You allow your class to have a pizza party. The pizza parlor has a special: The first pizza in an order costs $15.99, the second pizza in that order costs half the price of the first. The third pizza costs half the price of the second, and so on. If your class can only buy whole pizzas and has exactly $31.75 to spend (excluding tip and tax), what is the greatest number of pizzas that the class can purchase? Explain your reasoning.

(Source: “Menu of Problems”,  Mathematics Teaching in the Middle School, 12(5). December 2006/January 2007.)