The optimum fan pressure ratio is determined both numerically and analytically for separate-stream as well as mixed-stream bypass engines.  The results are applicable to civil and military engines.  The optimum fan pressure ratio is shown to be predominantly a function of the specific thrust and a weak function of the bypass ratio.  Two simple, explicit equations - one for each type of engines - have been derived that specify the optimum fan pressure ratio.  The predictions compare very well against numerical optimisation performed by a specialist computer program employing iterative and advanced search techniques and real gas properties.  The analytical relations accelerate the optimisation process and offer physical insight. It has been shown that the optimum fan pressure ratio achieves the condition V_jc/V_jh = h_KE   in a separate-stream engine and the condition  V₁₆₃/V₆₃ ~ h_KE   in a mixed-stream engine.  The condition   V₁₆₃/V₆₃ ~ h_KE  applies even under situations when significant departures from the normally assumed condition   p₀₁₆/p₀₆ ~ 1 can occur.