# Adding to the denominator of two fractions

Say I am working with one fraction larger than another and only in the domain of positive fractions.For example, let's say I have $$\frac{20}{20}$$ and $$\frac{50}{80}$$. Clearly $$\frac{20}{20}$$ is the larger fraction. If I add 1 to both denominators, $$\frac{20}{21}$$ still remains larger than $$\frac{50}{81}$$.

What I want to prove or disprove is that the difference between the original fractions and their respective new fractions (after adding 1) is always bigger (or equal to?) for the originally bigger fraction. So in this case $$\frac{20}{20} - \frac{20}{21}$$ is larger than $$\frac{50}{80} - \frac{50}{81}$$. Is that true in general cases as well if the one fraction is larger than the other and can I prove it if so? I've been trying a few examples on paper and it seems to hold but I don't really know how to make a formal proof of this.

This is not true in general. Take, for instance, the fractions $$\dfrac88$$ and $$\dfrac12$$. Again, the first one is the largest. But$$\frac88-\frac89=\frac19\text{ and }\frac12-\frac13=\frac16.$$And $$\dfrac19<\dfrac16$$.