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What is the process for finding common denominators

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5 Answers

Linda,
 
When you need to find a common denominator for two or more fractions, the first thing you need to do is find the least common multiple of all the denominators which is called the LCM.  The LCM will be your smallest common denominator.
 
You have to add the fractions 3/12 and 7/8 in this problem.   So first find the LCM of 12 and 8. There are a several ways to do this.  If you know how to split the numbers 8 and 12 into prime factors there is an easy way to find it, or alternatively you can just look at the multiples of 8 and 12
 
multiples of 8 are 8, 16, 24, 32..    
multiples of 12 are 12, 24, 36,..  
 
We see that 24 is the lowest number that is a multiple of both 12 and 8 (it's called the LCM of 12 and 8) so we can use 24 as a common denominator for the two fractions.  This method works with adding 3 or 4 fractions also.
 
So 3/12  = 6/24 by multiplying the numerator and denominator by 2.
And 7/8 = 21/24 by multiplying the numerator and denominator by 3.
 
3/12 + 7/8 = 6/24 + 21/24 = 27/24
 
 

Comments

LCD,and LCM are identical , and can be used interchangeably.
Least common denominator, or least common multiple, is what you must find first. Multiples of 12 are 12,24,36,48,60,72...  Multiples of 8 are 8, 16,24,32,40,48... When you list the multiples of each, you can see that the LCM is 24.
 
Change both of the fractions to where the denominator is 24 to both and change the numerators accordingly. Now you have 6/24 + 21/24. Leave the denominator as is and add the numerators and you get an answer of 27/24. However, when your answer is a fraction, you must put it in simplest form. You can reduce the fraction by dividing each number by 3, so your answer ends up being ....?
To find the LCM of 2 numbers , write down 2 numbers as product of its prime factors.
 
and Multiply the GCF only once to find the lowest common multiple.
 
 or  LCM ( A, B) = ( A. B) / GCF    (1)
 
 
   Example   8 = 2^3
                  12 = 2^2 . 3
                  
                 LCM = 2^3 . 3 = 24
                 CGF = 2^ 2 = 4
    
                 LCM = 12 . 8 / 4 = 96/ 4 = 24
 
        Equation ( 1) becomes very useful in Theory of Numbers, and advanced Mathematics.
I like to do it in-place so I keep all the work together (then if I'm looking for a mistake later I don't have to hunt for the work). Also, I just think about prime factors and how many of each I have in each denominator; then adjust to get the same set in all denominators. That's the concept and I think it gets lost among all the words defined to describe it.
 
3/12 + 7/8 =
 
3/(2*2*3) + 7/(2*2*2) =
 
3*2/(2*2*2*3) + 7*3/(2*2*2*3) =
 
(3*2 + 7*3)/(2*2*2*3) =
 
(6 + 21)/(8*3) =
 
27/24

Comments

Comment for Lynda:
 
So Steve is using the other method I mentioned for finding the LCM of the two numbers 12 and 8.
 
He factors 12 into prime factors:  12 = 3*2*2,  and factors 8 into prime factors: 8 = 2*2*2.  
 
Now the LCM of the two numbers 8 and 12 will need to include all the prime factors for each number.  So it will need to have 2*2*2 because that's in the number 8.  And it will need a 3 because that factor is in the number 12.  The LCM is 2*2*2*3 - it has the 3 2's from 8 and also the 2 2's and a 3 from 12.  
 
LCM = 2*2*2*3 = 24.  Then you do the addition just as before.
The idea is that to get the LCM, only multiply the GCF once.
Hi Linda;
The quickest technique is to multiply the denominators together...
(8)(12)=96
 
(3/12) + (7/8) =
[(3/12)(8/8)]+[(7/8)(12/12)]=
(24/96)+(84/96)=
108/96
54/48
27/24
 
However, what you may also notice about the denominators is...
(8)(3)=24 and
(12)(2)=24
Therefore, 24 is the lowest common denominator...
[(3/12)(2/2)]+[(7/8)(3/3)]
(6/24)+(21/24)
27/24