A vessel full of water weighs 40 kg. If it is one-third filled, its weight becomes 20 kg. What is the weight of the empty vessel?

Let W_v be the weight of the empty vessel and W_w be the weight of the water when the vessel is full.

1. From the first statement: A vessel full of water weighs 40 kg. This means: W_v + W_w = 40 kg (Equation 1)

2. From the second statement: If it is one-third filled, its weight becomes 20 kg. This means the vessel's weight plus one-third of the full water weight is 20 kg: W_v + (1/3)W_w = 20 kg (Equation 2)

To find the weight of the empty vessel (W_v), we can solve these two equations.

Subtract Equation 2 from Equation 1: (W_v + W_w) - (W_v + (1/3)W_w) = 40 - 20 W_w - (1/3)W_w = 20 (2/3)W_w = 20

Now, solve for W_w: W_w = 20 * (3/2) W_w = 30 kg

This is the weight of the water when the vessel is full. Now substitute W_w = 30 kg back into Equation 1: W_v + 30 = 40 W_v = 40 - 30 W_v = 10 kg

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

Analyzing the options: A) 10kg: This matches our calculated weight for the empty vessel. B) 15kg: If the empty vessel weighs 15kg, then full water weight is 40-15=25kg. One-third filled would be 15 + (1/3)*25 = 15 + 8.33 = 23.33kg, not 20kg. C) 20kg: If the empty vessel weighs 20kg, then full water weight is 40-20=20kg. One-third filled would be 20 + (1/3)*20 = 20 + 6.67 = 26.67kg, not 20kg. D) 25kg: If the empty vessel weighs 25kg, then full water weight is 40-25=15kg. One-third filled would be 25 + (1/3)*15 = 25 + 5 = 30kg, not 20kg.

The final answer is A.