AMC 10 · 2017 · #3

Grade 4 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border
area-rectanglesarea-difference identify-subproblems ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 2 insights 📊 Diagram
📘 View easy version →
Problem
A garden holds six identical 6-foot by 2-foot flower beds, set in 3 rows of 2. Every bed is separated from its neighbors and bordered along the outside by walkways exactly 1 foot wide. Find the total area, in square feet, covered by the walkways.

Pick an answer.

(A)
72
(B)
78
(C)
90
(D)
120
(E)
150

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

The walkways form an awkward plus-sign-and-border shape that is hard to measure directly. But the walkways are exactly whatever is left over once the beds are removed from the big outer rectangle. So Tool #16 (Count the Complement) turns a hard area into an easy subtraction: garden area minus bed area. Tool #1 (Draw a Diagram) lets us read off the outer rectangle's width and height by lining up beds and 1-foot strips, and Tool #7 (Identify Subproblems) splits the work into three clean pieces: outer area, total bed area, then the difference.

1STEP 1

Find the garden's width

Across one row: 2 beds at 6 ft each plus 3 one-foot strips (left, middle, right) give a width of 15 ft.

2 × 6 + 3 × 1 = 12 + 3 = 15 ft
2STEP 2

Find the garden's height

Down one column: 3 beds at 2 ft each plus 4 one-foot strips (top, bottom, and each gap) give a height of 10 ft.

3 × 2 + 4 × 1 = 6 + 4 = 10 ft
3STEP 3

Area of the whole garden

Treat the garden as one rectangle: 15 by 10 gives an area of 150 square feet.

15 × 10 = 150 ft²
4STEP 4

Area of all six beds

Each bed is 6 by 2, or 12 square feet, and six identical beds cover 72 square feet.

6 × (6 × 2) = 6 × 12 = 72 ft²
5STEP 5

Subtract to get the walkways

The garden is only beds and walkways, so the walkway area is 150 - 72 = 78 square feet, choice (B).

150 - 72 = 78 → (B)
Answer
78
The walkway area 78 is a bit more than the bed area 72, and together 78 + 72 = 150 rebuilds the full garden, so nothing is lost or double-counted. The answer sits comfortably between 72 (all-bed) and 150 (all-garden), exactly where a walkway-only total should land.
💡Key takeaway

When a leftover shape is awkward to measure, find the whole and subtract the easy parts you can see.

  • Find the garden's width
  • Find the garden's height
  • Area of the whole garden
  • Area of all six beds
  • Subtract to get the walkways

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