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. Alex scoops a full bucket of the black sand and pours it into the white sand. Alex stirs this mixture for hours (no homework, I guess) until the black and white sand are perfectly mixed. Alex 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..

Wait. Now I think it's impossible to solve because we don't know how much sand is in each sandbox and it says Alex scoops a full bucket of black sand and pours it in the white sand so it might overflow I guess…….
-Ricky A.

Ok. It says that he mixes the black and white sand together. So in one box, it has nothing and in the other it has 50% black sand and 50% white sand. Then, he puts some of it in the other box. Now there is the same amount of sand in each box but all of it is mixed. In conclusion, each box contains 50% white sand nd 50% black sand. I think they are equal.
Ricky A.

Well I guess that there equal amounts of black sand and white sand because as it says same amount of sand and it also says perfectly mixed so there were equal amounts of both sands the whole time. Alex just scoops a full bucket of the black sand and pours into the white sand and mixes and then puts the mixed sand back into the sandbox so it's equal again. The amount of black or white sand never changed…:)
~Anjiya N.

I agree with anjiya because Alex adds 1 bucket of black sand to the white sand, and after it is well mixed then he removes a bucket which contains half black sand and half white sand( if the sand is well mixed). So he brings half of whatever black sand he has put in the white sand so he has one half of a bucket of black sand remaining in the white sand and one half of a bucket of white sand in the black bucket…
-Aishwarya S.

I disagree with both of you becuase what if originally there were 5 buckets of sand in each sand box and then one bucket of the black sand, which is 6/6 buckets of sand, is added to the white sand? Then after the mixture of black and white sand is completely mixed perfectly the ratio for black sand to white sand will be 1:5 and the amount of white sand in the mixture is 5/6. If Alex puts one full bucket of the mixed sand into the white sand, there is only 5/6 bucket of white sand added. So, this is why i disagree and say that the amounts of sand added to each bucket was not the same, but that there was more black sand added than white sand.
-Priyanka C.

At first i thought they would be the same, but priyanka makes sense. Even if it is evenly mixed, you arent putting the same amount of black sand back in. So then there would be more black sand added than white.
-Alex H.

What Priyanka is saying could be possible but it says that Alex stirs the mixture for hours until the black and white sand are perfectly mixed. I think if it's perfectly mixed, then the amount of black and white sand are the same … but what Priyanka is saying could be possible…
~Anjiya N.

Like Priyanka said if there were 5 buckets then maybe there might be more black sand added, but what if there were only 2 buckets of the sand… then it would be the same
~Anjiya N.

WOW….Mr. Simmons can really find problems that just make our brains go up side down and all around… confused so much!!!!
~Anjiya N.

I believe that the most important piece of information here is the amount of sand a single full bucket contains. If they were equal to the sand boxes, then the amount of sand in each container would stay the same with the mixed black and white sand being 50% each. If they did not equal the total amount of sand each sand box contains then you could never really know what fraction you end up with. Due to the lack of information, I believe it is (at this moment) impossible to solve.
-Kevin W.

I agree with Kevin, but it's not entirely impossible to solve this at the moment…but the problem is not asking how much of the sand it is but how much is being moved. It's asking which movement of sand was greater, the black in the white or the white in the black.
~Anjiya N.

Well, the answer would and should be pretty much self explanatory if we were given the amount of sand moved. But one question to ask yourself is, if both boxes were filled with sand then the sand should overflow when poured on top wouldn't it? Just something to think about.
-Kevin W.

But I don't think the sand was filled all the way to the top of the sandbox… and usually sandboxes are filled with sand to the top but you never know.
~Anjiya N.

Alright, we're getting off topic now. Anyways, I think our answers differ because of our own definitions of the sand being "perfectly mixed". From the way I see it though, I think "perfectly mixed" means that both amounts of sand are equal. That being the case, that means that each each sandbox would have 1 bucket of black and white sand…..I'll come back to it later….
-Oscar D.

