There are two identical glasses. The first is filled with milk, while the second is filled with the same amount of coffee. From the first glass is poured a spoonful of milk in the second glass of coffee making a mixture and then a spoonful of the mixture is poured back to the glass with milk. Is there more milk in the coffee or coffee in the milk?

Solution.

Y is the amount of the milk or coffee.

X is the amount of the liquid in the spoon.

Step 1. From the first glass is poured a spoonful of milk in the second glass of coffee making a mixture.

As a result: First glass has Y-X milk; Second Glass has the mixture Y+X where Y is a coffee and X is a milk.

Step 2. A spoonful of the mixture is poured back to the glass with milk.

The spoonful of the mixture consists from X*Y / (X+Y) of the coffee and X*X / (X+Y) of the milk.

As a result: First glass has X*Y / (X+Y) coffee and Y*Y / (X+Y) milk; Second glass has Y*Y / (X+Y) coffee and X*Y / (X+Y) milk.

So, first glass has amount of coffee equal to the amount of the milk in the second glass and first glass has amount of milk equal to the amount of the coffee in the second glass.

Finally, the amount (volume) of the milk in the coffee *will be equal *to the amount (volume) the coffee in the milk.

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*Related*

the milk is slightly emptied by the first spoonful. The resulting coffee mixture is slightly diluted. The mixture is then added to the milk, but the concentration of coffee in the mixture will be slightly less than the concentration of the milk that was taken, so the milk will be less diluted than the coffee.

if you work on 10 spoons in each cup, then the milk becomes 9 spoons, and the coffee mixture becomes 11 but at 10/11 concentrate. Then the 10/11 concentrate is reintroduced to the milk and makes 1*10/11 out of 10, somewhere 91% give or take.

So the coffee is diluted to 89% and the milk is diluted to 91%. Can someone smart do the maths please>?

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