## Wednesday, September 23, 2009

### How Quickly Can You Do It?

Which is larger: one-third times one-third of a dozen dozen, or one-third dozen halved and cubed?

1. Math in public...

1/3 * 1/3 * (12*12)
12 * 12 = 144
1/3 * 144 = 48
1/3 * 48 = 16

[(1/3 * 12) * 1/2]^3
1/3 * 12 * 1/2 = 2
2^3 = 2 * 2 * 2 = 8

So I'd guess one-third times one-third of a dozen dozen

is larger

than one-third dozen halved and cubed.

2. Slightly different method, same answer: the first one is larger. If you wanted to do it really quickly, the first one is squared before dividing, but the second one is cubed after dividing. I think the time of division is the key.

3. (1/3)*(1/3)(12*12)=16
(1/3)(12)(.5)=8
So the first one would be larger

4. Wait, the second one should've looked like (1/3)(12)(.5)^3=8

5. A mathematician might have generalized and come to the wrong conclusion:
If we let X = 12/3, then the two formulas are
X^2 and (X/2)^3. For large enough X, the cubed is going to get much larger. But for X small enough, the first is going to be bigger. In this case, X = 4 is small enough for the first to be bigger. As an aside, at X = 8 the two functions are equal.

Enough lecturing... USAF Pilot, Charil and satyrnxoxo had it right. 16 is bigger than 8.

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