Set $x = c - d$ and $y = b - d$. We have
$$a - d = (c - d) + (a - c) = x + 3y$$
and
$$a - d = (b - d) + (a - b) = y + 2x,$$
hence $x + 3y = y + 2x$, which implies $x = 2y$. Now we get $a - d = 5y$ and
$$b - c = (b - d) - (c - d) = y - x = -y,$$
so
$$$$
\frac{a-d}{b-c} = -5.

We are given that
$$a - b = 2c - 2d, \text{ thus } a + 2d = b + 2c,$$
and
$$a - c = 3b - 3d, \text{ thus } a + 3d = 3b + c.$$
Multiply the first equation by 4 and the second one by 3, and subtract:
$$a - d = 4(a + 2d) - 3(a + 3d) = 4(b + 2c) - 3(3b + c) = -5b + 5c,$$
thus
$$$$
\frac{a-d}{b-c} = -5.