AMC 8 · 2008 · #11
Grade 4 countingPick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) is the cleanest move: sketch two overlapping circles (a Venn diagram) for the dog-owners and the cat-owners. The picture shows immediately that the overlap is double-counted when you add 20 + 26. Tool #7 (Identify Subproblems) then turns the picture into two simple sentences: "dog total + cat total counts every student, plus the overlap one extra time" and "20 + 26 - both = 39." That single subtraction gives the answer.
Draw two overlapping circles — dog-owners and cat-owners — whose union is the whole class of 39; label the overlap "both."
Drawing two overlapping circles and labeling each region is the Grade 2 "picture graph" idea — a visual way to sort the same kids into two groups.
2.MD.D.10Draw A DiagramAdd the two counts: 20 + 26 = 46 — but that counts the both-pet students twice, once in each circle.
Adding two two-digit numbers is Grade 2 arithmetic. The key insight is what the sum means, not the sum itself.
2.NBT.B.5Identify SubproblemsThe sum 46 overshoots the real total 39, and that extra 46 - 39 = 7 is exactly the overlap.
This is the Grade 4 "solve word problems using the four operations" move: the difference between the over-count and the real total is exactly the overlap.
4.OA.A.3Identify SubproblemsIn the choice list, 7 is option (A) — read the answer letter off directly.
Always read the answer letter off the choice list — small habit, saves bubble mistakes.
4.OA.A.3Identify SubproblemsWhen you add the two groups, the kids in both groups get counted twice — so the gap between 20 + 26 and the real total 39 IS the overlap. That's all this AMC 8 problem asks.
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