Imagine two identical cups - one filled with coffee and the other with milk, both at the same level. A teaspoon of coffee is now scooped from the coffee cup into the milk cup and stirred. A teaspoon of the mixture from the milk cup is then poured back into the coffee cup. Which cup then contains more foreign liquid?
Intuitively, one might think that there is more coffee in the milk cup, since the first teaspoon contained only coffee, while the returned teaspoon was a mixture of milk and coffee. The surprising answer is that there is the same amount of foreign liquid in both cups at the end. To do this, we analyze the amounts of coffee and milk at three points in time \(t_0, t_1, t_2\):
- \(t_0\) (at the beginning): Full cup of coffee(\(T\)) and no milk(\(0\)) in the coffee cup, no coffee(\(0\)) and full cup of milk(\(T\)) in the milk cup.
- \(t_1\) (after the first decanting): \(T-L\) coffee and \(0\) milk in the coffee cup, \(L\) coffee and \(T\) milk in the milk cup(\(L\) = teaspoon quantity)
- \(t_2\) (after the second decanting): \(T-L+L_2\) coffee and \(L_1\) milk in the coffee cup, \(L-L_2\) coffee and \(T-L_1\) milk in the milk cup(\(L_1\)= amount of milk in the teaspoon, \(L_2\) = amount of coffee in the teaspoon).
Since \(L_1+L_2 = L\), \(L-L_2 = L_1\), the amount of milk(\(L_1\)) in the coffee cup is equal to the amount of coffee(\(L-L_2\)) in the milk cup. This can be explained as follows: At the end of the experiment, the coffee cup has the same fill level as at the beginning. However, since milk was added, the same amount of coffee must have left the cup. This amount of coffee is now in the milk cup.