Maths Problem : Which is more contaminated
Discussion
Something an old teacher told me many years ago, and I can't seem to find any details on it online.
Puzzle/problem is based on having 100ml of liquid A, and 100ml of liquid B. Both are in separate containers of 150ml volume, each labeled A and B respectively to what they contain. You have a spoon that holds exactly 10ml of liquid.
The problem is based on deciding which is more contaminated after you first take 1 spoon from A and mix it into B, so it is evenly distributed/mixed. And then take 1 spoon from B and mix it into A.
For ease of argument, the liquids will mix.
Which is more contaminated, A or B?
I recall the answer being they are equally contaminated, but cannot recall the maths/logic to prove it. Can anyone assist?
Am I correct on this basis:
Initially:
A = 100ml of A
B = 100ml of B
1 10ml spoon of A is added to B and mixed evenly.
You now have
A = 90ml of A
B = 100ml of B + 10ml A
1 10ml spoon from B is added into A.
The spoon is made up (10/110)x100 of B, plus (10/100)x10 of A. This rounds to 9.1B and 0.9A.
After mixing back into A this leaves:
A = 90.9A + 9.1B
B = 90.9B + 9.1A
So both are equally contaminated with 9.1ml of the other liquid.
Am I proving this correctly?
Puzzle/problem is based on having 100ml of liquid A, and 100ml of liquid B. Both are in separate containers of 150ml volume, each labeled A and B respectively to what they contain. You have a spoon that holds exactly 10ml of liquid.
The problem is based on deciding which is more contaminated after you first take 1 spoon from A and mix it into B, so it is evenly distributed/mixed. And then take 1 spoon from B and mix it into A.
For ease of argument, the liquids will mix.
Which is more contaminated, A or B?
I recall the answer being they are equally contaminated, but cannot recall the maths/logic to prove it. Can anyone assist?
Am I correct on this basis:
Initially:
A = 100ml of A
B = 100ml of B
1 10ml spoon of A is added to B and mixed evenly.
You now have
A = 90ml of A
B = 100ml of B + 10ml A
1 10ml spoon from B is added into A.
The spoon is made up (10/110)x100 of B, plus (10/100)x10 of A. This rounds to 9.1B and 0.9A.
After mixing back into A this leaves:
A = 90.9A + 9.1B
B = 90.9B + 9.1A
So both are equally contaminated with 9.1ml of the other liquid.
Am I proving this correctly?
Edited by JatHanspal on Wednesday 14th November 17:24
JatHanspal said:
Something an old teacher told me many years ago, and I can't seem to find any details on it online.
Puzzle/problem is based on having 100ml of liquid A, and 100ml of liquid B. Both are in separate containers of 150ml volume, each labeled A and B respectively to what they contain. You have a spoon that holds exactly 10ml of liquid.
The problem is based on deciding which is more contaminated after you first take 1 spoon from A and mix it into B, so it is evenly distributed/mixed. And then take 1 spoon from B and mix it into A.
For ease of argument, the liquids will mix.
Which is more contaminated, A or B?
I recall the answer being they are equally contaminated, but cannot recall the maths/logic to prove it. Can anyone assist?
Theres some of A in B, and some of B in A. But they are both back to their original levels. So whatever is missing from A has been replaced exactly by whats in B and vice versa.Puzzle/problem is based on having 100ml of liquid A, and 100ml of liquid B. Both are in separate containers of 150ml volume, each labeled A and B respectively to what they contain. You have a spoon that holds exactly 10ml of liquid.
The problem is based on deciding which is more contaminated after you first take 1 spoon from A and mix it into B, so it is evenly distributed/mixed. And then take 1 spoon from B and mix it into A.
For ease of argument, the liquids will mix.
Which is more contaminated, A or B?
I recall the answer being they are equally contaminated, but cannot recall the maths/logic to prove it. Can anyone assist?
Thus they are equally contaminated.
If one liquid is water, and one is ethanol it could get complicated. I seem to remember from my A-level chemistry that if you mix, say, 100ml of water with 100ml of ethanol you don't finish up with 200ml of 50% ethanol, but somewhat less than 200ml, can't remember the exact numbers. Would this affect the above calculations? And what if the flasks were on a moving conveyor belt 
Edit to add: This explains it: https://www.carolina.com/teacher-resources/Interac...
