with different denominators is to rewrite each
fraction as an equivalent fraction in larger terms, such that both fractions then have the same
denominator. After we add (or subtract), sometimes the answer can be reduced to smaller terms.
We next address equivalent fractions.
E 120 Applies to: Exam 2 – Items 1, 2, 3, 4, 6, 7, 8, and 10. Secondary Exam 2 – Questions 1, 2, 3, 4, 6, 7, 8, 9, and 11.
1 2 3
4
, , , and
2 4 6
8
Equivalent fractions are fractions that represent the same value such as
1 of 2
2 of 4
3 of 6
4 of 8
How to express a fraction in larger terms: The value of a fraction is not changed if both its
numerator (top) and denominator (bottom) are multiplied by the same number.
1
1 2
2
=
=
2
4
2 2
Examples:
1
1 3
3
=
=
2
2 3
6
;
1
1 4
4
=
=
2
8
2 4
;
How to express a fraction in smaller terms: The value of a fraction is not changed if both its
numerator (top) and denominator (bottom) are divided by the same number.
2
2 2
1
=
=
4
4 2
2
Examples:
;
3
3 3
1
=
=
6
2
6 3
;
4
4 4
1
=
=
8
8 4
2
1
1 3
3
=
=
4
4 3
12
;
3
3 3
1
=
=
12
12 3
4
More examples of equivalent fractions:
2
2 4
8
=
=
3
12
3 4
(Note: 12
;
3 = 4)
1
1 4
4
=
=
3
3 4
12
(Note: 12
;
3 = 4)
(Note: 12
4 = 3)
(Note: 12
4 = 3)
Now you try it:
3
3
=
-- =
8
8
24
(Hint: 24
;
8 = 3)
2
2
=
-- =
3
3
15
(Hint: 15
3 = 5)
;
5
5
=
-- =
6
6
24
(Hint: 24
18
18
=
24
24
;
6 = 4)
(Hint: 24
-- =
4
4 = 6)
Answers: 9/24 10/15 30/24 3/4
To add or subtract two fractions it may be necessary for then to be expressed in larger terms because
they must have a common denominator, or in other words, the same denominator.
1
2
+
4
3
3
8
+
12
12
=
=
11
12
7
1
12
3
=
7
4
12
12
=
3
12
OR
1
4
Again, we can only add or subtract like quantities. In this case we were working with 1/12ths.
Methods of finding a common denominator are discussed next.
E 120 Applies to: Exam 2 – Items 1, 2, 3, 4, 6, 7, 8, and 10. Secondary Exam 2 – Questions 1, 2, 3, 4, 6, 7, 8, 9, and 11.
We can easily add or subtract like fractions (fractions with a common denominator).
A common denominator is a number that each denominator will go into evenly. A common
denominator can be found by simply multiplying all of the denominators, but it is best to work with the
lowest value we can which we call the least common denominator (Finding the L.C.D.is discussed on the next page).
Consider
1
1
1
+ +
. Since 2 x 3 x 12 = 72, then 72 is a common denominator.
2
3 12
1
1
1
+
+
2
3
12
1
1 36
36
=
=
2
2 36
72
;
1
1 24
24
=
=
3
3 24
72
1
1 6
6
=
=
12 12 6
72
;
We rewrite all three fractions as equivalent fractions having the common denominator.
=
36
24
6
+
+
72
72 72
=
66
72
66 6
=
72 6
=
11
12
(Note: We had to reduce the answer to smallest terms.)
The addition is easier by using the least common denominator (the L.C.D.) Note that 12 is the lowest
multiple of all three denominators, so the L.C.D. = 12. (12 is a multiple of 2, a multiple of 3, and 12)
1
1 6
6
=
=
2
2 6 12
1
1
1
+
+
2
3
12
;
1
1 4
4
=
=
3
3 4 12
;
1
is fine as it is
12
We rewrite all three fractions as equivalent fractions having the common denominator.
6
4
1
11
=
+
+
=
12 12 12
12
(Note: We did not have to reduce the answer.)
Now you try it: For now we will give you the L.C.D. to use for each problem.
