# 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.

 × 3 1 6 = 3 18 × 3
 × 2 5 9 = 10 18 × 2

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 =
 d. 5 9 − 1 3 ↓ ↓ 9 − 9 =
 e. 1 8 + 3 4 ↓ ↓ 8 + 8 =
 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:
 2 3 + 1 5 = 15 + 15
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.

 3 8 − 1 6 = 24 − 24
Check the multiples of 8 (the skip-counting 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.
 7 8 + 3 4 = 7 8 + 8
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 =
 b. 2 3 − 1 7 ↓ ↓ − =
 c. 3 10 + 1 2 ↓ ↓ + =
 d. 4 12 + 1 4 ↓ ↓ + =
 e. 1 2 − 2 7 ↓ ↓ − =
 f. 5 6 − 4 9 ↓ ↓ − =

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 ↓ ↓ − =

You will find free, printable worksheets for adding unlike fractions here.

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.