Edit to add: This explains it: https://www.carolina.com/teacher-resources/Interac...
Edited by Vizsla on Wednesday 14th November 17:47
FredAstaire said:
Theres some of A in B, and some of B in A. But they are both back to their original levels. So whatever is missing from A has been replaced exactly by whats in B and vice versa.
Thus they are equally contaminated.
I'd disagree. The spoon that went from A->B was all A, The spoon that goes from B->A is partly made up of A, so a little of A makes it back to the original container. Hence B is more contaminated.Thus they are equally contaminated.
xeny said:
FredAstaire said:
Theres some of A in B, and some of B in A. But they are both back to their original levels. So whatever is missing from A has been replaced exactly by whats in B and vice versa.
Thus they are equally contaminated.
I'd disagree. The spoon that went from A->B was all A, The spoon that goes from B->A is partly made up of A, so a little of A makes it back to the original container. Hence B is more contaminated.Thus they are equally contaminated.
By contamination if we are saying % of volume of each other then B would have a very small % larger volume of A in it than A has of B as some of the spoon from B would also have molecules of A in it where as the spoon that went into B has 100% A.
Reminds me of the maths required to replace gearbox oil by not flushing it but just draining the sump. It takes a lot more changes than you would think to get anywhere near 99% fresh oil.
Reminds me of the maths required to replace gearbox oil by not flushing it but just draining the sump. It takes a lot more changes than you would think to get anywhere near 99% fresh oil.
Henners said:
xeny said:
FredAstaire said:
Theres some of A in B, and some of B in A. But they are both back to their original levels. So whatever is missing from A has been replaced exactly by whats in B and vice versa.
Thus they are equally contaminated.
I'd disagree. The spoon that went from A->B was all A, The spoon that goes from B->A is partly made up of A, so a little of A makes it back to the original container. Hence B is more contaminated.Thus they are equally contaminated.
Try it with 100 hypothetical jelly beans in each hand. Move them about but end up with 100 in each hand. it's either 100 in each, or 99+1, or 98+2, or....
red_slr said:
If you have 100 red and 100 blue and take 10 red and mix with the 100 blue so you now have 110.
Then remove 10 from the 110 the statistical chance is you will not get the 10 red back out.
True, but whatever combination comes out, the opposite combination will be left in. So each side is equally 'contaminated' by the other.Then remove 10 from the 110 the statistical chance is you will not get the 10 red back out.
SpeckledJim said:
red_slr said:
If you have 100 red and 100 blue and take 10 red and mix with the 100 blue so you now have 110.
Then remove 10 from the 110 the statistical chance is you will not get the 10 red back out.
True, but whatever combination comes out, the opposite combination will be left in. So each side is equally 'contaminated' by the other.Then remove 10 from the 110 the statistical chance is you will not get the 10 red back out.
Doesn’t work with Jelly beans (or anything with discrete values). The OP stated perfect mixing. If you take it to the logical extreme and add all of A into B and then half of the result back to A you have exactly the same amount of A and B in each. It only works if you assume mixing...
RalphyM said:
Doesn’t work with Jelly beans (or anything with discrete values). The OP stated perfect mixing. If you take it to the logical extreme and add all of A into B and then half of the result back to A you have exactly the same amount of A and B in each. It only works if you assume mixing...
There's no difference (unless you're allowing for measurement error in non-discrete values?)Whether it's 1 out of a hundred, or 0.000000ml out of a litre, the principle is the same.
If you mix 2 equal quantities of anything and end up with the same equal quantities at the end, then they will contain equal proportions of each other.
Otherwise, something has to have been added or removed from the system, and the question doesn't allow for that, as I read it.
RalphyM said:
We’re saying the same thing. I was talking about the poster who used the jelly bean analogy. That gets into the realms of probabilities.
In the mixing scenario, they have to end up the same.
You misread it, I'm afraid. The jellybean analogy was merely to simplify the imagery, to help envisage matching 'units' moving between vessels.In the mixing scenario, they have to end up the same.
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