Weight of vessel full of water = 40 kg

Weight of vessel one-third filled with water = 20 kg

Weight of water in one-third filled vessel = $20$ - weight of empty vessel Let the weight of the empty vessel be $x$ kg.

Weight of water in one-third filled vessel = $20 - x$ Since the water fills one-third of the vessel, the weight of water in a fully filled vessel is $3 \cdot (20 - x)$.

Total weight of full vessel = $x + 3 \cdot (20 - x) = 40$ Solve for $x$: $x + 60 - 3x = 40$ $-2x = -20$ $x = 10$