Ok, I think I got it now. Since each sandbox would have 1 bucket of sand (possibility), Alex would add a full bucket of black sand to the white sand box. After Alex "perfectly mixes" it, the mixture should be 50% white sand and 50% black sand. That being the case, when we add the mixture back to the black box sand, we're only adding half a bucket of white sand. There, showing that my answer is thatthe amount of black sand added to the white sand is greater than the amount of white sand added back to the black sand.
-Oscar D.

that can only be if there is only one box of sand in the white box( before black sand added) for the black and white sand mixed to be 50/50, so that means you have taken all the sand in the black sand box and emptied it into the white sand box! so you will take one box worth of "perfecty mixed sand" from the white sand box and add it to the empty used to be black sand box, which still leaves you with 50% black sand and 50% white sand in the black box!!!!And the same amount fo the white box!
-Aishwarya S.

Exactly, so was your post an objection or an agreement? Both ways, the way I said it, you're adding a full bucket of black sand to the white sand box, then adding back a mixture of 50% white sand and 50% black sand back to the black sand. So, 1 bucket of black sand is more than half a bucket of white sand.
-Oscar D.

I think Oscar maybe correct. I think it is that the amount of black sand added is greater. I don't know… it's either that they're equal or that the amount of black sand is greater
~Anjiya N.

I think that the amount of black sand added is greater than the white sand added. I mean, you're taking one bucket full of black sand and adding it to the white sand. Then you're mixing it in equally, and you take another bucket of the combined sand. (50% black, 50% white) And dumping it back into the black sandbox. So it would actually be HALF a bucket of black sand and half a bucket of white sand. That's just what I think though.
-Cindy C.

I think that the key words are equally mixed. I dont think that means that is split 50/50. I think it means that every time you go to pick up a bucketful of sand their will be an equal number of black sand grains every single time. For example their will be 20 grains in the first bucketful out of 500 and then all the other bucketfuls will be equal to the this or very, very similar. Bu ti still think that the amount of black sand added to the white sandbox is greater.
-Alay M.

From what i have read, things are starting to make sense now. i agree with aishwarya when she said that what if each sandbox had only 1 bucket each to begin with and that the amounts of white and black sand added would be equal, but then i also realized that if both boxes originally started off with more than one bucket in each, then the amount of black sand added would be greater. i believe that we have reached a comclusion.
b=black sand added
w=white sand added
s=the unknown amount of sand that each box had originally
if s=1 then b=w
if s>1 then b>w
if you think that i have made a mistake or have to add anything, please do so as soon as possible so that we can fixx this!
~Priyanka

I haven't gotten around to reading all the responses, but I think there is more black sand added to the white than the reverse. I believe this because if you mix a bit of black sand with the white, and then you put it back in the black sandbox, you are adding a bucket of white sand minus some to make up for the black sand thats in it. And a whole bucket of black sand is added to the white.
-Jack C.

The only way it could be different is if the bucket's volume is half of the sandbox' volume.
-Jack C.

Wait wait wait….Aishwarya's explanation doesn't make sense from before. The question is asking the amount of sand added both times. Look, it doesn't matter how much sand each box contains. No matter what, Alex will always add a full bucket of black sand to the whtie sand box. After he mixes it and returns the sand, the white sand will probably still have traces of black sand in it, making it less than a full bucket. Therefore, there should always be more black sand added than white/black sand (mixture) added.
-Oscar D.

Even if it's not exactly 50% black sand and 50% white sand, it would still be less than a FULL bucket of black sand. Therefore, the answer must be there's more black sand added than white sand.
-Cindy C.

I would guess that the amount of black sand added to the white sand would be greater (I am not sure but we can get this together). This is because while both amounts are a full bucket, the problem says that the sand was perfectly mixed, and so there was bound to be some black sand with the white sand when it was added to the black. Therefore, it wasn't a full bucket of white sand, and the black sand bucket was full, and the black sand added to the white sand would be a greater amount.-Dhrumil P.

Ok, this problem seems way too easy, so the answer must be somthing totally unexpected. It seems simple enough though, a full bucket of black sand was poured into the white sand, while a full bucket of mixed sand was put into the black sand. No matter how much sand is in each pile or how much sand is put into the bucket each time, there will always be more black sand added to the white sand than the other way around. And it's impossible for it to come out equal because you always add the same amount of sand each time, and the second time the sand is mixed.
-Sarah H.

