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. Kahir scoops a full bucket of the black sand and pours it into the white sand. Kahir stirs this mixture for hours (the Internet is down) until the black and white sand are perfectly mixed. Kahir 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

Your Thoughts…

I'm first! <—is that inappropriate chat speak?
Well, I think there should be more of the black sand, since Kahir started out with a whole bucket full of black sand. Once he mixed it so that they were equal, he poured it back into the black sand thing (with both black and white sand) when the white sand is only half of the bucket while the black sand was the whole bucket.

i think i'm confused, but.. it's a thought.

-michelle l.

Well, i would say the amount of black of black sand added was greater because the bucket was only black sand and nothing else. The other bucket of sand that was added to the black sand had mostly white sand, but had some black sand mixed into it, so it isnt 100% one color, like the other bucket.

-ryan d.

the amount of blacks and is greater because a full scoop of black sand was dumped onto the white sand and the scoop of the white and black sand dumped on the black sand would be less. the only problem is that the solution seems too simple and obvious- so is it a trick question of some sort or not?

~Ann K.

i think i got the answer…….
i remember it from a book or something. :) its kinda hard to explain

first, pretend each pile has 10 pounds to start with. The bucket used to transfer sand holds 5 pounds.

the contants of the white pile will be on top and the black pile will be on bottom

10xW and 0xB
0xW and 10xB

transfer 5 W to other side

5xW and 0xB
5xW and 10xB

now is the slightly hard part
the {5xW and 10xB} on the bottom could also be written as {5xW, 5xB, and 5xB}
since a total of 5 pounds has to be taken back, 5/3 is taked from each part.
therefor, 5/3xW is taken back and 10/3xB is taken back
here is what you are left with

6+2/3xW and 3+1/3xB
3+1/3xW and 6+2/3xB

therefor, you can see that there is 3+1/3 of the foreign type of sand
the solution i got was that both will be the same

-Ayush S.

wow ayush…
well shouldn't it be more white sand? (agreeing with ryan. xD)
because if he picks up only one bucket of black which is all black right?
so its 100 black into however much white sand.
then his mixes it, and taking a mixture of the two he places the bucket back into the black sand box.
(man this guy must've been REALLY bored)
therefore, there must be more black sand moved.

-Peter F. :D

here's an easier way to understand
pretend each has 2 grains of sand the bucket can also carry 2 grains
take 2 grains from the left to the right
mix the 4 grains together
and if you randamly take 2 grains back, probability says you will probably take 1 black and 1 white
therefor, there will be 1 black grain and 1 white in each pile

-Ayush S.

well, if you think about it liek that ayush,
you may have a point.

-Peter F.

Okay here I go! Ayush it seems as though you are trying to solve for tha amount of sand in each bucket. The question, however, is asking wether the black sand added to the white sand is greater than, less than, or equivalent to the amount of white sand being added to the black sand. I therfore agree with peter and ryan. If the bucket of black sand contained 100% black sand, and is stirred in with the white sand, then the white sand is not 100% white (well that sounds weird). if you then take the mixture sand in a bucket and pour it into the black sand, then the poured sand was NOT 100% white. This means that the black sand added to the white sand is more than the white sand added to the black sand. that's just my thoughts. oh and i REALLY hope somebody understood that

-Conner J

Gahh! Ayush confused me.
I think Ann has a point about it being too easy… Mr. Simmons wouldn't give simple questions.
So Ayush might be right……. Strange as that would be :)
-Michelle L.

however conner, when taking the white sand back to the black area, some black sand goes with it.
what i think mr.simmons is asking is which is greater, the amount of white sand in the black sand or the amount of black sand in the white sand
im not sure what he is asking, it could be either one of us.

-Ayush S.

Yeah, I'm kind of confused on the way he writes the problem… Is he asking the total amount of sand added between each or how much there is after he dumps the sand back in? hmm.

-Michelle L.

The problem sort of depends on your view of what "perfectly mixed" means… it could mean 50% of each type of sand (black/ white), or a proportional value of how much back sand is in the white sand (ex- 1 grain of black sand and 10 grins of white sand). In my opinion, "perfectly mixed" means that there is 50% of each type of sand.
~Ann K.

I think examples are a good way to solve this problem… so just say that they're 50 grains of sand in each sandbox and the bucket can scoop out 10 grains of sand. To begin with, you scoop out 10 grains of black sand and mix it with the 50 white grains of sand to produce 10/60 black grains of sand (1/6) and 50/60 white grains of sand (5/6). Using the same bucket to scoop out the mixed sand, the grains in the bucket will be 1/6 black (1 2/3 grains) and 5/6 grains will be white (8 1/3 grains). So now in bucket 1, there are now 8 1/3 grains of white sand and 41 2/3 grains of black sand. In bucket 2, there are now 8 1/3 grains of black sand and 41 2/3 grains of white sand. Since 8 1/3= 8 1/3 and 41 2/3= 41 2/3, the amount of black sand added to the white sand and the amount of white sand added to the black sand are equal. I guess when I first saw the question, I didn't do any math and just decided that black sand added had to be greater, but it actually isnt, as shown above… if I confused you, just tell me.
~Ann K. :D

Thats exactly what i was saying ann :)


You confused me! Haha, but I'm guessing you're right, since Mr. Simmons wouldn't give us a problem that we could look at and know.
Unless you know, we're super smart…..
HAHA. sorry. I'll think about it, but I sort of get it now.

-Michelle L.

is it agreed the the final answer is equal
if anyone objects, speak now or forever hold your peace

-Ayush S.

When is the next problem going to be posted?

-Ryan D.

Was Ayush right? I wasn't here today.

-Michelle L.

The problem is, Ayush is right if you are talking about the amount of black/white sand in each box after all the transfers are finished. BUT the question is asking which color was ADDED more, so i still think it's the black one.

-Ryan D.

My friend said he gave the answer today. Oh wait, you guys were taking TAKS.
The answer's equal, apparently.

-Michelle L.

Told you

-Ayush S.

yay I'm right…
-Ann K.

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