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### Equations with Fractions

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

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

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!

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/

Associated Topics:
Middle School Algebra
Middle School Fractions

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