Suppose
you are planning a pizza party for 50 friends. The local pizza parlor sells a
12-inch medium pizza for $7.99 and a 16-inch 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!