Ok, I talked with Mr. Simmons today and I think the term "perfectly mixed" was just to throw us off. Basically, I think it just means that Alex thoroughly mixed both sands. The amounts of black and white sand don't have to be the same when he mixes it so he would probably end up with traces of black sand in the mixture. If you think about it in real life, what are the chances that a buck will be exactly half full of white sand and half full of black sand? And no, Sarah, sometimes the answer is just staring you in the face so you can keep it simple sometimes. Again, I stick with my answer of the black sand added to the white sand box is greater than the white sand added to the black sand box.
-Oscar D.

I TAKE BACK WHAT I SAID BEFORE AND I AGREE WITH OSCAR AND WHOEVER ELSE SAID THAT THERE WAS MORE BLAck(oops sorry i had caps lock on) than white because Sarah's logic makes sense. This problem was so easy! The answer is staring you in the face! No matter how much white sand you have, you will ALWAYS have a full bucket of black sand!
~Priyanka C.

This problem is pretty simple. When someone puts a bucket of black sand into a white sand, and mixes it, and then takes a bucket of the mixed sand and pours it into the black sand box, it only seems logical that it is more black sand addeed to white sand box. This is because when you add back the white and black sand, its not all white, there is some black, even if it is just specks.
-David B.

Hey guys! On very first thought it seemed sort of obvious to me that there was more black sand than white, but I wasnt sure. Do we have to have a mathematical approach? Well, I would think, knowing Mr.Simmons, that it is a trick problem. :)
~Aarthi S.

As Oscar said, I think the term "perfectly mixed" was there to throw us off. For all we know, it could just mean that it is in a set ratio. That doesn't neccessarly mean that it is the same amount.
~Aarthi S.

Like what David said this problem was very simple that there was more black sand than white. But I hope this isn't a trick question like what Sarah said. -Faysal S.

Sure. Either way, there is definately more black sand than white sand so when he asks us tomorrow, we will all be able to agree on an answer and that is the important thing that we have all come to a conclusion.
~Priyanka C.

I agree with David and Oscar. Mr. Simmons you just using the term perfectly mixed to throw us off. This problem seems easy, but I think there is a trick to it. David is right about there is more black sand than white because he pours a full bucket of black sand which means there should be more black than white. Even if he stirs the mixture for hours there should still be more black than white sand. I think there is more black sand than white sand in the bucket. -Taylor D.

Yall I think Mr. Simmons is trying to get us to think there is more black than white. What if he is tricking us. I'm starting to think Aarthi is right about having a mathematical approach. -Taylor D.

Thats funny everybody is thinking it might be a trick problem. Like Priyanka said, we should stick with our answer which is that there is more black sand than white sand. -Taylor D.

I agree with Sarah that the problem is sort of too simple. There has to be some kind of catch or trick to it. I think that is the amount of black sand in the white sand but you never know. There could some trick or something.
~Anjiya

I just thought of a weird way to prove our answer. Assume that both boxes contain 200 grains of sand. The bucket can hold 100 grains of sand, making the black sandbox have 100 and the white have 300. The mixture is 1/3 black sand. For the white sand to be greater than the black sand it has to be greater than 100(the amt in the black box) but that cannot be because the bucket holds only 100 grains. There cannot be equal amt because there are also some black grains in the white sandbox. In conclusion, my answer is there are more black grains than white. Does this make sense? I will get back to you and see if my method works with different numbers also, and that this wasn't just a coincidence. Does anyone agree with this method?
~Aarthi S

Umm. Yeah, I think that works with any number. Can someone else try it just to make sure that I didn't make a mistake, you know, if you have time.
~Aarthi S.

Hey Aarthi they makes sense… no wonder Mr. Simmons calls you genius. It is a good way to prove the answer. I just tried with other numbers and it works. GREAT JOB!
~Anjiya N.

I agree with Aarthi it seems to work. Logically I also believe the answer must be that there are more black grains of sand than white but maybe we should wait and ask Mr. Simmons about the true meaning of a few of the words that he used. We have had him as a teacher long enough to know that there is always a card up his sleeve, metaphorically speaking of course.
-Kevin W.

