2002 AMC 10A Problem 17

Keep a ledger of (coffee, cream) in each cup.

Start: cup 1 has $(4, 0)$, cup 2 has $(0, 4)$.

First pour: half of cup 1's coffee, $2$ oz, moves to cup 2. Now cup 1 is $(2, 0)$ and cup 2 is $(2, 4)$, six ounces total with coffee to cream in ratio $1 : 2$.

Second pour: half of cup 2's $6$ ounces, $3$ oz, goes back. Stirring means this $3$ oz has the same $1:2$ ratio, so it is $1$ oz coffee and $2$ oz cream. Cup 1 becomes $(2 + 1, 0 + 2) = (3, 2)$.

Cup 1 now holds $5$ ounces, of which $2$ are cream:
$$\frac{2}{5}.$$

The answer is $\boxed{\textbf{(D)}\ \frac25}$.

Mixture problems are bookkeeping: follow each ingredient separately through every transfer. The only real principle is that a stirred mixture is uniform, so any portion removed carries the same ingredient ratio as the whole. Writing the two amounts as an ordered pair after each step prevents losing track of the changing totals.

• Pouring half of the second cup's original 4 ounces (2 oz) back instead of half of its new 6 ounces (3 oz, in a 1:2 coffee-to-cream ratio).
• Answering the fraction of cream in the returned portion ($\tfrac23$) or of coffee in cup 1 ($\tfrac35$) instead of the fraction of cream in the whole first cup.