Finding the (Least) Common Denominator
This is a free lesson about finding a common denominator in fraction addition. The common denominator has to be a multiple of each of the denominators. The least common multiple of the denominators is the smallest possible common denominator, but it is not the only possible one. The lesson is meant for fifth grade.
Before adding or subtract unlike fractions, first
convert them into like fractions. 
When we add unlike fractions, we need to know
into what kinds of parts to convert them so that the converted fractions
will have the same denominator (or be like fractions).
We call this same denominator
the common denominator because all of the converted fractions
will
have this same denominator in common.
To do the actual conversion, use the principles for writing equivalent
fractions. 
Example: 
1
6 
+ 
5
9 



↓ 

↓ 



3
18 
+ 
10
18 
= 
13
18 

We use 18 as the common denominator. Why 18? You will find
out
soon, on the
next page. For now, notice this:
1/6 is converted into 3/18 and
5/9
is converted into 10/18
using equivalent fractions:Lastly, we add
3/18 and 10/18. 



1. You are given the common denominator. Convert the fractions using
the rule for writing equivalent
fractions. Then
add or subtract. Note: sometimes you need to convert only one fraction, not both.
a. 
1
3 
+ 
3
5 


↓ 

↓ 


15 
+ 
15 
= 

b. 
6
7 
− 
1
2 


↓ 

↓ 


14 
− 
14 
= 

c. 
1
6 
+ 
2
5 


↓ 

↓ 


30 
+ 
30 
= 



f. 
5
7 
− 
2
3 


↓ 

↓ 


21 
− 
21 
= 

g. 
2
5 
+ 
1
4 


↓ 

↓ 


20 
+ 
20 
= 

h. 
5
6 
− 
3
4 


↓ 

↓ 


12 
− 
12 
= 

i. 
3
4 
− 
3
7 


↓ 

↓ 


28 
− 
28 
= 

The
common denominator has to be a multiple of each of the
denominators.
In other
words, the common denominator has to be in the multiplication table of the individual
denominators.
Or, the common denominator has to be divisible by the individual denominators.
The individual denominators have to each “go into” the common denominator,
just like 5 goes
into 30.

Examples:


The common denominator must be a multiple of 5 and also a
multiple
of 3. Fifteen will work: it is in the multiplication table of 5 and also
of 3. 



Check the multiples of 8 (the skipcounting list): 0, 8, 16, 24, 32, etc.
Compare to the multiples of 6: 0, 6, 12, 18, 24, 30, etc.
We notice that
24 is the smallest number that is in both lists. 



We need a number that 4 can “go into” and that 8 can “go into.”
Actually, the smallest such number is 8 itself. So in this case, the 7/8
does not
need to be converted at all; you just convert the 3/4 into 6/8. 

2. Find a common denominator
(c.d.) that will work with these fractions.
fractions to add/subtract 
c.d. 
a. 4th parts and 5th parts 

b. 3rd parts
and 7th parts 

c. 10th parts and 2nd parts



fractions to add/subtract 
c.d. 
d. 4th parts
and 12th parts


e. 2nd parts
and 7th parts 

f. 9th parts and 6th parts 


3. Let’s add and subtract. Use the common denominators you
found above.
a. 
4
5 
+ 
1
4 


↓ 

↓ 


20 
+ 
20 
= 






You can always multiply
the denominators to get a common denominator. However, you can often
find a smaller number
than the denominator you get by multiplying the denominators.

7
10 
and 
1
15 

You could use 10 × 15 = 150,
but let’s look at the lists of multiples: Multiples of 10: 0, 10,
20, 30, 40, 50, ...
Multiples of 15: 0, 15, 30, 45, 60, 75 ...
So, 30 works as well, and it is smaller!
It is the
least (smallest) common denominator. 





2
7 
and 
1
6 

One possibility is 7 × 6 = 42, but
let’s check the multiples of 6 to make sure: Multiples of 6: 0, 6,
12, 18, 24, 30, 36, 42, 48, ...
None of those are in the multiplication table of 7, except 42.
So, 42 is the Least Common Denominator
(LCD). 





4. Find the
least common denominator
(LCD) for adding or subtracting these fractions. You may use
the space for writing out lists of multiples.

fractions 
LCD 
a. 
5
12 
+ 
3
8 


↓ 

↓ 



+ 

= 


b. 
7
4 
− 
9
11 


↓ 

↓ 



− 

= 


c. 
1
12 
+ 
1
9 


↓ 

↓ 



+ 

= 


d. 
7
8 
− 
4
9 


↓ 

↓ 



− 

= 



This lesson is taken from Maria Miller's book Math Mammoth Fractions 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
A selfteaching worktext for 5th grade that teaches fractions and their operations with visual models. The book covers fractions, mixed numbers, adding and subtracting like fractions, adding and subtracting mixed numbers, adding and subtracting unlike fractions, and comparing fractions.
Download ($3.50). Also available as a printed copy.
Learn more and see the free samples!
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