A fish tank with a length of 12 inches, width of 10 inches, and height of 15 inches is half-filled with water. How many metal cubes (which sink) with side length 2 inches must be dropped in the fish tank to raise the water at least 3.5 inches?

In this question, how do we know that we can ignore if this tank is already half filled?

The question asks how many cubes “must” be dropped, which is to say, what is the least number of cubes. The least number of cubes is satisfied when:

each cube displaces as much water as possible (when you drop the cube in, there is no part of the cube sticking above water).

Life is easiest if we then also assume that

the cubes can be placed in a way that does not cause any of the cubes to rise above the water

the starting water level is higher than the height of a cube.

Knowing that this is “half filled”, or ~7.5 inches tall, allows us to make these assumptions of convenience.

If you want to make this problem more difficult, consider the scenario where the water initially only fills 0.001 inch of the height. I would reason that in this case, you need 60 cubes rather than 53 to raise the water level by 3.5 inches (60 cubes perfectly filling the bottom 4 inches of the tank, which raises the tiny amount of water enough).

In the scenario where the water initially fills 0.05 inches of the height, I would reason that you need 59 cubes (one full layer of cubes, then one empty cube in an otherwise filled layer where all of the water sits).

If you want to avoid all of this, all you care about is that the water level is high enough that the cubes are fully submerged in water when dropped in. Half full simplifies things