Competition · AMC preparation · step 4 of 4
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 the two circles
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
Add 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 SubproblemsSubtract the true total
The 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.
The amount by which the dog count plus the cat count exceeds the class size is exactly the number of students who own both a dog and a cat.
▸ Why?
Adding the dog count and the cat count counts every both-pet student twice, once inside each count, while it counts every one-pet student only once.
▸ Why?
The dog count is the dog-only students plus the both-pet students, and the cat count is the cat-only students plus the both-pet students, so the both-pet students sit inside each of the two counts.
▸ Why?
Because the class size already counts each student once, the amount left after removing it from the doubled sum is one whole extra copy of the both-pet group.
▸ Why?
Every student is dog-only, cat-only, or both, with no one left out and no one in two groups, so the class size is those three groups added once each.
▸ Why?
Subtracting the class size from the sum undoes the one full count of everyone, leaving only the second copy of the both-pet students that the doubled sum added in.
Match against the choices
In 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.
- Draw the two circles
- Add the two counts
- Subtract the true total
- Match against the choices
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