Common Denominators
Carey's Weekly Feature

Why do we need common denominators?  This is not a question known to keep people up all night worrying.  However if you happen to be a student with a fraction phobia, then read on – there may be tidbits of information that will demystify one aspect of fractions.

The commonly seen answer to the questions above is “we need common denominators to add (or subtract) fractions.”  For some this is enough, but you may want more.  You may then ask “why are common denominators necessary when adding fractions?”  The answer is a direct conclusion from two fundamental concepts – when can we add and what are fractions?

We can add (or subtract) only the same type of things, a rule remembered as “you can’t add apples and oranges.”  What would be the result if you have 5 apples and were then given 1 orange.  You would have “5 apples and 1 orange.”    However if you renamed the fruit so that you began with 5 “fruit items” and were then given another 1 “fruit item” you could then add the same type of things together for a total of 6 “fruit items.”

Now consider fractions such as 5/9.  Its denominator 9 identifies the fraction’s type as ninths, such as when a pizza is cut into 9 equal sized slices; each slice is a ninth of the pizza.  The numerator 5 indicates how many of this type of fraction there are.  So if you had 5 slices of this pizza then you would have 5/9 of the pizza.  

Suppose you are given 1/3 of another pizza.  The total amount of pizza you have is 5—ninths of a pizza and 1—third of a pizza, different types of fractions that cannot be added any more that 5 apples and 1 orange can.  

In order to add, one or both fractions must be renamed so the fractions have common (the same) denominator.  When the larger 1/3 of a pizza slice is further cut up into 3 slices, each of these slices would be ninths.  1—third of a pizza has just been renamed as 3—ninths of a pizza.  

Now with a common denominator, 5—ninths of a pizza increased by 3—ninths of a pizza gives us a total of 8—ninths of a pizza.

Common denominators are needed when adding (or subtracting) fractions is an easy statement to memorise, but do you need to remember it by rote?  Not when you should already know when you can or cannot add and what the two numerals that make up the fraction signifies.  These two fundamental ideas used together allow anyone to explain why fractions need common denominators when added. 

Try Carey's quiz on this topic: Lowest Common Denominator

For another explanation of why we need common denominators try the following site.

http://forum.swarthmore.edu/dr.math/problems/thomas.06.04.01.html

http://forum.swarthmore.edu/dr.math/problems/curr2.8.96.html

For more information about fractions look at these sites.

http://ucsub.colorado.edu/~maybin/mtop/ms02/exp.html

http://cne.gmu.edu/modules/dau/algebra/fractions/frac3_frm.html

http://www.ricksmath.com/old/fractip2.html

http://www.math.com/school/subject1/lessons/S1U4L3DP.html

 

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