Equations with Fractions

Date: 04/21/97 at 21:15:10
From: Alisia Fu'e
Subject: Adding fractions

How do find the common denominator for a fraction that has a variable 
in it? This is what I know so far:

To solve the equation x/2 + x/3 = 1/6, first find the common 
denominator. After finding the commom denominator, distribute it to 
both sides of the equation:

6(x/2 + x/3) = 6(1/6)  

Right? If so, what next?

Thanks,
Alisia Fu'e

Date: 05/03/97 at 16:50:25
From: Doctor Kathryn
Subject: Re: adding fractions

Hi Alisia,

You have the right idea to start, but do you understand WHY you need 
to find a common denominator?  Whether trying to solve an equation or 
simplify an expression, you need to turn "unlike fractions" into "like 
fractions."  

Fractions can be thought of as parts/whole.  So 1/2 is one part of the 
two parts that make the whole. Similarly, 5/8 is five parts of the 
eight parts that make a whole.  Improper fractions can also be thought 
of the same way. 9/4 is nine parts of the four parts that make a 
whole.  So that must mean that there are two wholes and one part of 
four left over: 2 1/4!

Now, your question:

x/2 + x/3 = 1/6

You need a common denominator so that you can combine the two 
fractions.  How can you know how many pieces (OF EQUAL SIZE) you have 
if some are halves and some are thirds?  You need to find a common 
denominator, which is 6. Your shortcut works, but let's see why:

First let's change each fraction so that all have the same denominator 
(I often think that the purpose of getting a common denominator in 
equations is so that you can get rid of it. Watch!):

   x/2 =  3x/6
   x/3 =  2x/6
   1/6 =  1/6

So, we have: 3x/6 + 2x/6 = 1/6

Now we want to get rid of the denominator so that the equation is 
easier to solve.  This means that we multiply everything by the 
denominator (note that this does not change the equation):

6(3x/6 + 2x/6 = 1/6)  

Distribute the 6: 6(3x/6) + 6(2x/6) = 6(1/6)

Simplify each fraction (note that the 6's cancel): 3x + 2x = 1

Combine like terms: 5x = 1

Isolate the variable by dividing by 5: x = 1/5

I hope this helps!

-Doctor Kathryn,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/