Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and $\frac{l(\alpha x)}{l(x)}$ (for all rational $x$) are bounded with some constants depending on $\alpha$ only. 


Is this result new?



Lengths of continued fractions for the numbers with fixed ratio
It definitely is not new for the length, and I am nearly sure that is not for height either.


See, for example, 


Labhalla, Salah; Lombardi, Henri
Transformation homographique appliqu´ee `a un d´eveloppement en fraction continue fini ou
infini. (French) [Fractional linear transformations applied to finite and infinite continued
fractions]
Acta Arith. 73 (1995), no. 1, 29–41.


