AMC 10 · 2002 · #17
Grade 6 rate-ratioSarah 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?
Pick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A cup holds $4$ ounces of coffee; another same-size cup holds $4$ ounces of cream. Pour half the coffee into the cream cup, stir, then pour half of that mixed cup back. Find what fraction of the first cup is now cream.
Givens: Cup 1 starts with $4$ ounces of coffee; Cup 2 starts with $4$ ounces of cream; Both cups are the same eight-ounce size; First pour: half of Cup 1's coffee goes into Cup 2; Cup 2 is stirred thoroughly, so it becomes a uniform mixture; Second pour: half of the liquid now in Cup 2 goes back into Cup 1; Answer choices: (A) $\frac14$, (B) $\frac13$, (C) $\frac38$, (D) $\frac25$, (E) $\frac12$
Unknowns: The fraction of the liquid in Cup 1 that is cream after both pours
Understand
Restated: A cup holds $4$ ounces of coffee; another same-size cup holds $4$ ounces of cream. Pour half the coffee into the cream cup, stir, then pour half of that mixed cup back. Find what fraction of the first cup is now cream.
Givens: Cup 1 starts with $4$ ounces of coffee; Cup 2 starts with $4$ ounces of cream; Both cups are the same eight-ounce size; First pour: half of Cup 1's coffee goes into Cup 2; Cup 2 is stirred thoroughly, so it becomes a uniform mixture; Second pour: half of the liquid now in Cup 2 goes back into Cup 1; Answer choices: (A) $\frac14$, (B) $\frac13$, (C) $\frac38$, (D) $\frac25$, (E) $\frac12$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units, #4 Introduce a Variable
The process has two pours, so Tool #7 (Identify Subproblems) says handle them one at a time and just keep a running tally of how many ounces of coffee and how many ounces of cream sit in each cup. Tool #8 (Analyze the Units) keeps that tally honest: every number is an amount in ounces, and the final answer is (ounces of cream in Cup 1) divided by (total ounces in Cup 1). The one idea that makes the second pour easy is that a stirred cup is uniform, so pouring back half of it pours back exactly half of its coffee and half of its cream — the ratio is preserved. Tool #4 (Introduce a Variable) is only lightly needed: instead of unknown letters we track two concrete quantities, coffee and cream, through each step.
Execute — Answer: D
5.NF.B.4 Step 1 Pour half the coffee across
- Cup 1 has $4$ ounces of coffee.
- Half of that is $\frac12 \times 4 = 2$ ounces, which moves to Cup 2.
- Cup 1 is left with $2$ ounces of coffee (and no cream).
- Cup 2 started with $4$ ounces of cream and now also has the $2$ ounces of coffee, so Cup 2 holds $2$ ounces of coffee and $4$ ounces of cream, $6$ ounces in all.
💡 Taking half of a $4$-ounce amount just gives $2$ ounces, and that coffee simply joins the cream already in the other cup.
6.RP.A.1 Step 2 See Cup 2 as a fixed mixture
- Stirring makes Cup 2 uniform: its $6$ ounces are $2$ ounces coffee to $4$ ounces cream, a coffee-to-cream ratio of $2:4 = 1:2$.
- So in every ounce of this mixture, $\frac13$ is coffee and $\frac23$ is cream.
- This ratio is the same in any scoop you take out, which is what makes the next pour easy to split.
💡 A well-stirred drink tastes the same in every sip, so any portion of it has coffee and cream in that same $1:2$ mix.
6.RP.A.3 Step 3 Pour half of Cup 2 back
- Half the liquid in Cup 2 is $\frac12 \times 6 = 3$ ounces, poured back into Cup 1.
- Since Cup 2 is a $1:2$ coffee-to-cream mixture, those $3$ ounces split the same way: $\frac13 \times 3 = 1$ ounce of coffee and $\frac23 \times 3 = 2$ ounces of cream.
- Adding this to Cup 1 (which held $2$ ounces of coffee) gives Cup 1: $2 + 1 = 3$ ounces of coffee and $0 + 2 = 2$ ounces of cream.
💡 Because the cup is uniform, pouring back half of it pours back half the coffee and half the cream, so the $3$ ounces break into $1$ and $2$.
6.RP.A.3 Step 4 Take the cream fraction
- Cup 1 now holds $3$ ounces of coffee and $2$ ounces of cream, a total of $5$ ounces.
- The fraction that is cream is the cream amount over the total: $\frac{2}{5}$.
- That matches choice (D).
💡 A fraction of a cup is just the part you care about (cream) divided by everything in the cup.
5.NF.B.4 Cup 1 has $4$ ounces of coffee. Half of that is $\frac12 \times 4 = 2$ ounces, w 6.RP.A.1 Stirring makes Cup 2 uniform: its $6$ ounces are $2$ ounces coffee to $4$ ounces 6.RP.A.3 Half the liquid in Cup 2 is $\frac12 \times 6 = 3$ ounces, poured back into Cup 6.RP.A.3 Cup 1 now holds $3$ ounces of coffee and $2$ ounces of cream, a total of $5$ oun Review
Reasonableness: Check that the cream is conserved. There were $4$ ounces of cream to begin with; the second pour sent $2$ ounces into Cup 1, leaving $2$ ounces still in Cup 2 — total still $4$, so no cream was invented or lost. A size check also fits: Cup 1 ended with $5$ ounces of liquid and started with none of the cream, so its cream fraction must be below $\frac12$; $\frac25$ is just under half, which is sensible. The trap answer $\frac12$ comes from forgetting that Cup 1 also gained an ounce of coffee in the pour-back (total $5$, not $4$).
Alternative: Track by conservation instead of by pours (Tool #16, Change Focus). Both cups end with $5$ ounces because $3$ ounces moved right on pour one net... more directly: after everything, the coffee is conserved at $4$ ounces total and cream at $4$ ounces total across the two $5$-ounce cups. Cup 1 has $2$ ounces cream, so Cup 2 must have the other $2$ ounces cream. Then Cup 1's cream fraction is $\frac{2}{5}$ — the same answer (D) without re-deriving each pour amount.
CCSS standards used (min grade 6)
5.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a fraction (Taking half of the $4$-ounce coffee ($\frac12\times4=2$) and half of the $6$-ounce cup ($\frac12\times6=3$).)6.RP.A.1Understand the concept of a ratio and use ratio language (Describing the stirred Cup 2 as a fixed $1:2$ coffee-to-cream mixture, uniform in every portion.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Splitting the $3$ ounces poured back into $1$ ounce coffee and $2$ ounces cream, then reporting cream as $\frac{2}{5}$ of Cup 1.)
⭐ Keep a running count of coffee and cream ounces: a stirred cup is uniform, so pouring back half of it returns half of each, leaving Cup 1 with $2$ ounces cream out of $5$, which is $\frac25$.
⭐ Keep a running count of coffee and cream ounces: a stirred cup is uniform, so pouring back half of it returns half of each, leaving Cup 1 with $2$ ounces cream out of $5$, which is $\frac25$.
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