Correct Option

10kg

Let V represent the weight of the empty vessel and W represent the weight of the water when the vessel is completely full.

From the problem statement, two linear equations can be established:

The vessel full of water weighs 40 kg: V + W = 40 kg (Equation 1)
The vessel one-third filled with water weighs 20 kg: V + (1/3)W = 20 kg (Equation 2)

To determine the weight of the water (W), subtract Equation 2 from Equation 1:

(V + W) - (V + (1/3)W) = 40 - 20

W - (1/3)W = 20

(2/3)W = 20

W = (20 * 3) / 2

W = 30 kg

Now, substitute the calculated value of W (30 kg) into Equation 1 to find the weight of the empty vessel (V):

V + 30 = 40

V = 40 - 30

V = 10 kg

Therefore, the weight of the empty vessel is 10 kg.

Incorrect Options

Options 2, 3, and 4 are incorrect as they do not satisfy the conditions presented in the problem statement when substituted as the weight of the empty vessel.

If the empty vessel weighed 15 kg (Option 2), the total water capacity would be 40 - 15 = 25 kg. A one-third filled vessel would then weigh 15 + (1/3)*25 = 23.33 kg, which contradicts the given weight of 20 kg.
If the empty vessel weighed 20 kg (Option 3), the total water capacity would be 40 - 20 = 20 kg. A one-third filled vessel would then weigh 20 + (1/3)*20 = 26.67 kg, which contradicts the given weight of 20 kg. This option represents the weight of the vessel when one-third filled, not the empty vessel.
If the empty vessel weighed 25 kg (Option 4), the total water capacity would be 40 - 25 = 15 kg. A one-third filled vessel would then weigh 25 + (1/3)*15 = 30 kg, which contradicts the given weight of 20 kg.

Only an empty vessel weight of 10 kg is consistent with both conditions provided in the problem.