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Theoretical treatment of the Richtmyer-Meshkov instability in compressible fluids is a challenging task due to the presence of compressibility and nonlinearity. In this Letter, we present a quantitative theory for the growth rate and the amplitude of fingers in Richtmyer-Meshkov instability for compressible fluids based on the methods of the two-point Padé approximation and asymptotic matching. Our theory covers the entire time domain from early to late times and is applicable to systems with arbitrary fluid density ratios. The theoretical predictions are in good agreement with data from several independent numerical simulation methods and experiments. © 2018 American Physical Society.