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Finding a Common Denominator

Date: 02/21/2002 at 17:34:40
From: Victor
Subject: Fractions

Here is where I got stuck:

1. Find a common denominator for each pair of fractions.
    3/4, 5/8     3/4, 7/10     5/6, 3/5

2. Then write equivalent fractions using the common denominator for 
   the pairs of fractions in the exercise above.

Can you help me?

Date: 02/22/2002 at 00:10:47
From: Doctor Twe
Subject: Re: Fractions

Hi Victor - thanks for writing to Dr. Math.

One quick way to get a common denominator (though not always the 
_lowest_ common denominator) is to multiply the two denominators 

For example, one common denominator for 5/6 and 3/10 is 6 * 10 = 60. 
(Note that this is not the LCD; because 30 is also a common 
denominator and is lower than 60, it is the LCD.)

To convert a fraction to an equivalent fraction with a different 
denominator, you have to first ask yourself, "What do I need to 
multiply my fraction's denominator by to get the new denominator?" For 
example, to convert 5/6 to 60ths (in the form x/60), what do we need 
to multiply 6 by to get 60?

     ---   --- = ----
      6  *  ?     60

The answer for this example is 10.

     ---   ---- = ----
      6  *  10     60

Now if we multiply the denominator by 10, we also have to multiply the 
numerator by 10 to keep the value the same. (This is because 
10/10 = 1, and when we multiply any value by 1, or 10/10, we get our 
original value back.) So then we have:

      5  *  10     50
     ---   ---- = ----
      6  *  10     60

So 5/6 = 50/60.

We can do the same thing to convert 3/10 to 60ths. What number do we 
need to multiply 10 by to get 60? (6, of course.)

     ----   --- = ----
      10  *  6     60

So then we must also multiply the numerator by 6 to get our answer:

       3  *  6     18
     ----   --- = ----
      10  *  6     60

So our fractions, expressed with a common denominator, are 50/60 and 
18/60. In this form, we can add or subtract them.

Did you notice that the number we multiplied the first numerator (5) 
by was the second denominator (10), and the number we multiplied the 
second numerator (3) by was the first denominator (6)? This was not a 
coincidence. It was because we found our common denominator by 
multiplying the two denominators together in the first place. This 
leads us to a "shortcut" formula:

To convert a/b and c/d to a common denominator, we "cross multiply" 
the numerators and multiply the denominators, to get:

     a*d         c*b
     ---   and   ---
     b*d         b*d

Remember that this will not always get you the *lowest* common 
denominator for the two fractions, but it will quickly get you *a* 
common denominator.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum   
Associated Topics:
Elementary Fractions
Middle School Fractions

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