In binary mixtures with hydrophilic substances, water is usually the more mobile component and its relaxation time is shorter than those of the other components. An exception is the case of the mixture of 1-propanol with 45 mol % water, where the alpha-relaxation of water is slower than the alpha-relaxation of 1-propanol and even slower than the local relaxation of water confined in various spaces of nanometer size. This unusual result, so far obtained in a mixture of 1-propanol with water at a single composition, deserves confirmation by experiments in another mixture at more than one composition. Toward this goal, we have chosen mixtures of methanol with water at concentrations of water ranging from 10 to 40 mol % and investigated the dynamics of the slower water and the faster methanol components by broadband dielectric relaxation measurements. The alpha-relaxation time of the water component becomes shorter with increasing content of the faster methanol component in the mixture as expected and is much shorter than in the mixture of 1-propanol with 45 mol % water. In mixtures with lower water contents of 10-20 mol %, the alpha-relaxation of the methanol component has a narrow frequency dispersion and no resolved Johari-Goldstein beta-relaxation, indicating a low degree of intermolecular coupling or cooperativity of methanol. An increase of the content of the slower water component effectively enhances intermolecular coupling of the methanol component. Consequently, the alpha-relaxation of the methanol component becomes more cooperative, as evidenced by broadening of its frequency dispersion and the appearance of a resolved Johari-Goldstein beta-relaxation of methanol when the water concentration is higher than 30 mol %. The observations are rationalized by application of the coupling model.