Help with a fraction involving variables.

**beefdawg** · June 13th 2011, 07:03 AM

Hello everyone, I have this problem for homework:

(a-b)/a - (a+b)/b is the numerator. The denominator is: (a-b)/b + (a+b)/a. I know the answer is supposed to be -1, but I'm not sure how to get it.

**Prove It** · June 13th 2011, 07:05 AM

PLEASE rewrite this with brackets where they are needed, this is unreadable.

**Soroban** · June 13th 2011, 09:37 AM

Hello, beefdawg!

$\text{Simplify: }\;\frac{\dfrac{a-b}{a} - \dfrac{a+b}{b}} {\dfrac{a-b}{b} + \dfrac{a+b}{a}}$

$\text{Multiply by }\frac{ab}{ab}\!:\;\;\frac{ab\left(\dfrac{a-b}{a} - \dfrac{a+b}{b}\right)} {ab\left(\dfrac{a-b}{b} + \dfrac{a+b}{a}\right)} \;=\;\frac{b(a-b) - a(a+b)}{a(a-b) + b(a+b)}$

. . . . . $=\;\frac{ab - b^2 - a^2 - ab}{a^2-ab + ab + b^2} \;=\; \frac{-a^2 - b^2}{a^2+b^2} \;=\;\frac{-(a^2+b^2)}{a^2+b^2} \;=\;-1$

**beefdawg** · June 13th 2011, 06:56 PM

Aah I see. Thank you very much for the help...just a quick question though...why exactly do we multiply both sides by ab?

**jgv115** · June 13th 2011, 10:46 PM

It isn't necessary to multiply the fraction by ab, it probably just make it easier.

Instead, you can treat the numerator and the denominator as separate fractions, simplify then put it into one fraction

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