AMC 8 · 2011 · #6

Grade 4 arithmetic

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 💡 2 insights

Problem

In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle?

Pick an answer.

(A)

(B)

(C)

(D)

306

(E)

351

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A town has $351$ adults, and every adult owns at least one of: a car, a motorcycle, or both. $331$ adults own cars and $45$ adults own motorcycles. How many of the car owners do **not** own a motorcycle?

Givens: Total adults $= 351$; Every adult owns a car, a motorcycle, or both (no one owns neither); Car owners $= 331$; Motorcycle owners $= 45$; Answer choices: (A) $20$, (B) $25$, (C) $45$, (D) $306$, (E) $351$

Unknowns: The number of adults who own a car but not a motorcycle

Understand

Plan

Primary tool: #12 Draw a Venn Diagram

Secondary: #16 Change Focus / Count the Complement

The trigger words "car, motorcycle, or both" and "do not own a motorcycle" point straight at Tool #12 (Venn Diagram): two overlapping circles for Car and Motorcycle, with the every-adult-owns-at-least-one condition meaning the two circles cover all $351$ adults (no "neither" region). The question "car owners who do **not** own a motorcycle" is the left-only slice of the Venn diagram, which is exactly Tool #16 (Count the Complement) — instead of counting the slice directly, compute the overlap first and subtract from the car total.

Execute — Answer: D

#12 Draw a Venn Diagram 4.OA.A.3 Step 1

Set up two overlapping circles labeled $C$ (car owners) and $M$ (motorcycle owners).
Because every adult owns at least one, the union $C \cup M$ contains all $351$ adults — there is no "outside both circles" region.

$$|C \cup M| = 351, \quad |C| = 331, \quad |M| = 45$$

💡 Draw the picture first: two circles inside one big box, and label what each region has to add up to.

#12 Draw a Venn Diagram 4.NBT.B.4 Step 2

Find the overlap (adults who own both).
If we add the car total and the motorcycle total, the adults in both circles get counted twice, so $|C| + |M|$ exceeds the true total $|C \cup M|$ by exactly $|C \cap M|$.

$$|C \cap M| = |C| + |M| - |C \cup M| = 331 + 45 - 351 = 25$$

💡 The center of the Venn diagram ("both") is whatever has been double-counted — the excess of $C + M$ over the true total.

#16 Change Focus / Count the Complement 4.NBT.B.4 Step 3

Switch focus to the complement: instead of counting "car owners who also own a motorcycle" inside $C$, subtract that overlap from the car total to get "car owners who do **not** own a motorcycle."

$$|C \setminus M| = |C| - |C \cap M| = 331 - 25 = 306$$

💡 Inside the car circle there are two parts: "car + motorcycle" and "car only." Subtract the part we don't want from the whole.

#12 Draw a Venn Diagram 4.OA.A.3 Step 4

Match $306$ to the answer choices.

$$306 \Rightarrow \textbf{(D)}$$

💡 The Venn-diagram bookkeeping lands exactly on one of the listed choices.

[1] #12 4.OA.A.3 Set up two overlapping circles labeled $C$ (car owners) and $M$ (motorcycle owne

[2] #12 4.NBT.B.4 Find the overlap (adults who own both). If we add the car total and the motorcyc

[3] #16 4.NBT.B.4 Switch focus to the complement: instead of counting "car owners who also own a m

[4] #12 4.OA.A.3 Match $306$ to the answer choices.

Review

Reasonableness: Sanity-check by filling the Venn diagram: Car-only $= 306$, Both $= 25$, Motorcycle-only $= 45 - 25 = 20$. Sum $= 306 + 25 + 20 = 351$, which matches the total population. The motorcycle circle ($45$) is small relative to the car circle ($331$), so nearly every adult owns a car and very few own a motorcycle — making $306$ (almost all car owners) the expected magnitude.

Alternative: Tool #16 (Count the Complement) on its own: adults who do **not** own a motorcycle number $351 - 45 = 306$. Since every adult owns a car or a motorcycle, anyone without a motorcycle must own a car, so all $306$ of them are car-only. This shortcut avoids computing the overlap and confirms $\textbf{(D)}\ 306$.

CCSS standards used (min grade 4)

4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Modeling the two overlapping groups (cars, motorcycles) with the union-equals-$351$ constraint and chaining the addition and subtraction steps to answer the multistep question.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Carrying out $331 + 45 - 351 = 25$ for the overlap and $331 - 25 = 306$ for the final "car only" count.)

⭐ This AMC 8 problem is really a Grade 4 Venn-diagram puzzle: count the overlap, then take it away from the car circle.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 4)

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