The authors study finite continued fractions with bounded partial quotients. Let \(A,k,q\) be natural numbers, \(q>1\), \(1\leq A<q\), \((A,q)=1\) and let \([0,B_1, \dots, B_n(A)]\) be the representation of \(A\over q\) as a finite continued fraction. The authors prove that for any constant \(\gamma\) and any sufficiently large \(k\), if \(N(k,q)\) is the number of numbers \(A\) for which \(B_i\leq k\), \(i=1,2, \dots, n(A)\), then \[ N(k,q) \ll \varphi(q) q^{-{\gamma \over k\log k}}, \] where \(\varphi(q)\) is the Euler function. The proof is based on the properties of the Farey series. The authors sketch the proof of a similar theorem for so called almost arithmetic progressions.