Question 584850
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Let *[tex \Large B] represent the amount of black sand originally in the black sandbox, and let *[tex \Large b] represent the amount of black sand in one full bucket of black sand.  Likewise, let *[tex \Large W] and *[tex \Large w] represent analogous quantities of white sand.


Before anything happens, there are *[tex \Large B] units of black sand in the black sandbox and *[tex \Large W] units of white sand in the white sandbox.  After the one bucket of black sand is taken from the black sandbox and added to the white sandbox, the black sandbox will contain *[tex \Large B\ -\ b] units of black sand and the white sandbox will contain *[tex \Large W] units of white sand plus *[tex \Large b] units of black sand.


After the white sandbox has been "perfectly" mixed and a full bucket of mixed sand is taken out, the bucket will (because the mixture is "perfect") contain exactly *[tex \Large \frac{b}{2}\ +\ \frac{w}{2}] units of mixed sand and the formerly white but now mixed sandbox will contain *[tex \Large W\ +\ \frac{b}{2}\ -\ \frac{w}{2}] units of mixed sand (because half a bucket each of white and black were taken away -- the white from what was already there at the start and the black from the whole bucket of black that was added).


When the bucket of mixed is added back to the black sandbox, now there is *[tex \Large B\ -\ \frac{b}{2}\ +\ \frac{w}{2}] in the formerly black sandbox; half a bucket of black being returned leaving a deficit of half a bucket of black and half a bucket of new white sand that has been added.


So the amount of black added to the white is equal to the amount of white added to the black.


Super Double Plus Extra Credit:


Let's say you didn't bother spend the "hours and hours" to get the "perfect" mixture.  Instead, you just dump the bucket of black sand into the white sandbox and then scoop up a bucketful of the white sandbox contents at random such that anywhere from 0 to 100% of the bucket contains black sand.  After you dump the contents in the black sandbox, answer the "Which is greater..." question again.   


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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