A vessel is filled with liquid, 4 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

Initial ratio: water:syrup = 4:5, total = 9 parts. Water is 4/9, syrup is 5/9. Let x be the fraction drawn off. After drawing x and replacing with water: remaining water = (4/9)(1-x) + x, remaining syrup = (5/9)(1-x). For equal parts: (4/9)(1-x) + x = (5/9)(1-x). Solving: 4 - 4x + 9x = 5 - 5x, so 4 + 5x = 5 - 5x, giving 10x = 1, thus x = 1/10.