Lesson 7Finding Common Denominators

 Definition When two or more fractions have the same denominator, they are said to have a common denominator.

The concept of common denominators is extremely important when working with fractionsfractions cannot be added or subtracted unless they have the same denominators.

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.

Example

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

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.

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.

 Definition the lowest (or least) common denominator, or LCD, is the smallest possible integer 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).

Topic 1
Finding the LCD: Multiplication Method

 Procedure To find 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.

 Notes When two or more fractions have the same denominator, they are said to have a common denominator.

The method of multiplication is most useful when the denominators are fairly small values.

Use the division method (next topic) 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.

Use the multiplication method to find the LCD for these pairs of fractions.

Continue doing the examples and exercises until you can work them without making errors.

Topic 2
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:

 Procedure To find common denominators using the division method. Align the denominators in an "upside-down" 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.
 Problem Use the division method to determine the LCD for 5/12, 3/18 , 13/21 Procedure Align the denominators in an "upside-down" 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  213  )    6   9  21         2   3    7 2 · 3 · 2 · 3 · 7  = 252 Solution LCD = 2 · 2 · 3 · 1 · 3 · 7  = 252

Use the division method to find the LCD for these pairs of fractions.

 If you are having any trouble understanding the content of this lesson, you will benefit from a more detailed tutorial on the subject.