# 2002 AMC 12A Problems/Problem 10

The following problem is from both the 2002 AMC 12A #10 and 2002 AMC 10A #17, so both problems redirect to this page.

## Problem

Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? $\mathrm{(A) \ } \frac{1}{4}\qquad \mathrm{(B) \ } \frac13\qquad \mathrm{(C) \ } \frac38\qquad \mathrm{(D) \ } \frac25\qquad \mathrm{(E) \ } \frac12$

## Solution

We will simulate the process in steps.

In the beginning, we have:

• $4$ ounces of coffee in cup $1$
• $4$ ounces of cream in cup $2$

In the first step we pour $4/2=2$ ounces of coffee from cup $1$ to cup $2$, getting:

• $2$ ounces of coffee in cup $1$
• $2$ ounces of coffee and $4$ ounces of cream in cup $2$

In the second step we pour $2/2=1$ ounce of coffee and $4/2=2$ ounces of cream from cup $2$ to cup $1$, getting:

• $2+1=3$ ounces of coffee and $0+2=2$ ounces of cream in cup $1$
• the rest in cup $2$

Hence at the end we have $3+2=5$ ounces of liquid in cup $1$, and out of these $2$ ounces is cream. Thus the answer is $\boxed{\text{(D) } \frac 25}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 