If a toy boat is added to the side of a tank of water that is balanced on a central pivot, why doesn't the tank tip? Suppose we have the following situation:
A toy boat is placed in a tank of water that is perfectly balanced before. Which way, if at all, will the tank tip?

As the boat is placed in the water, by Archimedes' Law, it displaces a weight of water equal to its own weight, meaning there must be no net downwards force where the boat is added.
I assume, however, that slightly more of the displaced water would be distributed to the side opposite the boat, as the boat itself obstructs the water:

However, the correct answer is that there is still no resultant torque. How is this the case?
 A: Your  

by Archimedes' Law, it displaces a weight of water equal to its own
  weight

provides the answer: In your drawing you can replace the boat displacement with the equivalent amount of water.  Then you can see that floating the boat is the same as simply adding the boat displacement's amount of water to the tank without the boat, which only increases the depth of the water (with no boat) which would not effect the balance of the tank.
A: I think it should be pointed out that the book answer is only correct if it assumes the toy floats or can otherwise remain suspended in the water (not sink to the bottom). 
If the toy is heavy enough such that when submerged the downward force of gravity on the toy plus any water it may "contain" is greater than the upward buoyant force of the water (which equals the weight of the volume of water displaced), the toy will sink to the bottom and contribute a clockwise moment (torque) about the balance point.
I would also add the statement that the toy "displaces a weight of water equal to its own weight" can be misleading. If the toy is heavy enough to sink, adding more weight to the toy does not displace any more water.
Hope this helps.
A: The answer lies in the comment givenby user45664 above.
The total mass in the tank is increased by the amount of mass of the boaat.  However th boat floats, so we know that the amount of upward force on the boat is the same as the amount of downward force on it.  It is held in place by the water which is, of course, a fluid.
The total mass of the water in the tank is transmitted evenly through the water to the base of the tank as the water pressure across the whole tank is the same.
A: Add the boat slowly.
If you add the boat fast enough that it makes waves, the waves are an unbalancing force. Because you have added the boat faster than the water could adjust. That may or may not be enough to get the tank to tip over.
