Fraction Time

$\begin{aligned} &&\frac{a+1}{b+1} &= \frac{2a}{b} \\ &\implies& ab+b &= 2ab + 2a \\ &&\frac{a-1}{b-1} &= \frac{a}{2b} \\ &\implies& 2ab - 2b &= ab - a \end{aligned}$ Adding two of the first equation to the second yields $\begin{aligned} 4ab &= 5ab + 3a \\ 0&= ab + 3a \\ 0 &= a(b+3) \end{aligned}$ So either $a=0$ or $b=-3$ . If $a=0$ then $b=0$ , which is impossible. Thus $b=-3$ and we have $\begin{aligned} -3a - 3 &= -6a + 2a \\\ a &= 3 \end{aligned}$ Our original fraction was $\dfrac{a}{b} = \dfrac{3}{-3} = \boxed{-1}$

Let $x$ be the numerator and $y$ be the denominator. Then from the statements above, $\frac{x + 1}{y + 1} = \frac{2x}{y}$ and $\frac{x - 1}{y - 1} = \frac{x}{2y}$ .

Rearranging the first equation gives $xy = y - 2x$ , and rearranging the second equation gives $xy = 2y - x$ . Therefore, $y - 2x = 2y - x$ , and this solves to $y = -x$ .

Therefore, the original fraction $\frac{x}{y}$ simplifies to $\frac{x}{-x} = \boxed{-1}$ .