After drinking ⅜ of the water, there would be ⅝ remaining. ⅓ of this is ⁵⁄₂₄, leaving ⅔ of ⅝ which is ¹⁰⁄₂₄=⁵⁄₁₂.
An easy way of looking at this is to suppose he had 24 fluid ounces of water at the beginning of the day. So he drank ⅜ which is 9 fluid ounces, leaving 15. He drank a further ⅓ which is 5 fluid ounces leaving 10 fluid ounces. And 10 fluid ounces is ¹⁰⁄₂₄ of the amount of water he started with, which cancels down to ⁵⁄₁₂.