Is $\frac{1}{\lfloor \frac{a}{b} \rfloor}=\lceil \frac{b}{a} \rceil$??

Question

Is $\frac{1}{\lfloor \frac{a}{b} \rfloor}=\lceil \frac{b}{a} \rceil$? assuming that $\frac{a}{b}>1$.

Answer 1

In the case $\displaystyle\frac{a}{b}\ge2$ we get $\displaystyle\lfloor{\frac{a}{b}\rfloor}\ge2\Rightarrow\frac{1}{\lfloor{\frac{a}{b}\rfloor}}\le\frac{1}{2}$ and $\displaystyle\frac{b}{a}\le\frac{1}{2}\Rightarrow\lceil{\frac{b}{a}\rceil}=1$.