1
3
+
(LCD = 8)
2
8
=
8
+
18
+
18
=
72
-
72
=
2
2
=
-- =
9
9
18
;
1
1
=
-- =
6
6
18
5
5
=
-- =
8
8
72
;
1
1
=
-- =
9
9
72
18
5
1
(LCD = 72)
8
9
=
3
is fine as it is
8
;
3
=
8
8
2
1
+
(LCD = 18)
9
6
=
1
1
=
-- =
2
2
8
72
ANSWERS: 7/8 7/18 37/72
E 120 Applies to: Exam 2 – Items 1, 2, 3, 4, 6, 7, 8, and 10. Secondary Exam 2 – Questions 1, 2, 3, 4, 6, 7, 8, 9, and 11.
Finding the Least Common Denominator (L.C.D.)
Method 1: By inspection.
Please review the method by inspection on textbook (lessons 1-3) pages 102 through 106.
Note: The L.C.D. is always the largest denominator multiplied by a whole number such as 1, 2, 3, …
Basically, we are trying successively larger multiples of the largest denominator (as a guess) and then
seeing if the guess is evenly divided by the other denominator.
Example:
3
1
and
8
6
Try 8 x 1 = 8, does 8 divide evenly by 6?  NO
Try 8 x 2 = 16, does 16 divide evenly by 6?  NO
Try 8 x 3 = 24, does 24 divide evenly by 6?  YES, so L.C.D. = 24
Now you try it:
2
4
and
L.C.D. = ___
3
12
5
1
and
L.C.D. = ___
9
6
3
2
and
L.C.D. = ___
5
7
Answers: 12 18 35
Method 2 (New): Inspection with multiplication table assistance.
Please open your textbook (lessons 1-3) to page 32. The multiplication table can always help when
working with two fractions with denominators of 12 or less.
Basically, we are looking at successively larger multiples of the largest denominator (as a guess) and
then seeing if the guess is one of the multiples of the other denominator.
3
1
and
8
6
Is this a multiple of 6? 
Is this a multiple of 6? 
Is this a multiple of 6? 
Example:
Find the lowest multiple of 8 that is also a multiple of 6.
1x8=8
NO
2 x 8 = 16 NO
3 x 8 = 24 YES, so L.C.D. = 24
4 x 8 = 32
5 x 8 = 40
1x6=6
2 x 6 = 12
3 x 6 = 18
4 x 6 = 24
5 x 6 = 30
Now you try it using the multiplication tables (page 32):
2
4
and
L.C.D. = ___
3
12
5
1
and
L.C.D. = ___
9
6
3
2
and
L.C.D. = ___
5
7
Answers: 12 18 35
We can often use the multiplication tables to handle several small denominators.
5
1
1
Consider and
and
9
6
4
1x9=9
1x6=6
1x4=4
2 x 9 = 18
2 x 6 = 12
2x4=8
3 x 9 = 27
3 x 6 = 18
3 x 4 = 12
4 x 9 = 36
4 x 6 = 24
4 x 4 = 16
The L.C.D. is 36
5 x 9 = 45
5 x 6 = 30
5 x 4 = 20
6 x 9 = 54
6 x 6 = 36
6 x 4 = 24
7 x 9 = 63
7 x 6 = 42
7 x 4 = 28
8 x 8 = 72
8 x 6 = 48
8 x 4 = 32
9 x 9 = 81
9 x 6 = 54
9 x 4 = 36
E 120 Applies to: Exam 2 – Items 1, 2, 3, 4, 6, 7, 8, and 10. Secondary Exam 2 – Questions 1, 2, 3, 4, 6, 7, 8, 9, and 11.
Methods 1 and 2 are most useful with two denominators or three relatively small denominators.
5
9
6
Could you find the L.C.D. of
and
and
with methods 1 or 2 ?
14
20
35
Method 3: A general method for finding the least common denominator.
Please review the general method on textbook (lessons 1-3) pages 107 through 112.
Basically, we note how many times the denominators can be evenly divided by 2, then 3, then 5, etc.
The divisors to try in order are 2, 3, 5, 7, 11, 13, 17, 19, 23
Step 1: Write the denominators in a row (with proper spacing).
Step 2: Draw a division line under the row of numbers like this: )________
Step 3: Choose the smallest divisor from the above list so that at least one number divides evenly.
Step 4: Write the divisor on the left.
Step 5: For each number that divides evenly, writes the result below the line.
If a number does not divide evenly, just copy that number below the line.
Step 6: We now have a new row of numbers below. If they are not all 1’s then go back to step 2.
