Answer
$$-\frac{1}{2}$$
Work Step by Step
Recall, one never divides by a fraction. Rather, if we want to divide by a fraction, we multiply by the reciprocal of the fraction. Recall, the reciprocal of a fraction is that fraction with the numerator and the denominator switched. Thus, we find:
$$ \frac{x-y}{6}\div \frac{y-x}{3}\\ \frac{x-y}{6}\times \frac{3}{y-x}\\ \frac{\left(x-y\right)\times \:3}{6\left(y-x\right)}\\ \frac{x-y}{-2\left(x-y\right)}\\ -\frac{1}{2}$$