If one fraction has a smaller numerator and a larger denominator than another, will it always be the smaller fraction (and vice-versa)?

Question

E.g. take:

a/b and c/d

If

c < a and d > b

Will

c/d < a/b

?

Answer 1

If $a,b,c,d$ are all positive, this holds.

But beware if some of them are negative! eg take a=-1, b = -1, c=-5, d=-0.5 one has $a > c$ and $d>b$, yet $\frac{c}{d} = 10 > 1 = \frac{a}{b}$.

edit (building on Henry's remark above)
It is indeed enough that $b>0$ and $d>0$ : Since $b>0$ one has $\frac{a}{b} > \frac{c}{d} \Longleftrightarrow a > \frac{bc}{d}$.
But since $d>0$ one has $a > \frac{bc}{d} \Longleftrightarrow ad > bc$. Now, as $d>b$ and $a>c>0$, you get $ad>bc$ whence the claim.