AMC 8 · 2007 · #13

Grade 4 counting

Archetype Overlap / Complement Counting └ Sub-archetype Two-Set Inclusion-Exclusion: Solve for Missing Region

set-partitionlinear-equations-one-var identify-subproblemsconvert-to-algebra ↑ Prerequisites: set-partitionlinear-equations-one-var

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Problem

Two sets A and B have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. How many elements are in A?

Pick an answer.

(A)

503

(B)

1006

(C)

1504

(D)

1507

(E)

1510

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Venn Diagram

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|.

1STEP 1

Draw two overlapping circles; label the shared middle 1001, the A-only crescent a, the B-only crescent b.

middle = 1001, left crescent = a, right crescent = b

2STEP 2

A and B share the same middle 1001, so their outer crescents must match: a = b.

a + 1001 = b + 1001 → a = b

3STEP 3

The union counts each region once, so a + 1001 + b = 2007 with a = b gives 2a + 1001 = 2007.

a + 1001 + b = 2007 → 2a + 1001 = 2007

4STEP 4

Subtract 1001 from both sides and halve: each crescent is 503.

2a = 2007 - 1001 = 1006 → a = 503

5STEP 5

Set A is its crescent plus the middle: 503 + 1001 = 1504, choice (C).

|A| = a + 1001 = 503 + 1001 = 1504 → (C)

Answer

1504

Plug back into the Venn diagram. Left crescent 503, middle 1001, right crescent 503. Then |A| = 503 + 1001 = 1504 and |B| = 503 + 1001 = 1504, so |A| = |B| as required. The union is 503 + 1001 + 503 = 2007, matching the given. The intersection is 1001, also given. Magnitude check: |A| must be between 1001 (the intersection) and 2007 (the union), and 1504 sits squarely in that range. Choices (A) 503 and (B) 1006 are too small to contain the 1001 intersection.

💡Key takeaway

Drawing 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.

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