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Simplifying a Fraction -- and Oversimplifying It

Date: 03/03/2012 at 16:41:50
From: Jessica
Subject: simplify 

Simplify

   a^2/(a - 4) - a/(a - 4) - 12/(a - 4)

Since these fractions all had the same denominator, first I combined them:

   (a^2 - a - 12)/(a - 4)

I divided -12 by -4:

   a^2 - a + 3/a

Then I divided a^2 by a:

   a - a + 3 = +3

But the answer key says (a + 3), so I'm confused how they got that.



Date: 03/03/2012 at 17:23:15
From: Doctor Peterson
Subject: Re: simplify 

Hi, Jessica.

The original expression does combine, as you showed in your first step:

  a^2/(a - 4) - a/(a - 4) - 12/(a - 4) = (a^2 - a - 12)/(a - 4)

Simplifying has to result in an expression that has the same value. So
let's make sure that our expression evaluates to the same thing from one
step to the next by looking at this with a specific value for a.

I'll pick 5. Substituting that in gives

   5^2 - 5 - 12    8
   ------------ = --- = 8
       5 - 4       1

Now, what you did first was to divide only the 12 by the -4 and eliminate
the -4 from the denominator, giving

   5^2 - 5 + 3   23
   ----------- = --- = 4.6
         5        5

That's not the same thing, is it? I hope you can see why: you divided just
ONE of the terms in the numerator by 4, and you ADDED 4 to the
denominator. You just can't do that! Canceling in a fraction has to mean
DIVIDING the ENTIRE numerator and the ENTIRE denominator by the same
number, and that's not what you did.

Similarly, your second step was to divide just the a^2 (5^2 here) and the
a (5) by 5, giving

  5 - 5 + 3    3
  --------- = --- = 3
      1        1

That, too changed the value of the expression, because you didn't divide
the ENTIRE numerator by 5.

The only way to simplify a fraction is to FACTOR the numerator and
denominator, and remove a common factor from both (which is the same as
dividing them by that factor).

So the correct work looks like this:

   a^2 - a - 12   (a - 4)(a + 3)   a - 4
   ------------ = -------------- = ----- * (a + 3) = a + 3
       a - 4      (a - 4)          a - 4

By the way, notice that when a = 5, this DOES give the same value we got
above, namely 5 + 3 = 8.

Many teachers avoid using the word "cancel" entirely in algebra, because
so many students think it's just permission to cross off a number anywhere
in an expression. You didn't use the word "cancel," but still did the same
wrong thing. You have to always think about what cancellation means:
dividing the numerator and denominator by something.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Algebra
Middle School Fractions

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