Answer: Juice = 7/18 and water = 11/18.
As we're talking about halves and thirds, it's easiest to break down the glasses into anything that is divisible by 2 and 3, such as 6 units for the size of the smaller glass, the larger glass is therefore 12 units.
The first glass (size 6) contains 3 units of juice and 3 units of water.
The second glass (size 12) contains 4 units of juice and 8 units of water.
Once combined the tumbler (size 18) will therefore contain 7 units of juice and 11 units of water. QED.