Step 7: Multiply all of the divisors used (the numbers on the left). This is the L.C.D.
__________________________________________________________________________________
1
3
1
1
1
Example: and
2)6 8
Example: and and
2 ) 2 3 12
6
8
2
3
12
2)3 4
2)1 3 6
2)3 2
3)1 3 3
2 x 2 x 2 x 3 = 24
3)3 1
2 x 2 x 3 = 12
1 1 1
so L.C.D = 24
1 1
so L.C.D = 12
__________________________________________________________________________________
5
4
16
14
Example:
and
2 ) 12 21
Example:
and
3 ) 39 63
12
21
39
63
2 ) 6 21
3 ) 13 21
3 ) 3 21
7 ) 13
7
2 x 2 x 3 x 7 = 84
7) 1
7
3 x 3 x 7 x 13 = 819
13) 13
1
so L.C.D = 84
1
1
so L.C.D = 819
1 1
__________________________________________________________________________________
Now you try it.
1
3
5
Find the L.C.D. of
and and
9
8
12
We set up the first row for you.

) 9 8 12 (Place a 2 to the left in this line)
Divide the above row by 2 (where possible) 
)
(Bring down the 9, a 4, and a 6)
Divide the above row by 2 (where possible) 
)
(Bring down the 9, a 2, and a 3)
Divide the above row by 2 (where possible) 
)
(Bring down the 9, a 1, and a 3)
Divide the above row by 3 (where possible) 
)
(Bring down a 3, a 1, and a 1)
Divide the above row by 3 (where possible) 
(You should now have all 1’s)
Multiply 2, 2, 2, 3, and 3 to get the L.C.D. 
L.C.D. = ___
Consider
1
14
and
6
45
) 6 45
)
)
)
Consider
5
9
6
and
and
14
20
35
_
_
_
) 14 20 35
)
)
)
L.C.D = ___
_
_
_
L.C.D = ___
Answers: 72 90 140
E 120 Applies to: Exam 2 – Items 1, 2, 3, 4, 6, 7, 8, and 10. Secondary Exam 2 – Questions 1, 2, 3, 4, 6, 7, 8, 9, and 11.
(Optional): Results from adding or subtracting fractions should be expressed in simplest form.
A fraction is simplified when: 1) The fraction is expressed in smallest terms (lowest denominator).
2) An improper fraction is converted into a mixed number.
2
4
A proper fraction is a fraction whose numerator is less than its denominator such as
or
3
9
3
4
In an improper fraction the numerator is greater than or equal to the denominator such as
or
2
4
Note: An improper fraction has a value greater than or equal to one.
2
5
or 2
3
6
In lesson 3, simplified answers are stressed. In lesson 2, we concentrate on the basic operations.
A Mixed number is a whole number and a fraction, such as 1
More examples of adding and subtracting fractions:
2
1
+
(LCD = 12 by inspection)
3
12
=
=
5 2
(LCD = 8 x 9 = 72)
8 9
2 4
1
+
3 4 12
=
9
3
8
1
9 3
+
=
=
=
12
4
12
12
12 3
1
4
+
2
5
+
45
72
=
5
18
5 9 2 8
8 9 9 8
-
16
72
=
29
72
2 ) 4 5 18
2)2 5 9
1 45
2 36
5 10
=
+
+
4 45
5 36 18 10
3)1 5 9
3)1 5 3
=
167
45
72
50
+
+
=
180
180 180 180
5)1 5 1
1 1 1
L.C.D. = 2 x 2 x 3 x 3 x 5 = 180
Now you try it: Complete the following problems on scratch paper.
A:
1
3
+
= ?
6
8
E:
1
2
3
+
+
= ?
2
3
4
H: Add
B:
5
9
6
and
and
14
20
35
1
2
+
= ?
2
5
F:
C:
11
1
= ?
12
3
1
3
5
+
+
= ?
9
8
12
Answers: A = 13/24
G:
B = 9/10
E = 23/21 or 1 11/12
D:
5
4
= ?
12
15
1
1
7
+
= ?
2
3
12
C = 7/12
F = 65/72
D = 9/60 or 3/20
G = 3/12 or 1/4
H = 137/140
E 120 Applies to: Exam 2 – Items 1, 2, 3, 4, 6, 7, 8, and 10. Secondary Exam 2 – Questions 1, 2, 3, 4, 6, 7, 8, 9, and 11.