Some while back, Ben Orlin of the brilliant Maths With Bad Drawings blog posted a puzzle he’d set for some eleven-year-olds:

Which is larger, $\frac{3997}{4001}$ or $\frac{4996}{5001}$? Hint: they differ by less than 0.000 000 05.

He goes on to explain how he solved it (by considering the difference between each of the fractions and one), but I wanted to share my alternative way: find a common denominator.

Wait! Come back! Don’t worry, I’m not going to be multiplying 4001 by 5001 (although it’s not that hard). Instead, I’m going to define $k$ to be 1000.

Why would I do that? It’s because then I have $\frac{4k-3}{4k+1}$ versus $\frac{5k-4}{5k+1}$. And those are easy to work with.

The first one is equivalent to $\frac{(4k-3)(5k+1)}{(4k+1)(5k+1)}$; the second is $\frac{(5k-4)(4k+1)}{(5k+1)(4k+1)}$

Now it’s just a case of expanding the tops: the first numerator is $20k^2 -11k - 3$, while the second is $20k^2 - 11k -4$, which means the second fraction is clearly smaller, by $\frac{1}{(4k+1)(5k+1)}$. That works out to $\frac{1}{20k^2 + 21k + 1}$, or $\frac{1}{20,021,001}$, which looks like it’s the 0.000 000 05 Ben mentioned!