AMC 10 · 2021 · #2
Grade 4 countingPick an answer.
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.
Count blue inside blue pairs
Count the students inside the blue pairs.
2 students per pair × number of pairs — Grade 3 multiplication within 100.
3.OA.C.7Identify SubproblemsFind the leftover blue students
Every leftover blue is in a mixed pair.
Take away the ones already used — Grade 2 subtraction word problem.
2.OA.A.1Identify SubproblemsCount yellow in mixed pairs
Each mixed pair holds one yellow.
1 yellow per mixed pair — Grade 3 multiplication/division word problem.
3.OA.A.3Draw A DiagramCount the leftover yellow
The rest are all in yellow pairs.
Subtract the mixed users from the yellow total — Grade 4 multi-digit subtraction.
4.NBT.B.4Identify SubproblemsDivide by two
Halving gives 32 pairs.
2 students → 1 pair, so divide by 2 — Grade 3 division within 100.
Two students make one pair, so dividing by two turns a head count into a pair count.
▸ Why?
Each pair uses exactly two students, so the students split evenly with nothing left over.
▸ Why?
Every student belongs to exactly one pair, so counting students counts each pair twice.
This AMC 12 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!