Infer inequality from another inequality I have four strictly positive real numbers, $a$, $a'$, $b$, $b'$. I know the exact values of $a'$ and $b'$. I do not know the values of $a$ and $b$, but I know the value of their ratio $a/b$. Say then that it holds that $$\frac {a'}{b'} > \frac {a}{b}$$ **Can I say anything about the sign of** $$c=a'b' - ab$$ or at least obtain some computable numerical bounds for $c$? This has tormented me for a while. I guess either the answer is "no, everything is possible", or there is an obvious approach I cannot see.

No, everything is possible.

Note that expanding or simplifying one of your fractions won't change which of the the fractions is larger or smaller (since it doesn't change the value of the fraction at all), but it can change the sign of $c$, since only one of the two terms will change.

Example: $$ \frac{2}{1} > \frac 11\\\ c = 2\cdot 1 - 1\cdot 1 = 1>0 $$ but $$ \frac21 > \frac 22\\\ c = 2\cdot 1 - 2\cdot 2 = -2 < 0 $$