If $\frac{x-y}{z-y}=-10$, then the value of $\frac{x-z}{y-z}$ is
(A) 11
(B) -10
(C) 9
(D) -9
(E) 10
Solution 1

Since the problem asks us to find the value of $\frac{x-z}{y-z}$, then this value must be the same no matter what $x, y$ and $z$ we choose that satisfy $\frac{x-y}{z-y}=-10$.

Thus, if we can find numbers $x, y$ and $z$ that give $\frac{x-y}{z-y}=-10$, then these numbers must give the desired value for $\frac{x-z}{y-z}$.

If $x=10, y=0$ and $z=-1$, then $\frac{x-y}{z-y}=-10$.

In this case, $\frac{x-z}{y-z}=\frac{10-(-1)}{0-(-1)}=\frac{11}{1}=11$.

Solution 2

Manipulating,

$$
\frac{x-z}{y-z}=\frac{(x-y)+(y-z)}{y-z}=\frac{x-y}{y-z}+\frac{y-z}{y-z}=-\frac{x-y}{z-y}+1
$$

Since $\frac{x-y}{z-y}=-10$, then $\frac{x-z}{y-z}=-(-10)+1=11$.

ANswer: (A)