Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #13
Grade 4 counting
Pick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem hands us a Venn diagram and asks about union and intersection sizes — Tool #12 (Draw a Venn Diagram) is the literal fit. Sketch two overlapping circles, label the middle (intersection) 1001, and let the two crescents be the "only-A" and "only-B" regions. Since |A| = |B|, those crescents have equal size. Tool #7 (Identify Subproblems) then splits the union of 2007 into three pieces — both crescents and the middle — and the symmetry gives each crescent in one short calculation. Add one crescent back to the middle to get |A|.
Draw the Venn diagram
Draw two overlapping circles; label the shared middle 1001, the A-only crescent a, the B-only crescent b.
Sorting the union into three disjoint regions is the Grade 2 "picture graph" idea — a visual way to break a set into non-overlapping parts.
2.MD.D.10Draw A Venn DiagramTranslate the equal sizes
A and B share the same middle 1001, so their outer crescents must match: a = b.
If two totals are equal and they share the same chunk, the leftover chunks are equal too — Grade 3 "explain patterns in arithmetic."
3.OA.D.9Identify SubproblemsWrite the union as three parts
The union counts each region once, so a + 1001 + b = 2007 with a = b gives 2a + 1001 = 2007.
Splitting a union into non-overlapping pieces is the standard Grade 4 multi-step word-problem move — count each region once.
The union splits into three non-overlapping regions — the left crescent, the shared middle 1001, and the right crescent — so their counts add to 2007, and because the two crescents are equal this becomes 2a + 1001 = 2007.
▸ Why?
Every element of the union lies in exactly one of three regions — only in A, in both, or only in B — with nothing left out and nothing counted twice, so the three region counts add back to the union's count of 2007.
▸ Why?
The two crescents have the same size, so the union is the shared middle plus two equal crescents, which we write as 2a + 1001.
▸ Why?
Set A is its left crescent joined to the middle and set B is its right crescent joined to the same middle, so A counts a + 1001 and B counts b + 1001.
▸ Why?
The two sets are given as equal in size, so a + 1001 and b + 1001 are the same number, and taking the shared 1001 back off each side leaves the crescents a and b equal.
Solve for the crescent
Subtract 1001 from both sides and halve: each crescent is 503.
After removing the shared middle, the leftover 1006 is split evenly between the two equal crescents, so each gets half.
4.OA.A.3Identify SubproblemsRead off the size of A
Set A is its crescent plus the middle: 503 + 1001 = 1504, choice (C).
From the Venn picture, A is its own crescent plus everything it shares with B.
4.OA.A.3Draw A Venn DiagramDrawing the Venn diagram turns this AMC 8 problem into a Grade 4 split-and-add: the union 2007 is the middle 1001 plus two equal crescents, so each crescent is 503 and |A| = 503 + 1001 = 1504.
- Draw the Venn diagram
- Translate the equal sizes
- Write the union as three parts
- Solve for the crescent
- Read off the size of A
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