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We study fractions associated to Ford circles which are extracted by meanscontinuous curves. We show that the extracted fractions have similar propertiesto Farey sequences, like the Farey sum, and we prove that every orderedsequence that satisfies the Farey sum and has two adjacent fractions, can beextracted from Ford circles through continuous curves. This allows us to relatesequences of fractions that satisfy the Farey sum and continuous curves. Wefocus on the fractions $F_{1/m}$ extracted from inclined lines with positiveslopes of the form $1/m$ and define jumps as the cardinality increments ofthese fractions with respect to $m$. We relate the expression for every jump tothe prime omega function in terms of $m$ and find a cardinality formula relatedto the M\"obius function, which we approximate with three tractable expressionsthat grow in a log-linear way considering estimations of sums related toEuler's totient function or the graph of the lattice points corresponding tothe fractions $p/q$.

Properties and approximations of fractions associated to Ford circles extracted by inclined lines