Alright people, don't over analyze the problem. Don't be intimidated by Mr. Simmons! It's what he wants us to think. Just stick to an answer that you think is logical (after everyone contributing to this wiki) and make sure that you know it's correct. Really, this problem could be simpler than all of us thought. Like when Mr. Simmons was explaining to us in class about how a rectangle's perimeter can be the same but the area would be different, it took us several guesses until we thought of saying it was slowly turning to a square. Same scenario. Have confidence in your answer and defend it (unless it's truly obvious that your wrong).
-Oscar D.

Aarthi your so smart! Oscar your right Mr. Simmons just wants us to think that. We all need to stick to the answer we got. :) Taylor D.

Oscar's right. Why else do you think I spent time on the all-crazy way to check our answer? To defend our answer, of course! I think we all agree on more black than white, right. By the way, is Mr. Simmons grading this by any chance? I wasn't here on monday and tuesday, so I need a little help. Since our answer works "mathematically" and "logically" I would be pretty darn surprised if it wasn't correct.
~Aarthi S.

Do we all agree that the black sand added to the white sand is greater?-Dhrumil P.

Yes Dhrumil, I agree with the answer, there was more black sand added than white sand. Jus think of it as this - there are 4 buckets of each sand. 4B(Black) and 4W(White) so after you mix 1 bucket of black sand with the white there will always be some black sand and each bucket of white sand. Therfore, when you put 1 bucket of white sand back in, you are putting less than 100% pure white sand in. - Omar Z.

Also, Oscar you are right….this isn't truly a very hard problem, he knows that we overthink alot and go past the basic answer. So lets just stick with the answer - There was more black sand added to the white sand than white sand added to the black sand. - Omar Z.

One last thing guys. I read some of the stuff at the top and the question isn't about how much of each colored sand is in the sandbox, the question is Which color of sand that was added to the other sand box was more? Basically, this is asking which bucket of sand that was added to the other sandbox was more "pure"? - Omar Z.

I agree that there is more black sand added than white. People from other classes have been trying to convince me that it is even because you cant have a fraction of a grain, but thats not even part of it. If the sand is mixed, you arent going to have a full bucket of white sand.
-Alex H.

Oh my gosh i just realized the names were based off of who's in the class. I'm first! YAY!!!!!! sorry … MORE BLACK THAN WHITE!!!!!!!!!!
-Alex H.

I totally agree with what Omar and Oscar think because Mr. Simmons knows that we will over think the problem and not take the obvious solution in out minds. The answer has to be more black sand than white because when you put any random numbers in you will realize that it has to be _the amount of black sand added to the white sand is greater
-Nabeel M

I also agree with Oscar, OMar and everyone else because say you add 100% of the black over to the 100% white so then u get 50 /50 right ? Then you have to move that back right? So bascically you move the 100% black WITH some white grains so THERE HAS TO BE MORE BLACK SAND. do i make sense? Reno R.

You make perfect sense, Reno.
My answer is more black than white!!
~Aarthi S.

Ok then. It seems about all of us agree that there is more black sand added to the white sand box than white sand added to the black sand box. Now we can just wait for our other classmates to post something, or prepare for another debate in case someone else finds another answer.
-Oscar D.

I agree
-Jack C.

Does anyone think this problem can be solved algebraically? I dont think so but does anyone else?
Reno R.

I think that this problem CAN be solved algebraically…I simply don't know what the formulas are. This is very similar to the solution problems that we do for homework.
Cindy C.

i just realized something even though this is kind of over. If you have black sand in the bucket, you're putting it back in when its mixed with the white so wouldn't it be even? Just saying.
Alex H.

Well first I see that Alex scooped a full bucket of black sand and pored it into the white sand. Then after he mixed the two he poured the mixed sand back into the black sand. By doing this he baisically poured the black sand that he first took out back into its sand box so therefore i think that there is more black sand than white because when he mixed the two and poured it into the black sand box, there was still the white sand in the mixture. Saraphin D.

I think we mite have to prove it to mr.simmons in an algebraic way o else he mite no except it
idk but thts what i think

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