2002 AMC 10A Problem 17

P16-20~3 minPrint

Ratios, Percents & Averages Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Keep track of coffee and cream separately. Stirred liquid keeps the same mix, so the 3 ounces poured back are 1 ounce of coffee and 2 of cream.

Solution

Mixing problems are about careful bookkeeping. Write down how much coffee and how much cream is in each cup after every pour.

	cup 1 coffee	cup 1 cream	cup 2 coffee	cup 2 cream
start	$4$	$0$	$0$	$4$
after pour 1	$2$	$0$	$2$	$4$
after pour 2	$3$	$2$	$1$	$2$

Pour 1. Half the coffee in cup 1 is $2$ ounces. It goes into cup 2, which now has $2$ ounces of coffee and $4$ of cream: $6$ ounces in all.

Stirring. After stirring, every bit of cup 2 is the same mix: $2$ parts coffee to $4$ parts cream, which is $1$ part coffee to $2$ parts cream.

Pour 2. Half of cup 2 is $3$ ounces (half of $6$ , not half of $4$ ). With the $1$ to $2$ mix, those $3$ ounces are $1$ ounce of coffee and $2$ ounces of cream. They go into cup 1, which now has $2 + 1 = 3$ ounces of coffee and $0 + 2 = 2$ ounces of cream.

4. Answer. Cup 1 holds $3 + 2 = 5$ ounces, and $2$ of them are cream:
$\frac{2}{5}.$

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

Why this works

Track each ingredient on its own through every step; a table makes it hard to lose anything. The one physical fact used is that a well-stirred mixture is the same everywhere, so any amount poured out has the same mix as the whole cup. Also notice that the total in a cup changes after each pour, so "half" means half of what is there now.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(D)}\ \frac25}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)