I was reading a text book and came across the following:
If a ratio $a/b$ is given such that $a \gt b$, and given $x$ is a positive integer, then $\frac{a+x}{b+x} \lt\frac{a}{b}\quad\text{and}\quad \frac{a-x}{b-x}\gt \frac{a}{b}.$
If a ratio $a/b$ is given such that $a \lt b$, $x$ a positive integer, then $\frac{a+x}{b+x}\gt \frac{a}{b}\quad\text{and}\quad \frac{a-x}{b-x}\lt \frac{a}{b}.$
I am looking for more of a logical deduction on why the above statements are true (than a mathematical "proof"). I also understand that I can always check the authenticity by assigning some values to a and b variables.
Can someone please provide a logical explanation for the above?
Thanks in advance!