NOTES: A few words on common denominators
When do you use common denominators?
 They’re necessary for adding or subtracting fractions.
 They’re useful for comparing fractions. You can also compare fractions by using common
sense or by the “multiplying up” method described on the A quick overview of fractions handout.]
Why do you need common denominators?
 Imagine you have a flat cake cut into equal-sized pieces.
The denominator indicates how many pieces the cake is cut into.
If the cake is cut into 4 pieces, each piece is
If the cake is cut into 8 pieces, each piece is
1
4
1
8
of the cake.
of the cake.
 Adding and subtracting basically involve counting. When we count, we count
things of the same sort. If I eat 3 of the small
1
8
1
1
8
8
+ +
=
3
8
1
8
sized pieces, I can count:
I just add the numerators to show how many I’ve eaten.
I leave the denominator alone since that just indicates the size of the pieces.
What if I then also eat a big
1
4
sized piece? I’ve now eaten
3
8
1
+ of the cake.
4
But I can’t say I’ve eaten 3+1=4 of anything. One big piece doesn’t count the
1
same as a small piece. (Just try giving an 8-year old twin one sized piece and the
other twin one
1
8
4
sized piece. Each gets 1 piece, but see if they think it’s fair.)
In order to count pieces fairly, I must be dealing with same-sized pieces.
That’s what having common denominators indicates: same-sized pieces.
Once I recognize that one
1
4
piece is the same size as two
then I can count the total in terms of same-sized pieces:
3
8
1
8
pieces,
1
3
2
5
4
8
8
8
+ = + =
 If fractions have common denominators, it’s easy to compare them since we’re
comparing same-sized pieces. For example,
5
8
3
> .
8
Eating 5 is clearly consuming more than eating 3 of the same-sized pieces.
TAKE-HOME MESSAGE: If you have common denominators,
you’re dealing with same-sized pieces of cake.
D. Stark 11/29/2016
1
So, how do I get common denominators? Choose your method.
METHOD 1:
easy common denominator
METHOD 2:
least common denominator (LCD)
PROS: easy; good for beginners
CONS: may require tricky reducing
PROS: requires less reducing
CONS: takes practice to find the LCD
Multiply the 2 denominators.
The result is a common denominator.
Find the least common multiple (LCM) of
the 2 denominators. The result is the
least common denominator.
1
2
EXAMPLE:
+
6
9
EXAMPLE:
1
+
6
2
9
The easy common denominator is
6  9 = 54
1
+
6
2
9
=
=
9
1
54
12
6
2
+
54
21
54
=
+
4
1
12
=
=
=
3
18
4
18
7
18
If 1 denominator is a factor of the
other, the bigger one is the
common denominator. You don’t
have to use the listing or ladder
method in this “nice case.”
=
9
7
TIP FOR THE LCD METHOD:
3
The least common denominator is the
least common multiple (LCM) of 6 and 9,
which is 18.
18
But how do I figure out the LCM of 6 and
9? Use the listing method or the ladder
method discussed on the LCM handout.
listing: list multiples
6: 6, 12, 18, 24, …
9: 9, 18, 27, …
9
ladder:
12
1
LCM =
3  2  3 = 18
12
10
12
=
5
6
D. Stark 11/29/2016
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