how many mL of ethanol would be required in 225 mL of water to make a 38% ethanol by volume solution?

Let y mL be the volume of ethanol.

y / (y + 225) = 38/100
100y = 38 (y +225)
100y = 38y + 8550
62y = 8550
y = 138

Volume of ethanol = 138 mL

38% ethanol means that the solution is 38 parts ethanol to 62 parts water. (Notice that 38 + 62 = 100.) So each part is 225 mL / 62 = 3.629 mL.  We need 38 of these parts, and 38 * 3.629 mL = 137.9 mL. We need 137.9 mL of ethanol.