Elementary Theory of Numbers – Addition to the denominator of two fractions

Let's say that I work with a fraction larger than another and only in the field of positive fractions. For example, let's say that I've $ frac {20} {20} $ and $ frac {50} {80} $. Clearly $ frac {20} {20} $ is the largest fraction. If I add 1 to both denominators, $ frac {20} {21} $ still bigger 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 larger (or equal to?) For the larger fraction at the origin. So in this case $ frac {20} {20} – frac {20} {21} $ is taller than $ frac {50} {80} – frac {50} {81} $. Is this true also in general cases if one fraction is larger than the other and can I prove it if yes? I have tried some examples on paper and this seems to be valid, but I do not really know how to make a formal proof of it.