A Solution to the Pizza Party Problem

Suppose you are planning a pizza party for 50 friends. The local pizza parlor sells a 12” medium pizza for $7.99 and a 16” large pizza for $12.99.  Which pizza is the better deal?  How many pizzas should be ordered if a medium pizza serves about 3 people?  Is your answer any different if you think that most of your friends will not eat the pizza crust?  Explain the reasoning for your answers.

Solution:

G:  Generally it is more economical to buy a larger-quantity product, often called the “economy size.”  However, it is possible that the medium pizzas are “on sale” at a special discount price that is lower than usual.  About 50/3 = 17 medium pizzas would be needed to feed 50 people, and the number of large pizzas would be less than this, maybe about 12 if each large pizza serves 4 people.  The large pizzas also seem like the better option if a minimum amount of crust is desired since small pizzas are mostly crust.

O:  This is a ratio problem that can be solved with simple algebra and the formula for the area of circle:  A = (PI)r2.  The price per unit area is the best way to compare the cost of the two pizzas.

A:  The cost per unit area of the medium pizza is:   $7.99/PI(6 in)2 = $0.071/in2
      The cost per unit area of the large pizza is:     $12.99/PI(8 in)2 = $0.065/in2
      So a large pizza is about a 10% better buy than a medium.

Assuming that one medium pizza will adequately feed 3 people, then 50/3 = 16.7, so 17 medium pizzas would be needed, at a cost of $135.83.
However, we want to find the number of large pizzas that equal the same total amount of pizza:
    N(large)A(large) = N(medium)A(medium)

Therefore,    N(large)  =  [PI(6 in)2 / PI(8 in)2 ](50/3) = 9.4
    So 10 large pizzas are needed, at a cost of $129.99 (a savings of $5.84 over the price for medium pizzas).

The amount of pizza crust is proportional to the circumference of the pizza pies:  C = (PI)D.
 The medium pizzas would have about 16.7(12 in.)(PI) = 630 in. of crust.
 The large pizzas would have about 9.4(16 in.)(PI) = 470 in. of crust.
    Since the large pizzas have less total crust, they are still the better option.

L:  As we suspected, the large pizzas are the best option, both in terms of cost and minimum crust.  In solving this problem, it is important to distinguish between the area of the circle and the circumference.  Careful attention to units helps avoid confusing these two formulas.
Several assumptions were used in solving this problem.  Depending on how hungry your friends are and what other food is being served, the assumption of one medium pizza feeding 3 people may not be valid.  We also assumed that the specified prices are fixed and no coupons or other discounts apply.  Most importantly, this problem was not complicated by the challenging task of deciding what toppings should be ordered on the pizzas!