AMC 10 · 2013 · #2

Grade 4 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

unit-conversionarea-rectanglesrate dimensional-analysisidentify-subproblems ↑ Prerequisites: unit-conversionarea-rectangles

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Problem

Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?

Pick an answer.

(A)

600

(B)

800

(C)

1000

(D)

1200

(E)

1400

AMC 10 2013 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 rectangular garden measures $15$ steps by $20$ steps, and each step is $2$ feet long. The garden yields half a pound of potatoes for every square foot. Find the total pounds of potatoes expected.

Givens: The garden is a rectangle measuring $15$ steps by $20$ steps; Each step is $2$ feet long; The yield is half a pound of potatoes per square foot; Answer choices: (A) $600$, (B) $800$, (C) $1000$, (D) $1200$, (E) $1400$

Unknowns: The total number of pounds of potatoes the garden is expected to produce

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

The units carry the whole problem: steps become feet, feet by feet become square feet, and pounds-per-square-foot times square feet becomes pounds. Tool #8 (Analyze the Units) keeps that chain straight so no conversion is skipped. Tool #7 (Identify Subproblems) splits the work into three clean steps — convert, find area, apply the rate. Tool #3 (Eliminate Possibilities) flags the built-in trap: stopping at the area $1200$ (choice (D)) means forgetting the half-pound rate.

Execute — Answer: A

#8 Analyze the Units 4.NBT.B.5 Step 1

Convert steps to feet

The side lengths are in steps, but the yield is per square foot, so change steps into feet first.
Each step is $2$ feet, so multiply each side by $2$: the $15$-step side is $15\times2=30$ feet and the $20$-step side is $20\times2=40$ feet.

$$15\text{ steps}\times2\tfrac{\text{ft}}{\text{step}}=30\text{ ft},\quad 20\times2=40\text{ ft}$$

💡 Multiplying the number of steps by the feet in one step turns a step-count into a real length.

#7 Identify Subproblems 4.MD.A.3 Step 2

Find the garden's area

The garden is a rectangle, so its area is length times width.
Multiply the two feet measurements: $30\times40=1200$.
Because both lengths are in feet, the area is in square feet.

$$30\text{ ft}\times40\text{ ft}=1200\text{ ft}^2$$

💡 Length times width tiles the rectangle with unit squares, and each square foot is one unit of yield.

#3 Eliminate Possibilities 4.NF.B.4 Step 3

Apply the yield rate

Each square foot gives half a pound, so multiply the $1200$ square feet by $\tfrac12$: $1200\times\tfrac12=600$.
That is $600$ pounds, choice (A).
Notice that $1200$ itself is choice (D) — that is the trap for anyone who finds the area but forgets the half-pound rate.

$$1200\text{ ft}^2\times\tfrac12\tfrac{\text{lb}}{\text{ft}^2}=600\text{ lb}\;\Rightarrow\;\textbf{(A)}$$

💡 Half a pound for each square foot means the total pounds is just half the number of square feet.

[1] #8 4.NBT.B.5 The side lengths are in steps, but the yield is per square foot, so change steps

[2] #7 4.MD.A.3 The garden is a rectangle, so its area is length times width. Multiply the two f

[3] #3 4.NF.B.4 Each square foot gives half a pound, so multiply the $1200$ square feet by $\tfr

Review

Reasonableness: Check the size: $1200$ square feet at half a pound each should give clearly less than $1200$ pounds, and $600$ is exactly half — sensible. The four larger choices ($800$, $1000$, $1200$, $1400$) all sit at or above the raw area, which would only happen if the rate were one pound or more per square foot, not a half. The most tempting wrong answer, $1200$, is the area in square feet with the halving skipped. So $600$ is the only value that respects every unit.

Alternative: Halve first, then multiply. Since the yield is $\tfrac12$ pound per square foot, a $30\times40$ garden gives $\tfrac12\times30\times40$. Group it as $(\tfrac12\times40)\times30=20\times30=600$ pounds — the same answer with easier arithmetic and no large product to reduce.

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 (Converting each side from steps to feet ($15\times2=30$, $20\times2=40$) and multiplying $30\times40$ for the area.)
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems (Using area $=$ length $\times$ width to get $30\times40=1200$ square feet.)
4.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a whole number (Multiplying the $1200$ square feet by the rate $\tfrac12$ pound per square foot to get $600$ pounds.)

⭐ Match your units before you multiply: turn steps into feet, then feet into square feet, and only then apply the pounds-per-square-foot rate.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Convert steps to feet

Find the garden's area

Apply the yield rate

Review

CCSS standards used (min grade 4)

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