AMC 8 · 2015 · #15

Grade 6 arithmeticlogic

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

set-partitioncomplementary-counting complementary-countingidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 3 insights

Problem

At Euler Middle School, 198 students voted on two issues. 149 voted yes on the first issue, 119 voted yes on the second issue, and exactly 29 voted no on both. How many students voted yes on both issues?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

149

AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Venn Diagram

The trigger words "both" and "against both" (neither) point straight at Tool #12 (Venn diagram): two overlapping circles for issues A and B, plus an outside region for the 29 students who voted no on both. Tool #7 (Identify Subproblems) splits the work into two clean steps — first find how many voted yes on at least one issue (the union), then use that union to pin down the overlap. With those two subproblems in hand the Venn diagram fills in by simple subtraction; no algebra heavier than x = a + b - u is needed.

1STEP 1

Draw circles A and B inside a box holding all 198 voters; the 29 who voted no on both sit outside both circles.

outside both circles = 29

2STEP 2

Everyone except those 29 is in at least one circle, so the union is 198 - 29 = 169.

|A ∪ B| = 198 - 29 = 169

3STEP 3

By inclusion-exclusion, |A| + |B| counts the overlap twice, so it exceeds the union by exactly |A ∩ B|.

|A ∪ B| = |A| + |B| - |A ∩ B| ⟹ |A ∩ B| = |A| + |B| - |A ∪ B|

4STEP 4

Substitute: |A ∩ B| = 149 + 119 - 169 = 99, choice (D).

|A ∩ B| = 149 + 119 - 169 = 268 - 169 = 99 → (D)

Answer

Sanity-check the four Venn regions with the answer 99: both = 99, only A = 149 - 99 = 50, only B = 119 - 99 = 20, neither = 29. Total = 99 + 50 + 20 + 29 = 198. The four pieces partition the 198 voters exactly, so the answer holds. Also, |A ∩ B| = 99 is at most min(|A|, |B|) = 119, which a real intersection must satisfy.

💡Key takeaway

Once the Venn diagram is drawn, this AMC 8 problem only needs Grade 6 "write one equation, solve for the unknown" — the overlap pops out as 149 + 119 - 169 = 99.

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