AMC 10 · 2006 · #13
Grade 6 rate-ratioJoe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
Pick an answer.
AMC 10 2006 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: Two people each start with $12$ ounces of coffee in a $16$ ounce cup. Joe first drinks $2$ ounces of coffee, then adds $2$ ounces of cream. JoAnn first adds $2$ ounces of cream and stirs, then drinks $2$ ounces of the mixture. Find the ratio of the cream now in Joe's cup to the cream now in JoAnn's cup.
Givens: Each cup begins with $12$ ounces of pure coffee (cup holds up to $16$ ounces); Joe: drink $2$ oz of coffee, then add $2$ oz of cream; JoAnn: add $2$ oz of cream, stir well, then drink $2$ oz of the mixture; Answer choices: (A) $\frac{6}{7}$, (B) $\frac{13}{14}$, (C) $1$, (D) $\frac{14}{13}$, (E) $\frac{7}{6}$
Unknowns: The ratio (cream in Joe's cup) : (cream in JoAnn's cup)
Understand
Restated: Two people each start with $12$ ounces of coffee in a $16$ ounce cup. Joe first drinks $2$ ounces of coffee, then adds $2$ ounces of cream. JoAnn first adds $2$ ounces of cream and stirs, then drinks $2$ ounces of the mixture. Find the ratio of the cream now in Joe's cup to the cream now in JoAnn's cup.
Givens: Each cup begins with $12$ ounces of pure coffee (cup holds up to $16$ ounces); Joe: drink $2$ oz of coffee, then add $2$ oz of cream; JoAnn: add $2$ oz of cream, stir well, then drink $2$ oz of the mixture; Answer choices: (A) $\frac{6}{7}$, (B) $\frac{13}{14}$, (C) $1$, (D) $\frac{14}{13}$, (E) $\frac{7}{6}$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units, #4 Introduce a Variable
The question compares two independent cups, so Tool #7 (Identify Subproblems) splits it into 'how much cream ends up in Joe's cup' and 'how much cream ends up in JoAnn's cup', solved separately and then divided. Tool #8 (Analyze the Units) keeps the bookkeeping straight by tracking ounces of cream against ounces of total liquid, which is the whole game here. The turning point is JoAnn's stirred drink: because the mixture is uniform, drinking a fraction of the total liquid removes that same fraction of the cream — a ratio idea. Tool #4 (Introduce a Variable) is a light backup for naming the amounts, but the two subproblems are concrete enough to compute directly.
Execute — Answer: E
4.OA.A.3 Step 1 Find the cream in Joe's cup
- Joe drinks $2$ ounces first, while his cup still holds only coffee, so that drink removes coffee and no cream at all.
- Then he pours in $2$ ounces of cream.
- Nothing leaves the cup after the cream goes in, so every bit of that cream stays.
💡 Adding cream last means none of it can be sipped away, so Joe keeps the full $2$ ounces.
6.RP.A.1 Step 2 Set up JoAnn's mixture
- JoAnn adds $2$ ounces of cream to her $12$ ounces of coffee before drinking, giving $12+2=14$ ounces of liquid, of which $2$ ounces are cream.
- After stirring, the cream is spread evenly, so cream makes up $\frac{2}{14}=\frac{1}{7}$ of every sip.
💡 A well-stirred drink has the same cream-to-liquid ratio everywhere, so one number describes the whole cup.
6.RP.A.3 Step 3 Remove JoAnn's stirred sip
- JoAnn drinks $2$ ounces of the $14$-ounce mixture, which is $\frac{2}{14}=\frac{1}{7}$ of everything in the cup.
- Because it is uniform, that sip carries away $\frac{1}{7}$ of the cream too: $\frac{1}{7}\times 2=\frac{2}{7}$ ounce of cream leaves.
- The cream left is $2-\frac{2}{7}=\frac{14}{7}-\frac{2}{7}=\frac{12}{7}$ ounce.
💡 Drinking one-seventh of a stirred cup drinks one-seventh of the cream inside it.
6.NS.A.1 Step 4 Compare the two amounts
- Now divide Joe's cream by JoAnn's cream.
- Joe has $2$ ounces; JoAnn has $\frac{12}{7}$ ounce.
- Dividing by a fraction means multiplying by its reciprocal: $2\div\frac{12}{7}=2\times\frac{7}{12}=\frac{14}{12}=\frac{7}{6}$.
- So the ratio is $\frac{7}{6}$, which is choice (E).
💡 The ratio just asks how many times bigger Joe's cream is, so divide one amount by the other.
4.OA.A.3 Joe drinks $2$ ounces first, while his cup still holds only coffee, so that drin 6.RP.A.1 JoAnn adds $2$ ounces of cream to her $12$ ounces of coffee before drinking, giv 6.RP.A.3 JoAnn drinks $2$ ounces of the $14$-ounce mixture, which is $\frac{2}{14}=\frac{ 6.NS.A.1 Now divide Joe's cream by JoAnn's cream. Joe has $2$ ounces; JoAnn has $\frac{12 Review
Reasonableness: Both cups received the same $2$ ounces of cream, but JoAnn drank some of hers back out while Joe never did, so Joe should have more cream and the ratio should be greater than $1$. The result $\frac{7}{6}\approx 1.17$ is indeed just above $1$, matching that expectation, and it rules out choices (A), (B), and (C), which are all $\le 1$. JoAnn losing only $\frac{1}{7}$ of her cream ($\frac{2}{7}$ oz) is a small loss, so the ratio should be only slightly above $1$, which fits $\frac{7}{6}$ better than a larger value.
Alternative: Work with exact ounces and skip fractions of a fraction: JoAnn's cup has $14$ oz total with $2$ oz cream, so the cream is $\frac{2}{14}=\frac{1}{7}$ of it. After she drinks $2$ oz, $12$ oz of the same uniform mixture remain, so the cream left is $\frac{1}{7}\times 12=\frac{12}{7}$ oz. Joe still has $2=\frac{14}{7}$ oz, so the ratio $\frac{14/7}{12/7}=\frac{14}{12}=\frac{7}{6}$ again gives (E).
CCSS standards used (min grade 6)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Tracking Joe's cup through its steps to see the added $2$ ounces of cream are never removed.)6.RP.A.1Understand the concept of a ratio and use ratio language (Describing JoAnn's stirred cup as cream being $\frac{2}{14}=\frac{1}{7}$ of the total liquid.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Concluding that drinking $\frac{1}{7}$ of a uniform mixture removes $\frac{1}{7}$ of the cream, leaving $\frac{12}{7}$ oz.)6.NS.A.1Interpret and compute quotients of fractions and solve word problems (Dividing $2$ by $\frac{12}{7}$ to get the final ratio $\frac{7}{6}$.)
⭐ In a well-stirred drink the cream spreads out evenly, so drinking one-seventh of the cup drinks one-seventh of the cream — and because Joe added his cream last, he never sips any of his away, leaving him with slightly more.
⭐ In a well-stirred drink the cream spreads out evenly, so drinking one-seventh of the cup drinks one-seventh of the cream — and because Joe added his cream last, he never sips any of his away, leaving him with slightly more.
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