In 3-dimensional contact geometry, every contact rational surgery is equivalent to a sequence of $\pm 1$-surgeries on a link, determined by the continued fraction expansion of a function of the slope.

I think the result is originally due to Ding and Geiges, and is explained here (section 5). A similar scheme appears in the classification of tight contact structures on lens spaces, due to Honda (see here). Ozbagci and Stipsicz give a pleasant exposition of both (and many other) results in their book "Surgery on contact 3-manifolds and Stein surfaces".