# Simplification question regarding a quotient

If I have an expression such as $\frac{(1-a)}{a(a-1)}$ am I allowed to distribute a negative sign only to the denominator, and if so why. How does the negative sign distribute over both factors that is $(a)(a-1)$? Is that not allowed?

• You could substitute $(1-a) = -(a-1)$ and then cancel out the $(a-1)$ of the numerator with the one in the denominator, leaving $-1/a$ – Sak Apr 1 '14 at 19:00
• What a blessing you are. Thank you. I was under the impression that people were multiplying the denominator by -1; which made no sense to me. Nonetheless, so that I can have a deeper understanding, is there any other way to arrive at -1/a? – Prologue Apr 1 '14 at 19:13

Both ways are correct since they lead to the same result: $$\require{cancel} \dfrac{(1-a)}{a(a-1)}=\dfrac{-\cancel{(a-1)}}{a\cancel{(a-1)}}=-\dfrac1a \\ \color{grey}{\text{or }}\dfrac{(1-a)}{a(a-1)}=\dfrac{\cancel{(1-a)}}{-a\cancel{(1-a)}}=\dfrac1{-a}=-\dfrac1a.$$
Best wishes, $\mathcal H$akim.