AMC 10 · 2012 · #1

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

multi-digit-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: mental-arithmetic

📏 Short solution 💡 1 insight

Problem

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2012 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: Every third-grade classroom holds $18$ students and $2$ pet rabbits. Across all $4$ classrooms, find how many more students there are than rabbits.

Givens: Each classroom has $18$ students; Each classroom has $2$ pet rabbits; There are $4$ third-grade classrooms; Answer choices: (A) $48$, (B) $56$, (C) $64$, (D) $72$, (E) $80$

Unknowns: The total number of students minus the total number of rabbits, across all $4$ classrooms

Understand

Restated: Every third-grade classroom holds $18$ students and $2$ pet rabbits. Across all $4$ classrooms, find how many more students there are than rabbits.

Givens: Each classroom has $18$ students; Each classroom has $2$ pet rabbits; There are $4$ third-grade classrooms; Answer choices: (A) $48$, (B) $56$, (C) $64$, (D) $72$, (E) $80$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

The question bundles two counts (students and rabbits) over $4$ rooms, so Tool #7 (Identify Subproblems) splits it into clean pieces: total students, total rabbits, then their difference. Tool #3 (Eliminate Possibilities) catches the classic trap of stopping at $72$ (the student total, choice (D)) instead of subtracting the rabbits.

Execute — Answer: C

#7 Identify Subproblems 4.NBT.B.5 Step 1

Count all the students

Each of the $4$ classrooms has $18$ students, so multiply to find the whole-grade total.
$18 \times 4 = 72$ students.

$$18 \times 4 = 72$$

💡 Same amount in each room means one multiplication does the work of adding four times.

#7 Identify Subproblems 4.NBT.B.5 Step 2

Count all the rabbits

Each classroom has $2$ rabbits, and there are $4$ classrooms, so $2 \times 4 = 8$ rabbits in all.

$$2 \times 4 = 8$$

💡 The rabbits scale up by the same factor of $4$ as the students do.

#3 Eliminate Possibilities 3.NBT.A.2 Step 3

Subtract to find how many more

"How many more students than rabbits" is a difference, so subtract the rabbit total from the student total: $72 - 8 = 64$.
Note this is not $72$ (that would ignore the rabbits), so choice (D) is a trap; the answer is (C).

$$72 - 8 = 64 \;\Rightarrow\; \textbf{(C)}$$

💡 "How many more" always means take away the smaller count from the larger one.

[1] #7 4.NBT.B.5 Each of the $4$ classrooms has $18$ students, so multiply to find the whole-grad

[2] #7 4.NBT.B.5 Each classroom has $2$ rabbits, and there are $4$ classrooms, so $2 \times 4 = 8

[3] #3 3.NBT.A.2 "How many more students than rabbits" is a difference, so subtract the rabbit to

Review

Reasonableness: Sanity check with the per-room shortcut: each room has $18 - 2 = 16$ more students than rabbits, and $16 \times 4 = 64$ — the same answer, which confirms $64$. The value also sits sensibly below the student total of $72$, since removing $8$ rabbits should trim only a little. Choice (D) $72$ is exactly the un-subtracted student count, the trap the question is testing.

Alternative: Work one classroom at a time first: $18 - 2 = 16$ extra students per room. Then multiply by the $4$ rooms: $16 \times 4 = 64$. Doing the subtraction before the multiplication gives the identical result, matching (C).

CCSS standards used (min grade 4)

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number (Scaling the per-classroom counts to all $4$ rooms: $18 \times 4 = 72$ students and $2 \times 4 = 8$ rabbits.)
3.NBT.A.2 Fluently add and subtract within 1000 (Taking the difference $72 - 8 = 64$ to answer "how many more.")
3.OA.D.8 Solve two-step word problems using four operations within 100 (Structuring the problem as multiply-then-subtract to reach the final comparison.)

⭐ Scale each count up to all the rooms first, then subtract — "how many more" always means take the smaller total away from the bigger one.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Count all the students

Count all the rabbits

Subtract to find how many more

Review

CCSS standards used (min grade 4)

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