AMC 10 · 2006 · #11

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
Two people each drink and add cream in opposite orders, starting from the same cup. Find the ratio of the cream left in the two cups.

Pick an answer.

(A)
$\frac 67$
(B)
$\frac {13}{14}$
(C)
1
(D)
$\frac {14}{13}$
(E)
$\frac 76$
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

Drinking first removes no cream, so that cup keeps all of it.

Joe's cream=2 oz
2STEP 2

Set up JoAnn's mixture

The other cup is stirred, so the cream is a fixed fraction throughout.

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

Remove JoAnn's stirred sip

The sip removes that share, leaving 12/7.

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

Compare the two amounts

Dividing gives 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

More like this

Same archetype — closest grade level first.