AMC 10 · 2002 · #10

Grade 6 rate-ratio
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Ratio Shift — Two State
fraction-arithmeticratio-proportion identify-subproblems ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights
Problem
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.

Pick an answer.

(A)
$\frac{1}{4}$
(B)
$\frac13$
(C)
$\frac38$
(D)
$\frac25$
(E)
$\frac12$
How to solve
Strategy Identify Subproblems

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.

1STEP 1

Pour half the coffee across

Half of 4 is 2 ounces of coffee moved across, leaving Cup 1 with 2 ounces and Cup 2 with 6 ounces.

Cup 1: 2 oz coffee Cup 2: 2 oz coffee + 4 oz cream = 6 oz
2STEP 2

See Cup 2 as a fixed mixture

Stirred, Cup 2 is 2 coffee to 4 cream, so every part is 1/3 coffee and 2/3 cream.

coffee:cream = 2:4 = 1:2 → 1/3 coffee, 2/3 cream in any part
3STEP 3

Pour half of Cup 2 back

Pouring back 3 ounces returns 1 coffee and 2 cream, so Cup 1 holds 3 coffee and 2 cream.

1/2(6)=3 oz back = 1_coffee + 2_cream → Cup 1: 3 oz coffee + 2 oz cream
4STEP 4

Take the cream fraction

Cream over total is 2/(2+3) = 2/5, choice (D).

cream/total = 2/(2+3) = 2/5 → (D)
Answer
2/5
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 1/2; 2/5 is just under half, which is sensible. The trap answer 1/2 comes from forgetting that Cup 1 also gained an ounce of coffee in the pour-back (total 5, not 4).
💡Key takeaway

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 2/5.

  • Pour half the coffee across
  • See Cup 2 as a fixed mixture
  • Pour half of Cup 2 back
  • Take the cream fraction

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