Math Riddle

00-Riddle.txt

There are each a cup of milk and a cup of coffee (same amounts).
First you transfer one spoon of milk into the coffee.
After achieving an optimal mixture, transfer one spoon of the milky coffee into the milk.

Is there more coffee in the milk or more milk in the coffee?

01-Solution.txt

Start with:

V amount of milk
V amount of coffee

Transfer a spoonful of milk, S

V-S milk
V coffee + S milk ;  a V:S coffee:milk proportion

Cup 2 now contains V+S liquid.  S amount of milk, V amount of coffee.
S/(V+S) is the fraction of milk in the mixture,
V/(V+S) the fraction of coffee.
(These fractions add up to one: S/(V+S) + V/(V+S) = (S+V)/(V+S) = 1)
So the fraction of coffee can also be expressed in terms of the fraction of milk:
1 - S/(V+S) coffee

A spoonful of cup 2 would contain:

S * (S/(V+S)) milk
S * (1 - S/(V+S)) coffee
S - (S*S)/(V+S) coffee

Transfer a spoonful from cup 2 back to cup 1:

Cup 1 therefore has (V-S) + S*(S/(V+S)) of milk, and S*(V/(V+S)) coffee
Cup 2 has V - S - (S*S)/(V+S) coffee, and S - S*(S/(V+S)) milk:

We see that each cup has the same amount of their original liquids:
V - S - S*S/(V+S)

Answer: Neither cup has a greater proportion of one substance than the other.

02-Solution-with-values.txt

Start with:

100 ml milk
100 ml coffee

Transfer a spoonful of milk (5ml):

95 ml milk
100 ml coffee + 5ml milk ;  a 20:1 coffee:milk proportion

Transfer a spoonful from cup 2 back to cup 1:

95 ml milk + 20:1 proportion of 5 ml of mixture
100 ml mixture of 20:1.

Cup 1 therefore has 95 + 0.24ml of milk + 4.76ml coffee
Cup 2 has 95.24 ml coffee, 4.76ml milk.

Answer: Neither cup has a greater proportion of one substance than the other.