Division of Two Strict Inequalities

Suppose $a,b,c,d\in\mathbb{R}$ and $0<a<b<1$ and $1<c<d<2$.

Consider dividing the first inequality by the second one. Then,

$$0<\frac{a}{c}<\frac{b}{d}<\frac{1}{2}.$$

This time, divide the second one by the first one. Then,

$$\infty>\frac{c}{a}>\frac{d}{b}>2.$$

My question: are these two divisions valid?

Solution 1:

As requested in comments

The suggested divisions do not always work:

• e.g. with $a=0.6, b=0.7, c=1.2, d=1.75$ you get $a<b$ and $c<d$ but $\frac{0.6}{1.2} = 0.5 > 0.4= \frac{0.7}{1.75}$ and $\frac{1.2}{0.6} = 2 < 2.5= \frac{1.75}{0.7}$

• while with $a=0.3, b=0.7, c=1.2, d=1.75$ you get $a<b$ and $c<d$ but $\frac{0.3}{1.2} = 0.25 < 0.4= \frac{0.7}{1.75}$ and $\frac{1.2}{0.3} = 4 > 2.5= \frac{1.75}{0.7}$

In general with $0<a<b$ and $0<c<d$, you can say $0 < \frac1d < \frac1c$ and so by multiplication $$\frac{a}{d} < \frac{b}{c} \text{ and similarly }\frac cb < \frac da$$ which are not quite the same as your original assertions