Gear Ratios in Number Theory? A machine shop is manufacturing a pair of gears that need to be in a ratio as close to
$1.1839323$ as possible, but they can’t make gears with more than $50$ teeth on them. How
many teeth should be on each gear to best approximate this ratio?
I can't figure out a number-theoretic approach to solve this, or the steps to get to a solution in such a manner. Can someone see a solution I can't?

Express the value as a continued fraction and then simplify it. The continued fraction for 1.1839323 is:
1; 5, 2, 3, 2, 5, 95, 2, 11, 1, 3, 2
45/38   = [1;5,2,3] = 1.1842105263157894     error +0.0002782263157894427 (0.02350%)
The best ratio with values under 50 is 45 to 38.
