AMC 10 · 2006 · #13

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
ratio-proportionfraction-arithmeticpercentage identify-subproblems ↑ Prerequisites: ratio-proportion
📏 Medium solution 💡 3 insights
Problem
Joe and JoAnn 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.

Pick an answer.

(A)
$\frac{6}{7}$
(B)
$\frac{13}{14}$
(C)
1
(D)
$\frac{14}{13}$
(E)
$\frac{7}{6}$

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

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.

1STEP 1

Find the cream in Joe's cup

Joe sips first, while his cup is still pure coffee, so no cream leaves — the 2 ounces he pours in afterwards all stays.

Joe's cream=2 oz
2STEP 2

Set up JoAnn's mixture

JoAnn stirs 2 ounces of cream into her 12 of coffee, so the cup holds 14 ounces and cream is 214=17\frac{2}{14}=\frac{1}{7} of every sip.

total=12+2=14 oz, cream fraction=2/14=1/7
3STEP 3

Remove JoAnn's stirred sip

Her 2-ounce sip is 17\frac{1}{7} of the uniform cup, so it carries off 17\frac{1}{7} of the cream, leaving 127\frac{12}{7} ounce.

cream drunk=1/7 × 2=2/7; cream left=2-2/7=12/7 oz
4STEP 4

Compare the two amounts

Divide Joe's cream by JoAnn's: 2÷127=2⋅7122 \div \frac{12}{7} = 2 \cdot \frac{7}{12}, which is 76\frac{7}{6} — choice (E).

2/ 12/7 =2×7/12=14/12=7/6 → (E)
Answer
7/6
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 7/6≈ 1.17 is indeed just above 1, matching that expectation, and it rules out choices (A), (B), and (C), which are all ≤ 1. JoAnn losing only 1/7 of her cream (2/7 oz) is a small loss, so the ratio should be only slightly above 1, which fits 7/6 better than a larger value.
💡Key takeaway

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.

  • Find the cream in Joe's cup
  • Set up JoAnn's mixture
  • Remove JoAnn's stirred sip
  • Compare the two amounts

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