Suppose you ate 1/3 of a pizza the first night and 1/4 of what remained every night after that. how many nights would it take until the pizza was half gone?

Ok, so you eat 1/3 of the pizza the first night. That would leave 2/3 of the pizza uneaten. 2/3 is equal to about 0.67.

You're now going to eat a quarter of what's left of the pizza. A quarter of 2/3 is 2/12 which is 1/6. Subtract 1/6 from 2/3... 4/6 - 1/6 = 3/6 = 1/2. The pizza is already half gone.

At the rate you've decided to eat the pizza, it's going to be half gone after 2 nights.