There are two cups of different liquids. Both liquids are equal in volume. If I take a tea spoon of one liquid and put it in the other, then mix that until the entire liquid is equally concentrated, then take a tea spoon of that cups mixture and put it in the first cup, which cup has a greater concentration of its original liquid and why?
Hello, Obsidantion!
This is a classic (very old) problem.There are two cups of different liquids. Both liquids are equal in volume.
If I take a teaspoon of one liquid and put it in the other,
then mix that until the entire liquid is equally concentrated,
then take a teaspoon of that cups mixture and put it in the first cup,
which cup has a greater concentration of its original liquid and why?
We have equal amounts of the two liquids, O's and X's.Code:| | | | | | | | | | | | |OOOOO| |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| *-----* *-----* A B
A specific amount is tranferred from A to B.Code:| | | | | | |OOOOO| | | |OOOOO| | | |XXXXX| | | |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| *-----* *-----* A B
The combination is thoroughly mixed.
Code:| | | | | | |OXOXO| | | |XOXOX| | | |XXXXX| | | |XXXOX| |OOOOO| |OXOXX| |OOOOO| |XOXXX| |OOOOO| |XXOXX| *-----* *-----* A B
The same amount is transferred from B to A.
Code:| | | | | | | | | | | | |OXOXO| |XXXXX| |XOXOX| |XXXOX| |OOOOO| |OXOXX| |OOOOO| |XOXXX| |OOOOO| |XXOXX| *-----* *-----* A B
If we let the liquids "separate", we see the obvious.Code:| | | | | | | | | | | | |XXXXX| |OOOOO| |OOOOO| |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| |OOOOO| |XXXXX| *-----* *-----* A B
The amount of X's in cup A must be equal to the amount of O's in cup B.
Neither cup has a greater concentration of its original liquid.
I just want to add something more,
think of 2 extreme cases
1. the spoon has capacity of 0 units
2. the spoon is able to take all of the liquid from the first cup to the second
then think of limit ideas approach those two extreme cases;
my result is whatever capacity of the spoon, same original liquid rate for them.
Both cups have an equal concentration of their original liquid. Congrats to those who got it right.
Here's one way of explaining why that is:
You have two cups of different liquids (white and grey).
You take an amount of one (the right) and put it in the other (the left). I haven't mixed the liquids in the left cup so that I can represent the volume of each liquid that the mixture contains clearly.
You then take that same amount from that one (the left) and put it into the first (the right). In the diagram bellow I haven't reduced the volume of the left cup so it no longer represents its true volume, but it does still represent its true concentration.
If you increase the volume of the mixture in the right cup and all of the components of that mixture proportionately, the concentration of the liquid will remain the same. Imagine that the right cup's mixture is increased in volume (so that the concentration remains the same) to that of the left cup's volume (length 'a' increases to 'b'). The volume of the white liquid within the right cup will also increase. Length 'c' of the white liquid will increase to 'd'. This is because, as the represented upper section of the right cup was taken from the left cup, they have the same concentration and so 'a' is the same proportion of 'b' as 'c' is of 'd'.
If length 'c' of the white liquid in the right cup increases to 'd' and length 'a' of the whole volume of the liquid in the right cup increases to 'b', we are left with the same representation of concentration in both cups. (We know that the volumes of the represented upper sections of each cup are the same because they both represent the same amount that we took from the cups at stages 2 and 3).
-Hope that made sense.
If 5 units transferred from 100 units cup A to 100 units of cup B.
After mixing, and transferring 5 units from B (which suppose now contain 2 units of A and 3 units of B)
Total of liquid A in cup A = 97 units
Total of liquid B in cup B = 97 units