Problem 1

## Here's The Problem…

Two sandboxes each contain the same amount of sand, but one is filled with black sand and other is filled with white sand. Jessica scoops a full bucket of the black sand and pours it into the white sand. Jessica stirs this mixture for hours (she could be helping her mother) until the black and white sand are perfectly mixed. Jessica then scoops a full bucket of the mixed sand and pours 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? — Mr. Simmons

The amounts added to both are equal because you add a whole bucket of black sand initially and then mix it and put a half bucket of white into black and a half bucket of black. So, they are equal because you put back the half bucket of black sand.
-Courtney W.

I think the amount of black sand is greater because the first time you were adding a whole bucket of black sand, and the second time it was a mixture of both black and white. (which would be half black and half white). But the solution seems to simple…
[Courtney- Would do you mean by "so they are equal because you put back the half bucket of black sand?" I thought they wouldn't be equal because its you're comparing a whole bucket and half bucket…]
~Jane L.

ok, based on my math i think they are equal.
i'll put up the reasoning after i finish.
-Jane Lee

Ya I think it's equal if you count the sand that goes back into the black sand after you add the mixture but if you just count the original amount it would be black.. but like jane said that would be too simple.
-Courtney W.

Hm, I've heard of this one before. (Though I don't guarantee that my answer is actually correct.)
I think I have to agree with Courtney and Jane. The initial amount of black sand that was mixed with the white didn't matter in the end because all of the sand ended up being mixed together anyway.
-Stacyyy T. ((:

Whoa. Wait a sec. It says that:
Jessica scoops a full bucket of the black sand and pours it into the white sand. [and then later she mixed it for like 249872894 hours until the black and white sand are perfectly mixed.] Then, Jessica then scoops a full bucket of the mixed sand and pours it into the black sand
Did Jessica start out with only one full bucket or more? Is it just me not understanding this problem correctly or am I overanalyzing it or what?! :O
-Stacyyy T. ((:

I think that it'll be equal because the amount of sand that was transferred is the same. She took one bucket of black sand and poured it into the white. She took the same quantity of the mixture (probably the same bucket?) and poured it back into the black sand. So that proves that they're equal (I agree with Courtney, Jane, and Stacy).
-Sai Aparna K. :)

Ya I think it's definately equal because she uses the same bucket and the first obvious answer would be that black would be more.. and it's obviously not white.. so i think we got this! (hopefully) :)
-Courtney W.

yayy! yea i hope its right too… :D Stacy- I think you're overanalyzing this ;) but yea she starts out with 2 sandboxes… and then takes a bucket and transfers stuff…
-Sai Aparna K.

Let's all hope that the amount being equal is the answer! When I made an example of each sandbox containing 100 grains and each full bucket contain 10 grains, my answers proved that the amount was equal. The first time, there were 91 white and 9 black. If I use those numbers and try pouring the one bucket of white into the black, I get 91 black and 9 white. So, I think that clearly proves that they're equal.
~Jane Lee :D

Well, when I originally read the problem, I thought it was equal, but the question says "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?" so its not asking about at the end how much of each is in the othe, but when they are actually being poured which one has more going into the other. Then it would be balck. But if it were asking Which is greater the black IN the hwite or the white IN the balck then it would be equal. You could interpret the question either way and i might be wrong (i probably am..) but its just a thought.
~~ Maha A. (:

I also think that there will be an equal amount in the end. I agree with Jane - examples are a good idea. First I solved it with algebra, and then to test it, I used examples. The numbers I used were 500 and 100. They worked pretty well. So lets do this together. If I scoop 100 grains of black sand into the white sand, I get 600 total grains of sand in the 2nd bucket. Now, after mixing, i scoop 100 grains of sand back into the black bucket. Since it is mixed evenly now, 100/600 (1/6) of the grains are black and 500/600 (5/6) are white. If I use the same bucket to scoop out 100 grains, 1/6 will be black (16 2/3 grains) and 5/6 will be white (83 1/3 grains).
So now, in bucket 1, I have 416 2/3 grains of black sand and 83 1/3 grains of white sand. In bucket 2, I now have 416 2/3 grains (500 original minus the 83 1/3 we mover to bucket 1) of white sand and 83 1/3 grains of black sand left. Since 416 2/3 = 416 2/3 and 83 1/3 = 83 1/3, there are the same amounts!!! YAY!
~~~Robert C.

woah! great example robert! (:
-Aparna K. :D

random. but. WEHAVEABEASTLYCLASSS. :DDD
okay. im done. :))
-Stacyyyyy T. ((:

Thanks Aparna. We only have a beast class cause everyone works hard and stuff. This is getting kinda off topic, but since we did the problem, what highschool are you guys going to?
~~~Robert

haha u asked me this in class.. but im going to kempner. :P :P :P izzy's going there too (: are you going to austin? btw- so freakishly scared about tomorrow! D:
-Aparna :D

Way to go 3rd period! we got it right to begin with! haha! and by the way.. AUSTIN! :) im going to miss everyone sooo much next year :(
-Courtney W.

page revision: 29, last edited: 02 May 2009 02:50