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Chapter 3—Fractions
37 Finding Common Denominators
When you complete the work for this section, you should be able to:  Demonstrate your ability to adjust two fractions so that they have the same denominator.
 Express the importance of having to multiply the numerator and denominator by the same number.
 Define least (or lowest) common denominator.

Definition When two or more fractions have the same denominator, they are said to have a common denominator. 
In order to add or subtract fractions, they must have the same denominator.  Fractions such as 1/8, 3/8, and 5/8 all have the same denominator: 8.
 Fractions such as 1/10, 3/10, 7/10, 9/10, and 10/10 all have the same denominator: 10
However, the world is rarely so kind as to offer us sets of fractions that happen to have the same denominator.  Fractions such as 1/8, 3/4, and 5/16 do not have the same denominator.
 Fractions such as 1/10, 3/5, and 1/2 do not have the same denominator.
 
Fractions can be added or subtracted only when they have the same denominator. When you are given a set of fractions that do not happen to have the same denominator, it is necessary to expand the fractions in such a way that forces them to have the same denominator. Any set of fractions can be forced to have the same denominator.
This lesson shows how you can find multipliers that allow you for force fractions to have a common denominator. This process is known as finding a common denominator. Here is a group of examples and exercises that will help prepare you for mastering the task of finding common denominators.
Example 1
Problem: Expand ^{2}/_{3} so that it has a denominator of 18.
Procedure:
 In order to expand 3 to 18, you have to multiply the 3 by a factor of 6.
"Whatever you do to the denominator, you must do to the numerator."  So multiply both the numerator and denominator by 6

 ^{2}/_{3} · ^{6}/_{6} = ^{12}/_{18}
Solution: ^{2}/_{3} = ^{12}/_{18}
Examples and Exercises
Changing the Value of Denominators Use these interactive examples and exercises to strengthen your understanding and build your skills:  
There are several different ways to force a group of fractions to have the same denominator—a common denominator. A couple of the methods are rather informal and are simply intended to get the job done. Others are more formal and are designed to work, even for the most esoteric combinations of fractions. But in any event, the idea is to expand one or both of the fractions in such a way that they end up having the same denominator.
The Lowest (or Least) Common Denominator
There can be an endless number of different common denominators for a set of fractions. Suppose you need to find common denominators for 1/4 and 2/5. For those two fractions, you can use 20 as a common denominator; but you can also use 40, 60, 100, 120, ... . The list is endless. However, your work with fractions and common denominators can be simpler and neater when you choose to use the lowest (or least) common denominator, or LCD.
What is the LCD for a set of fractions? It is the smallest value you can find for the common denominator. The LCD for ^{1}/_{4} and ^{2}/_{5} is 20. There are many, many other common denominators, but the LCD is the smallest and, therefore, the simplest and tidiest to use. (And besides, a lot of teachers and exams require you to use the LCD for a set of fractions).
Finding the LCD  Multiplication Method
Procedure Creating common denominators using the multiplication method.  Multiply the denominator of the first fraction by 1, 2, 3, 4, and 5
 Multiply the denominator of the second fraction by 1,2,3,4, and 5
 Compare the sets of multiples. Are there any that are equal? If yes, use them as the common denominator. If not, continue from step 1, multiplying by larger integer values.

Note The method of multiplication is most useful when the denominators are fairly small values. Use the division method when the denominators have fairly large values. 
Example: Find a common denominator for 3/4 and 1/5
For the denominator of the first fraction:  4 · 1 = 4
 4 · 2 = 8
 4 · 3 = 12
 4 · 4 = 16
 4 · 5 = 20
 For the denominator of the second fraction:  5 · 1 = 5
 5 · 2 = 10
 5 · 3 = 15
 5 · 4 = 20
 5 · 5 = 25

This shows that the LCD for 3/4 and 1/5 is 20.
Expanding both fractions:
 3/4 · 5/5 = 15/20
 1/5 · 4/4 = 4/20
Now the fractions have the same denominator.
Examples and Exercises
Finding LCDs by the Multiplication Method Use the multiplication method to find the LCD for these pairs of fractions.  
Finding the LCD  Division Method
The division method for finding LCDs is very reliable and easy to use—once you figure out how to use it.
Procedure Division method for finding LCDs:  Align the denominators in an "upsidedown" division box.
 Find the smallest integer (greater than 1) that divides evenly into at least two of the denominators.
 Bring down the results of the division and any remaining numbers that cannot be divided evenly.
 Repeat steps 2 and 3 until there are no integers greater than 1 that can be divided evenly into two or more of the numbers.
 Multiply all the divisors and remaining numbers to get the LCD.

Example 1
Problem Use the division method to determine the LCD for ^{3}/_{4} and ^{7}/_{10}  
Procedure  
 Align the denominators in an "upsidedown" division box.
 ) 4 10 
 Find the smallest integer (greater than 1) that divides evenly into at least two of the denominators.
In this example, 2 divides evenly into 4 and 10.  2 ) 4 10 
 Bring down the results of the division and any remaining numbers that cannot be divided evenly.
In this example, both can be divided evenly, so there are no numbers to "bring down."  2 ) 4 10 2 5 
 Repeat steps 2 and 3 until there are no integers greater than 1 that can be divided evenly into two or more of the numbers.
There are no more in this example.  2 ) 4 10 2 5 
 Multiply all the divisors and remaining numbers to get the LCD.
 2 ) 4 10 2 5LCD = 2 · 2 · 5 = 20 
Solution LCD for ^{3}/_{4} and ^{7}/_{10} is 20  
Example 2
Problem Use the division method to determine the LCD for ^{5}/_{12}, ^{3}/_{18} , ^{13}/_{21}  
Procedure  
 Align the denominators in an "upsidedown" division box.
 ) 12 18 21 
 Find the smallest integer (greater than 1) that divides evenly into at least two of the denominators.
In this example, 2 divides evenly into 12 and 18.  2 ) 12 18 21 
 Bring down the results of the division and any remaining numbers that cannot be divided evenly.
In this example, 2 divides evenly into 12 and 18, so bring down the results of this division. However, 2 does not divide evenly into 21, so bring down the 21 without any change.  2 ) 12 18 21 6 9 21 
 Repeat steps 2 and 3 until there are no integers greater than 1 that can be divided evenly into two or more of the numbers.
 Repeat Step 2 2 ) 12 18 21 3 ) 6 9 21 Repeat Step 3 2 ) 12 18 21 3 ) 6 9 21 2 3 7 
 Multiply all the divisors and remaining numbers to get the LCD.
 2 ) 12 18 21 3 ) 6 9 21 2 3 7 2 · 3 · 2 · 3 · 7 = 252 
Solution LCD = 2 · 2 · 3 · 1 · 3 · 7 = 252  
Examples and Exercises
Finding LCDs by the Division Method Rewrite these fractions with their LCD.  