AMC 8 · 2018 · #9

Grade 3 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border

area-rectanglesperimeterspatial-visualization area-differenceidentify-subproblems ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic

📏 Medium solution 💡 3 insights

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Problem

Monica is tiling a rectangular living room that is 12 feet by 16 feet. She places a one-foot-wide border of 1 ft × 1 ft square tiles along all four edges, and fills the interior with 2 ft × 2 ft square tiles. How many tiles does she use in total?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

120

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The floor is a compound region: a thin border of small tiles wrapped around a big inner rectangle of large tiles. Tool #7 (Identify Subproblems) is perfect — solve the border count and the inner count as two independent area problems, then add. Tool #1 (Draw a Diagram) makes the split visible: sketch the 12 × 16 rectangle, shade the one-foot border, and label the inner rectangle as 10 × 14. That picture also makes it obvious that the inner sides are even, so 2 × 2 tiles tile it perfectly.

1STEP 1

Sketch the room and frame off the one-foot border; it shrinks each side by 1 ft, so the inner rectangle is 14 by 10 feet.

inner length = 16 - 2 = 14 ft, inner width = 12 - 2 = 10 ft

2STEP 2

Subproblem A — border area is whole minus inner; each 1-ft tile covers 1 sq ft, so the tile count equals that area: 52.

A_border = 16 × 12 - 14 × 10 = 192 - 140 = 52 sq ft → 52 border tiles

3STEP 3

Subproblem B — the inner area 140 divided by each 2-ft tile's 4 sq ft gives 35 tiles (check: 7 × 5).

\frac{140 sq ft}{4 sq ft/tile}

= 35 tiles and 7 × 5 = 35 tiles ✓

4STEP 4

Combine the subproblems — add the 52 border tiles and 35 interior tiles for a total of 87.

52 + 35 = 87 → (B)

Answer

Sanity check the magnitude: the total floor area is 192 sq ft. If every tile were the small 1 × 1 kind, Monica would need 192 tiles — close to choice (E) 120 but too many. If every tile were the big 2 × 2 kind, she would need 192 ÷ 4 = 48 tiles (choice A). Our answer 87 sits between 48 and 192, exactly where a mix-and-match floor should land, and it is the only choice that splits as 52 + 35 with both pieces matching the border-vs-inner geometry. Choice (B) is right.

💡Key takeaway

This AMC 8 problem only needs Grade 3 area and multiplication you already know!

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