## Friday, July 13, 2012

### A Pizza Puzzle

Summer is the time of picnics, various camps, late sport activities, music in the park, take out dinners and of course... pizza.  In our Pizza Palace a Large pizza costs 1.5 as much as a Small pizza. And, the diameter of a Large pizza is 1.5 times than that of a Small pizza. Is it a better deal to buy three Small pizzas, two Large or the same?

Your answers are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.

Ilya said...

While some may disagree, let’s assume that value of a pizza is proportional to its total area. :-). Also, for simplicity in area calculations let's use radius (and so the same ratio of 1.5 will apply).

So, a better deal would be one where price per area is less. Price ratio of three small to two large pizzas is (3*P / 2*(1.5*P)) = 1. Area ratio is (3*(Pi*R^2)) / (2*(Pi*R^2*1.5^2)) = 2/3. So you get less area for the same price with three small pizzas, therefore 2 large ones is a better deal.

Jerome said...

Area_Large = 1.5 Area_Small
2*large = 3 * Small

Price Small * 1.5 = Price Large
3 Small = 2 Large

The price is the same either way.

Checking on a brand name pizza parlor this ratio is no wheres near the correct one.

A 10 inch and 12 inch pizza have roughly the ratio of 1 to 1.5 in area. The price of the 10 is 15 dollars (a medium is usually a 10) and the price of a large is just 2 dollars more.

A consumer is nuts to get the medium.

Tom said...

2 large pie are squared

Wang said...

This is a matter of comparing the price of the pizza vs. how much pizza you get in terms of area

The small pizza has a diameter of x

The large pizza therefore has a diameter of 1.5x

The area of the small pizza is pi*(x/2)^2 = pi*x^2/4

The area of the large pizza is pi*(3/2*x/2)^2 = pi * 9x^2 / 16

The large pizza is pi*(9x^2/16) / pi*(x^2/4) ==
9/4 ~ 2 times bigger.

The large pizza costs 1.5 times the cost of the small pizza. Let's say the small pizza costs M; therefore the large pizza costs 1.5*M.

You are getting pi*(x^2/4) / M (pizza per dollar) for the small pizza

You are getting pi*(9x^2/16) / (1.5*M) or pi*(18x^2/48) / M (pizza per dollar) for the large pizza.

Is it a better deal to get 3 small pizzas or 2 large ones?

3 small pizzas: 3 * pi * (x^2/4) / M = pi*(3x^2/4) / M

2 large pizzas: 2*pi*(18x^2/48)/ M = pi*(36x^2/48)/M = pi*(3x^2/4) / M

Both offer you the same amount of pizza per dollar therefore both are the same.

Anonymous said...

I took a random measure of diameter of 10" for the small pizza and 15 for the large. The formula for area of a circle is pi r squared. The small pizza had an area of 78.5sq. in. The large was 176.625 sq. in. I multiplied 1.5 for the increase in price of a large pizza times the sq. in. of the small pizza and that equaled 117.76 sq. in. so the increase in price of the large on is not enough to make up for the increase in diameter and area of the large pizza since 117.76 is no where near 176.625 sq. in.

Gurubandhu

Beny Ben said...

hello Ben here, on my google account. The answer would be the same either way the size and cost would be exactly the same if you buy 3 small pizzas or 2 large pizzas. since the cost and size is 1.5 compared to the small pizzas 2 large pizzas would both have 1.5 and 1.5 + 1.5 = 3 the equivalent to 3 small pizzas cost and size.

anne-marie said...

The deal would be the same. The size and cost of two large pizza are equivalent to the size and cost of three small pizzas. ( 2x1.5=3)

Anonymous said...

kj submitted her answer as an image. Click "back" button after viewing to return.

Annie said...

The deals are the same. The cost per each unit area is the same. The diameter of the lg. pizza is 1.5 times the diameter of the small pizza. Using the equation A (circle) = 3.14d^2/4 shows us that the area of the lg. pizza is 1.5 x the area of the small pizza. Since the cost for the lg. pizza is also 1.5 times that of the small it's a 1:1 ratio. Letting x = area (or cost) of the small pizza, for 3 small the area (or cost) is 3x and for the lg. the area (or cost) is 1.5 (2x) = 3x, the same!

(d^2 I'm assuming means "d squared" judging from others answers. In any case, that's what I mean! )

Maria said...

OK, it seems that choosing a pizza is not as simple as it looks. Perhaps restaurants should have online ordering pizza area calculators. Some of you are making a money-wasting choice!

Check out kj's drawing above.

Yes, 2 large pizzas cost the same as 3 small ones. On this we all agree (because large costs 1.5 x small).

But the area of the pizza is proportional to a square of the radius. So the area of 2 big pizzas: 2 x Pi x R^2 = 2 x Pi x (D/2)^2 = Pi/2 x D^2

Area of 3 small pizzas: 3 x Pi x r^2 = 3 x Pi x (d/2)^2 = 3Pi/4 x d^2

now, we know that D = 1.5 d
substitute:

Area of 2 big pizzas: Pi/2 x (1.5d)^2 = 1.125 x Pi x d^2

Area of 3 small pizzas: 0.75 x Pi x d^2

Two large pizzas are a better choice!
Puzzle point only for those who chose well.

Jerome said...

Every so often, I don't read a question. Today is the day I reconfirm that. I took the 1.5 to be an area. Ie the area of the large is 1.5 times as large. That ratio is roughly the difference between a 10 and 12 inch pizza.

Area 10 inch = pi*(5^2)
Area 12 inch = pi*(6^2)

Area 12/Area 10 = 36/25 = 1.44

It is instructive to note that the large pizza chains (in the US) sell pizza this way. My conclusion stands: you would have to be crazy to buy a medium pizza.

Sure wish I'd seen that word diameter.