Competition · AMC preparation · step 4 of 4

AMC 10 · 2021B · #4

Grade 4 counting
Archetype Overlap / Complement Counting └ Sub-archetype Two-Set Inclusion-Exclusion: Solve for Missing Region
pair-countingparitymulti-digit-arithmeticset-partition identify-subproblemscomplementary-counting ↑ Prerequisites: multi-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem
57 students wear blue and 75 wear yellow. All 132 are split into 66 pairs. Exactly 23 pairs are blue-blue. How many pairs are yellow-yellow?

Pick an answer.

(A)
23
(B)
32
(C)
37
(D)
41
(E)
64

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

Tool #7 (Subproblems) chops the count by pair type: blue-blue → blue students used; mixed → blue students left over = yellow students used; yellow-yellow → yellow students left, halved. Tool #1 (Diagram) helps draw three labelled bins (BB, BY, YY) and watch each student get placed. Tool #3 (Eliminate) gives a quick parity safety net: total pairs = 23 + (mixed) + (yellow-yellow) = 66, so mixed + yellow-yellow = 43, ruling out (A) 23 at a glance.

1STEP 1

Count blue in blue pairs

23 blue-blue pairs × 2 students each lock up 46 blue students.

23 × 2 = 46 blue students in blue-blue pairs
2STEP 2

Find blue in mixed pairs

Leftover blue students: 57 - 46 = 11, each forced to pair with a yellow.

57 - 46 = 11 blue students in mixed pairs
3STEP 3

Match each mixed pair

One blue + one yellow per mixed pair, so 11 pairs use 11 yellow students.

11 mixed pairs → 11 yellow students in mixed pairs
4STEP 4

Find the leftover yellow

Yellow left for yellow-yellow pairs: 75 - 11 = 64 students.

75 - 11 = 64 yellow students in yellow-yellow pairs
5STEP 5

Halve the leftover yellow

Halve them: 64 ÷ 2 = 32 yellow-yellow pairs — choice (B).

64 ÷ 2 = 32 → (B)
Answer
32
Total pair tally: 23 blue-blue + 11 mixed + 32 yellow-yellow = 66 ✓. Total student tally: 46 + 11 + 11 + 64 = 132 ✓. Both totals match the givens, so the count is consistent.
💡Key takeaway

This AMC 10 problem only needs Grade 4 multi-digit subtraction — track the 46 blue in BB pairs, the 11 leftover blue in mixed pairs, and the 64 leftover yellow making 32 YY pairs!

  • Count blue in blue pairs
  • Find blue in mixed pairs
  • Match each mixed pair
  • Find the leftover yellow
  • Halve the leftover yellow

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