# Math Problem - 8



## nitrocan (Aug 14, 2008)

This is more likely a logic problem. (you can think of it mathematically as well)

You have 2 of the exact same cups, with exact same volume of milk in one of them and coffee in the other one. You take one spoon of milk, and stir it into the coffee until its completely homogenized. Then from that mixture, you take one spoon and then stir it into the milk.

Now, does the milk have more coffee in it, or does the coffee have more milk? (don't think like a chemist because thats not what the question is asking)


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## AvGalen (Aug 14, 2008)

That depends on the size of the cup compared to the size of the spoon. Just imagine what happens when the spoon would be bigger than the cup



Spoiler



But if we assume that the size of the spoon is pretty small compared to the size of the cup the coffee has more milk because
1) You took a full spoon of milk away
2) You returned a mixture of coffee and milk


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## nitrocan (Aug 14, 2008)

well yes the spoon is not bigger than the cup and can fit in, but the comparison is not given.


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## brunson (Aug 14, 2008)

Actually, I'm pretty sure that they have the same ratio. I'll work on the proof.


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## fanwuq (Aug 14, 2008)

I'd say is if the spoon is smaller than half the cup, eventually, they will both be half milk and coffee. I haven't taken chemistry or calculus yet. Going to take both once school starts in a few days.


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## Lucas Garron (Aug 14, 2008)

I don't like coffee.


Spoiler



Assuming reasonably assumable things: however much coffee in the coffee cup is missing (and replaced by milk) is in the milk cup, replacing exactly as much milk. Same amounts.

Can I have a problem that I haven't seen before?


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## kratos94 (Aug 14, 2008)

Haha thats interesting... hokay here comez a wild guess... HOkay given that the coffee originally is counted completely as coffee(the milk added to originally make it is not taken into account). AND that the amount of milk added to the coffee during the question is counted as part of the coffee now. The milk would have more coffee in it. Here is my proof(sorry this may be a little retarted >.<)

1. When the milk is taken and mixed in with the coffee the milk becomes a part of the coffee itself, and what is coffee but some weird stuff which i don't know(sorry im not a fan of coffee or tea or anything like that) and some milk(i think.. >.<). Which gives us 1 cup of coffee + 1 spoon of coffee. While the milk is now equal to 1 cup of milk - 1 spoon of milk. Then we take 1 spoon of coffee and put that into the milk. This gives us...(drumroll please) 1 cup of coffee and 1 cup of milk - 1 spoon of milk + 1 spoon of coffee. The coffee has no milk and the milk has some coffee. Alright hope at least someone understands that...


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## JBCM627 (Aug 14, 2008)

Agree with lucas. The most obvious check of this is taking a spoonful of nothing.


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## badmephisto (Aug 14, 2008)

I love coffee! 
Lucas thats a nice way of looking at it. Pretty obvious once you read it.
Solved mathematically:



Spoiler



start:
1st cup: A of milk
2nd cup. A of coffee

after 1st transfer:
1: A-s milk
2. s of milk, A of coffee

after 2nd transfer:
1. A-s+s/2 = A-s/2 milk, s/2 coffee
2. s-s/2 =s/2 milk, B-s/2 coffee

same amount...



fun note, it doesnt matter that there is same amount of milk in 1st cup as there is amount of coffee in 2nd cup. This is a little more obvious from the 'math' solution. This one was pretty fun!  I didn't believe it for the first few seconds


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## cmhardw (Aug 14, 2008)

Without having read any other spoilers (yet):



Spoiler



I see 3 possible cases:

case 1: The volume of the spoon is less than or equal to the volume of one of the cups individually. In this case I get that the volume of coffee in the milk cup after both spoonfulls is V*S/(V+S) where V is the volume of the cup and S is the volume of the spoon. The volume of milk in the coffee is S - S^2/(V+S) = V*S/(V+S) after both spoonfulls. Thus the volume of milk in the coffee cup exactly equals the volume of coffee in the milk cup. If the volume of the spoon equals the volume of the cup the above formulas still hold and each reduces to half the volume of each cup is the other liquid, which is still saying that each cup contains the same amount of the other liquid.

case 2: The volume of the spoon is greater than the volume of the cup, but less than the combined volume of both cups. After both spoonfulls the volume of coffee in the milk cup is S/2 where S is the volume of the spoon. The volume of milk in the coffee cup is 1/2 * (2V - S) = V - S/2. In this case there is more coffee in the milk cup than there is milk in the coffee cup.

case 3: The volume of the spoon is greater than or equal to the combined volume of both cups. In this case there is more coffee in the milk cup than milk in the coffee cup, because the coffee cup is empty.



I did not expect this result, that's cool.

Chris


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## AvGalen (Aug 14, 2008)

I didn't expect the "normal" result, but it seems so obvious now


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## cmhardw (Aug 14, 2008)

AvGalen said:


> I didn't expect the "normal" result, but it seems so obvious now



I still don't understand the "normal" result on an intuitive level, even though I know it's true. I have to think of it mathematically written out to understand it. You guys are your crazy mathematical intuitions 

Chris


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## guusrs (Aug 14, 2008)

Guys,
You don't have to solve this mathematically, only logically.
(When using the same spoon) it's obvious the cups contain the same amounts of liquid before and *after* the mixture. Any milk removed is in the coffee cup and vice versa. So the "polution" is the same for both liquids.


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