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AMC 8 · 2011 · #2

Grade 4 geometry-2d
Archetype Arithmetic Expression Decomposition
area-rectanglesmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem

Karl's rectangular vegetable garden is 2020 feet by 4545 feet, and Makenna's is 2525 feet by 4040 feet. Whose garden is larger in area?

Pick an answer.

(A)
Karl's garden is larger by 100 square feet.
(B)
Karl's garden is larger by 25 square feet.
(C)
The gardens are the same size.
(D)
Makenna's garden is larger by 25 square feet.
(E)
Makenna's garden is larger by 100 square feet.
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Toolkit + CCSS Solution

Understand

Restated: Karl has a rectangular vegetable garden that is $20$ feet by $45$ feet. Makenna has one that is $25$ feet by $40$ feet. Which garden has the larger area, and by how many square feet?

Givens: Karl's garden is a $20 \text{ ft} \times 45 \text{ ft}$ rectangle; Makenna's garden is a $25 \text{ ft} \times 40 \text{ ft}$ rectangle; Answer choices compare whose garden is larger and by how many square feet

Unknowns: Which gardener has the larger area; The positive difference between the two areas, in square feet

Understand

Restated: Karl has a rectangular vegetable garden that is $20$ feet by $45$ feet. Makenna has one that is $25$ feet by $40$ feet. Which garden has the larger area, and by how many square feet?

Givens: Karl's garden is a $20 \text{ ft} \times 45 \text{ ft}$ rectangle; Makenna's garden is a $25 \text{ ft} \times 40 \text{ ft}$ rectangle; Answer choices compare whose garden is larger and by how many square feet

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

The question bundles three small jobs into one sentence: find Karl's area, find Makenna's area, then compare. Tool #7 (Identify Subproblems) makes that structure explicit — solve each rectangle separately, then subtract. Tool #8 (Analyze the Units) is a quick sanity check that $\text{ft} \times \text{ft} = \text{ft}^2$, so the answer's unit "square feet" lines up automatically and we don't need to convert anything.

Execute — Answer: E

#7 Identify Subproblems 3.MD.C.7 Step 1
  • Compute Karl's area using the rectangle formula $A = \ell \times w$.
  • A handy trick: $20 \times 45 = 2 \times 45 \times 10 = 90 \times 10$.
$$A_K = 20 \times 45 = 900 \text{ ft}^2$$

💡 Multiplying side lengths to get area is the Grade 3 rectangle-area standard.

#7 Identify Subproblems 3.MD.C.7 Step 2
  • Compute Makenna's area the same way.
  • A handy trick: $25 \times 40 = 25 \times 4 \times 10 = 100 \times 10$.
$$A_M = 25 \times 40 = 1000 \text{ ft}^2$$

💡 Same area formula, just plugged into Makenna's rectangle — Tool #7's second subproblem.

#7 Identify Subproblems 4.NBT.A.2 Step 3
  • Compare the two areas.
  • Since $1000 > 900$, Makenna's garden is larger.
$$A_M = 1000 > 900 = A_K$$

💡 Comparing multi-digit whole numbers is a Grade 4 place-value skill.

#8 Analyze the Units 4.NBT.B.4 Step 4
  • Subtract to find by how much Makenna's garden is larger.
  • The units stay as square feet throughout.
$$1000 - 900 = 100 \text{ ft}^2 \;\Rightarrow\; \textbf{(E)}$$

💡 Whole-number subtraction with the unit "square feet" carried along — Grade 4 standard algorithm.

[1] #7 3.MD.C.7 Compute Karl's area using the rectangle formula $A = \ell \times w$. A handy tri
[2] #7 3.MD.C.7 Compute Makenna's area the same way. A handy trick: $25 \times 40 = 25 \times 4
[3] #7 4.NBT.A.2 Compare the two areas. Since $1000 > 900$, Makenna's garden is larger.
[4] #8 4.NBT.B.4 Subtract to find by how much Makenna's garden is larger. The units stay as squar

Review

Reasonableness: Both gardens have a perimeter of $2(20+45) = 130$ ft for Karl and $2(25+40) = 130$ ft for Makenna — the same fence length. With a fixed perimeter, a rectangle's area grows as its shape gets closer to a square. Makenna's $25 \times 40$ is closer to a square than Karl's $20 \times 45$, so Makenna's area should be larger. The $100 \text{ ft}^2$ gap is small compared to the $\sim 1000 \text{ ft}^2$ scale, which matches the answer choices.

Alternative: Tool #14 (Wishful Thinking — what if both gardens were squares with the same perimeter $130$ ft?) gives a square of side $32.5$ ft and area $1056.25 \text{ ft}^2$, the theoretical maximum. Karl's $900$ and Makenna's $1000$ both fall below it, and Makenna's is closer to that max — confirming Makenna wins by the difference of squares pattern: $25 \times 40 - 20 \times 45 = (32.5-7.5)(32.5+7.5) - (32.5-12.5)(32.5+12.5) = (32.5^2 - 7.5^2) - (32.5^2 - 12.5^2) = 12.5^2 - 7.5^2 = 156.25 - 56.25 = 100$. Same answer (E).

CCSS standards used (min grade 4)

  • 3.MD.C.7 Relate area to multiplication and find areas of rectangles (Computing each garden's area as length times width: $20 \times 45 = 900$ and $25 \times 40 = 1000$ square feet.)
  • 4.NBT.A.2 Read, write, and compare multi-digit whole numbers (Comparing $1000$ and $900$ to decide whose garden is larger.)
  • 4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Subtracting $1000 - 900 = 100$ to find by how many square feet Makenna's garden is larger.)

⭐ This AMC 8 problem only needs the Grade 3 "area $=$ length $\times$ width" rule plus Grade 4 subtraction — find each area, then take the difference.

⭐ This AMC 8 problem only needs the Grade 3 "area $=$ length $\times$ width" rule plus Grade 4 subtraction — find each area, then take the difference.

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