The Least Common Denominator

 

  1. The Least Common Multiple

    A multiple of a number is a whole number times that number.  For example, some multiples of 6 are 

            6, 12, 18, 24, 30, and 36

    If two numbers are given then a common multiple of the two numbers is a number that is a multiple of both.  Of all the common multiples of two numbers, there is a smallest one which we call the least common multiple.  

    Example

    Find the least common multiple of     6 and 9.  

    Solution

    One way of solving this problem is to write out multiples of each and see what is common to the list:

            6, 12, 18, 24, 30, 36, ...        multiples of 6

            9, 18, 27, 36, 45, ...            multiples of 9

    We see that the numbers 18 and 36 are both common multiples of 6 and 9.  The least common multiple is the smallest which is 18.


    Example

    Find the least common multiple of     8 and 32.

    Solution

    Instead of listing many multiples of each, we just notice that     32 is a multiple of 8 and hence 32 is a common multiple.  It is the first multiple of 32.  We can conclude that 32 is the least common multiple of 8 and 32.  

    In general, the least common multiple of two numbers with one the multiple of the other is just the larger number.



    Exercise
    1. Find the least common multiple of 15 and 54.

      Hold mouse over the yellow rectangle for the solution  270

    2. Find the least common multiple of 9 and 81.

      Hold mouse over the yellow rectangle for the solution  81


      As you saw from the Exercise A, writing out many multiples of each number can be tedious.  There is an alternate method that may save time.  The strategy is based on the following idea.  A multiple of a number is a multiple of each of the prime divisors.  

    Steps in Finding the LCM

    1. Write the prime factorization of each number
    2. List the primes that occur in at least one of the factorizations
    3. Form a product using each prime the greatest number of time it occurs in any one of the expressions

    Example

    Find the LCM of     45 and 21

    Solution

    1. 45  =  9 x 5 =  3 x 3 x 5
      21  =  3 x 7
    2. 3, 5, and 7
    3. 3 x 3 x 5 x 7   The prime 3 occurs two times as it does in 3 x 3 x 5
      =   9 x 5 x 7  =  45 x 7 =  315

    Exercises 

    Find the LCM of 

     

    1. 18 and 40
      Hold mouse over the yellow rectangle for the solution  360
    2. 12 and 15
      Hold mouse over the yellow rectangle for the solution  60
    3. 27 and 10
      Hold mouse over the yellow rectangle for the solution  270

     

  2. The Least Common Denominator 

    We define the     least common denominator of two fractions as the least common multiples of the denominators.

    Examples

    Find the least common denominator of 
    1. 3/4 and 9/10
    2. 5/6 and 10/11

    Solutions

    1. We find the least common multiples of 4 and 10

              4  =  2 x 2            10  =  2 x 5

      So the least common denominator is

              2 x 2 x 5  =  20
    2. We find the least common multiples of 6 and 11

                    6  =  2 x 3        11  is prime

      So the least common denominator is 

              2 x 3 x 11  =  66

    Exercises

    Find the least common denominator of

    1. 3/14 and 2/63
      Hold mouse over the yellow rectangle for the solution  126
    2. 8/25 and 23/100
      Hold mouse over the yellow rectangle for the solution  100

  3. Building Up Fractions With a Least Common Denominator

    We have already learned how to simplify a fraction by dividing through by a common factor.  Sometimes it is convenient to be able to work this process in reverse.

    Example

    Build up the fraction to answer the question

            5           ?
                  =                    
            6          24

    Solution

    We see that 

            24  =  6 x 4

    so

            5           5 x 4            20
                  =                  =                     
            6           6 x 4            24


    Exercise

    Build up the fraction to answer the question

            3           ?
                  =                    
            7          35
     Hold mouse over the yellow rectangle for the solution  15


    Example

    Which number is larger:      5/8 or 9/14?

    Solution

    Since the denominators are different, these numbers are difficult to compare.  Our strategy is to build up each fraction to fractions with the least common denominator.  We first find the least common denominator:

            8  =  2 x 2 x 2        14  =  2 x 7

    The least common denominator is 

            2 x 2 x 2 x 7 =  56

    The next step is to notice that 

            8 x 7  =  56        and         14 x 4  =  56

    We write

            5            5 x 7           35
                   =                 =                     
            8            8 x 7           56

    and

             9             9 x 4           36
                    =                  =                     
            14           14 x 4          56

    Since

            35           36
                    <                     
            56           56

    We conclude that

            5           9
                  <                    
            8          14

    Exercise

    Which is larger:  3/10 or 7/25?
    Hold mouse over the yellow rectangle for the solution  3/10



    Example

    Write the three fractions  1/6, 5/8 and 3/10 as equivalent fractions with the LCD as the denominators.

    Solution

    We have

            6  =  2 x 3         8  =  2 x 2 x 2        10  =  2 x 5

    So the least common denominator is 

            2 x 2 x 2 x 3 x 5  =  8 x 3 x 5  =   24 x 5  =  120

    We write

            1           1 x 20            20
                  =                   =                        
            6           6 x 20            120

            5           5 x 15            75
                  =                    =                     
            8           8 x 15            120

            3             3 x 12              36
                    =                     =                       
           10           10 x 12            120


    Exercise

    Write the three fractions  2/15, 4/9 and 3/25 as equivalent fractions with the LCD as the denominators.

    Hold mouse over the yellow rectangle for the solution  30/225, 100/225, and 27/225

 

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