Question 584850: Two sandboxes, one with black sand and one with white sand, each contain the same amount of sand. You scoop a full bucket of black sand and pour it into the white sand. Trying to avoid doing your algebra homework, you stir the mixture for hours and hours until the black and white sand are perfectly mixed. You then scoop a full bucket of the mixed sand and pour it into the black sand so that both sandboxes once again have the same amount of sand.
Which is greater, the amount of black sand added to the white sand or the amount of white sand added to the black sand, or are they equal? Be sure to fully explain your reasoning with a mathematical example.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Let  represent the amount of black sand originally in the black sandbox, and let  represent the amount of black sand in one full bucket of black sand. Likewise, let  and  represent analogous quantities of white sand.
Before anything happens, there are units of black sand in the black sandbox and 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  units of black sand and the white sandbox will contain units of white sand plus 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  units of mixed sand and the formerly white but now mixed sandbox will contain  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  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
My calculator said it, I believe it, that settles it
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