Question

Dear Mitch,

Would you please reveal the answer to the middle school Thanksgiving math question about the two different sized pies. And if you would explain it a little, that would be awesome because my friend and I have two different answers, and I'm having trouble explaining to him why mine is the right one and his is way wrong.

Thank You,

Vincent D.

Seattle, WA

Answer

Dear Vincent,

Yes, of course.

First, the problem, in case someone missed it:

For Thanksgiving dessert, a bakery made special pies. They come in two sizes: small and large. Both sizes are circular in shape, and both sizes are the same thickness, density, and fluffiness.

The small pie has a diameter of 10", and the large has a diameter of 20".

If the 10" pie is priced at $3.00, how much should the 20" pie sell for? (Assume the bakery is managed by someone who uses a logical system for pricing the items in his shop).

Here are your choices:

(A) $6.00

(B) $8.00

(B) $10.00

(D) $12.00

(E) $15.00

First, I'll tell you the most popular answer. Then I will explain how to arrive at the correct one.

Most popular: Choice A, $6.00. Why? I can only assume it is because 20 inches is twice the 'size' as 10 inches, and two times three dollars is six dollars.

But Circles don't really work like that.

First, you should always pause for a moment to ask yourself this: What is the question really ABOUT? Here, the question is about the amount of food you are getting in one pie compared to another. Since the problem states that the pies are the same thickness, that factor becomes what math people call a 'constant', meaning it doesn't change, and so your thinking should be simplified from the math of 'volume' to the math of 'area'. (Area is always one step easier than volume because you only have to consider two dimensions, not three.)

So now you go to the formula for the area of a circle. That is the famous

π r².

R stands for radius, which is the measurement from the circle's center to the edge, or half the diameter. So the area of the 10 inch diameter pie is π5² = 25π.

Here there is the natural temptation to start converting 25π into a 'number' by inserting 3.14... for the other constant here, pi, but DON'T. HOLD OFF and let's see if that's necessary.

Now, the larger one has a diameter of twenty inches, so its radius is 10 inches. That pie has an area that measures π10² = 100π

100 π is FOUR TIMES THE SIZE of 25π

Therefore, if the small pie sells for $3.00, the large should sell for four times that amount, or $12.00 (choice D).

Hope that Helps,

Mitch