AMC 8 · 2011 · #2

Grade 4 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

area-rectanglesmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

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?

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.

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

How to solve

Strategy Identify Subproblems

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 ft × ft = ft², so the answer's unit "square feet" lines up automatically and we don't need to convert anything.

1STEP 1

Multiply Karl's sides with A = length × width: 20 × 45 gives 900 square feet.

A_K = 20 × 45 = 900 ft²

2STEP 2

Multiply Makenna's sides the same way: 25 × 40 gives 1000 square feet.

A_M = 25 × 40 = 1000 ft²

3STEP 3

Compare the two areas: 1000 ＞ 900, so Makenna's garden is the larger one.

A_M = 1000 ＞ 900 = A_K

4STEP 4

Subtract the areas to size the gap: 1000 - 900 = 100 square feet → (E).

1000 - 900 = 100 ft² → (E)

Answer

Makenna's garden is larger by 100 square feet.

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 × 40 is closer to a square than Karl's 20 × 45, so Makenna's area should be larger. The 100 ft² gap is small compared to the ∼ 1000 ft² scale, which matches the answer choices.

💡Key takeaway

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

More like this