Two boxes have the same volume. one box has a base that is 222 inches by 222 inches. the other box has a base that is 444 inches by 444 inches. how many times as tall is the box with the smaller base?

It is 4 times as tall.

Let h₁ be the height of the box with the smaller base and h₂ be the height of the box with the larger base.

The volume of the box with the smaller base is given by 2(2)(h₁).  The volume of the box with the larger base is given by 4(4)(h₂). Since they are equal:

2(2)(h₁)=4(4)(h₂)
4h₁=16h₂

Dividing both sides by 4,
4h₁/4 = 16h₂/4
h₁ = 4h₂

The height of the box with the smaller base is 4 times the height of the box with the larger base.