We present a simple method for determining equations of state (densities) of liquids up to ~10 GPa based on a precise measurement of the volume of the gasket hole in a diamond anvil cell. Key ingredients are the use of a highly sensitive optical pressure gauge, in our case SrClF:Sm 2+ and ruby, the preparation of gaskets which maintain a well-defined geometry of the sample chamber, and the application of corrections due to elasticity of the diamonds. We show that, following a certain procedure, the method allows under favorable conditions measurements of the equations of state with a precision of 1-3%. Result on the equation of state of the 4:1 methanol-ethanol mixture as well as of Daphne fluids 7474 and 7676 are presented. Measurements on water to 5.5 GPa allow a direct evaluation of the precision of the method.