Pizza cost comparison using mental math

I was recently at Jupiters Pizza in Champaign, IL, and we were trying to figure out what to order.

We were deciding between getting individual 9 inch pizzas at $7 a piece, or splitting a couple of medium 14 inch at $14 a piece.

For fun, I wanted to know which pizza was a better value in terms of total area (as is customary, the size refers to the diameter of the pizza). Usually it is the case that larger pizzas are better values, but it is not always the case, so I like to verify.

As I was slowly making the calculation on my cell phone calculator, my friend quickly calculated the 14 inch pizzas were a better deal.

How did he figure it out so fast? Here’s the neat trick he used.

The mental math my friend did

My friend used the following logic: he would compare the ratio of the cost to the ratio of the pizza size.

The ratio of cost is easy: the medium pizza is 2 times as expensive (14/7).

The question is: do you get more or less than 2 times pizza? Rather than think about circular areas and radii, my friend compared the ratio of diameters squared:

Ratio of areas = (14/9)2 = (1 + 5/9)2 = (1.55…)2 > (1.5)2 = 2.25

My friend explained that for 2 times the cost, you are getting more than 2.25 the amount of pizza. Therefore, the medium is a better value.

(Now I’ll admit, the above calculation relies on some other tricks too, like knowing 5/9 is 0.555… (similar trick for other integers from 1 to 8: like 4/9 is 0.4444), and knowing that 15 squared is 225. But I’d hope that arithmetic is part of one’s civic education).

Why does the trick work?

The textbook way to solve the problem is to compute the total areas of each pizza, and then compare the ratio of the areas.

But when you take the ratio of two areas, certain terms will cancel out, like the factor of π. Additionally, there’s no need to calculate the radius of each pizza and then divide: the factor of 1/2 on the diameter also cancels out.

Here is a proof of why the ratio of diameters squared is the same as the ratio of areas squared. Let’s say the larger pizza has radius R and diameter D versus the smaller pizza has radius r and diameter d

So there you go, the ratio of the diameters squared is the same as that of the total areas.

Long way to solve the problem

Here’s the way I was goign to solve the problem in three steps:

1. Figure out the radius of each pizza (d / 2)
2. Calculate the area each pizza, but ignore the π term (r2)
3. Calculate the unit cost of each pizza (cost/area)
4. Compare the unit costs

This method will give the same exact answer, of course, but it takes a lot more time and gives precision that is not necessary.



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  • http://ericherboso.com Eric Herboso

    But the crust! If you enjoy crust, you’ll miss out on so much crust area by getting the medium.

    The crust width does not vary between small and medium, so small pizzas have a much higher crust-to-noncrust ratio.

    As a crust lover, I feel this MUST be taken into account.

  • Guest

    For what it’s worth, the x/9 trick works for x=9 as well, though that’s less of a “helpful trick” and more of a “mathematical curiosity”.

  • jmhajek

    As a crust hater, I agree that this MUST be taken into account!

  • John

    I’m a little surprized to see this here as a “trick”. Most, if not all, who visit this site on a regular basis seem to be quite adept at math. I would have thought they would all know that an area is proportional to the square of any linear dimention that can be used to define the size (for a circle, radius, diameter, circumferance can all be used; for a rectangle, either edge, the diagonal, or the circumferance could be used – and these are by no means exhaustive lists).

    So, all you need to do is compare 9^2=81 to 14^2=196. 2*81=162 so clearly the 14″er is the better deal. Since 13^2 is 169, even 13″ at twice the price is better (although close enough that the crust lovers would no doubt opt for the 9″ pies).

    Calculating the squares of 2 digit numbers isn’t that taxing, but here are a few “tricks” that might make some even less difficult.

    This one is only for computer people – 16^2=(2^4)^2=2^8 (8 bits = 1 byte) =256.

    The sqare of n*10+5= n*(n+1)*100+25 (so 25^2=625, and 65^2=4225).

    (n+1)^2=n^2+2n+1 or n^2+n+(n+1) (so 21^2=400+40+1=400+20+21, and 66^2=4225+130+1=4225+65+66 – and 59^=3600-60-59=2500-100+1).

    This one is probably a little less usefull, and is an extention of the last. (n+2)^2=n^2+4n+4 (so 32^2=900+120+4 – also 2^10=1024 or 1k for computer folk- and 23^2= 625-100+4)

    P.S. I’m not at all impressed with the new DISQUS format.

  • Rast2

    I did it like this:
    first pizza: 9*9/7
    second pizza: 14*14/1 =14
    9*9/7 : 14
    9*9 : 14*7
    9*9 < 14*7
    so the second pizza is a better value.





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