Show $\frac{3997}{4001}>\frac{4996}{5001}$ I wish to show that $$\frac{3997}{4001}>\frac{4996}{5001}.$$
Of course, with a calculator, this is incredibly simple. But is there anyway of showing this through pure analysis? So far, I just rewrote the fractions:
$$\frac{4000-3}{4000+1}>\frac{5000-4}{5000+1}.$$

Begin by observing that $\frac{3996}{4000}=\frac{4995}{5000}$, and think of these fractions as $\frac{\mbox{wins}}{\mbox{games played}}$ for chess players $A$ ($3996$ wins) and $B$ ($4995$ wins). One additional win will do more to improve player $A$’s win percentage than it will player $B$’s.
