Wednesday, November 18, 2009

The Milk and the Tea

Consider two identical cups, one half-full of tea, the other half-full of milk.  You take a teaspoon of milk from the milk cup and put it in the tea cup.  Then you take a spoonful out of the tea cup and put it in the milk cup.  Now: is there more tea in the milk cup, or more milk in the tea cup? 

There is the same amount of milk in the tea cup as there is tea in the milk cup.  There are a couple of approaches to this problem.  One approach involves using fractions and hard numbers - easier conceptually, but requires a lot of calculation.  Note: it may help to draw out the steps as they are explained.  The relative volume of the spoon and cups are arbitrary, so to make the math easy, let 1 spoonful = 1/2 a cup.  Secondly, let each cup contain 6 ounces of liquid. Then a spoonful is 1/2*6=3 ounces. So when we take a spoonful out of the milk cup, that is 3 ounces. So we take 3 ounces of milk out of the milk cup and put it into the tea cup.  Now the tea cup contains its original 6 ounces tea plus 3 ounces milk.  And the ratio of tea to milk in this cup is 2 parts tea, 1 part milk.  So when we mix it up and take a 3-ounce spoonful of out of the tea cup, it will contain 2 ounces of tea and 1 ounce of milk, leaving 4 ounces of tea and 2 ounces of milk left in the tea cup.  When we add this spoonful to the 3 ounces that we left in the milk cup originally, we will have 4 ounces of milk and 2 ounces of tea in the milk cup.  Thus the two cups mirror each other in relative quantities of milk and tea. 

Another approach to this is to keep in mind that each cup must have the same volume in the end as it did in the beginning.  In the milk cup, we removed a spoonful and then added a spoonful.  In the tea cup, we added a spoonful, then removed a spoonful.  The key part here is this: in the end, whatever milk is missing from the milk cup MUST be replaced by tea.  Likewise, whatever tea is missing from the tea cup MUST be replaced by milk.  So if in the end of the problem the milk cup has two FEWER ounces of milk than it started with, it must have two MORE ounces of tea than it started with.  This is the only way for it to have the same amount of liquid in the end as it did in the beginning.

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