Problem 9 · 2018 AMC 8
Medium
Geometry & Measurement
area-decompositionperimeter
Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?
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Answer: B — 87 tiles.
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Hint 1 of 2
Two different tile sizes means two different regions: a one-foot-wide picture frame around the edge, and the rectangle left inside it. Handle them separately, and watch the corners — that's where over-counting hides.
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Hint 2 of 2
The technique: for a one-wide border, "walk the perimeter" but subtract the 4 corner squares you'd otherwise count twice; then the interior is just the room minus 1 foot off every side.
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Approach: border + interior
- Border (1×1 tiles): the four sides total 12 + 16 + 12 + 16 = 56, but each of the 4 corner squares sits on two sides and got counted twice, so subtract 4: 56 − 4 = 52 border tiles.
- Interior: peeling off the 1-foot border shrinks the room by 1 foot on each side, leaving 10 ft × 14 ft = 140 sq ft. Each 2×2 tile covers 4 sq ft, so 140 ÷ 4 = 35 tiles.
- Total: 52 + 35 = 87.
- You'll see it again: the corner double-count (subtract 4) shows up in every "border around a rectangle" problem — sidewalks, picture frames, fence posts.
Another way — count border tiles by the inner rectangle:
- Another clean way to size the border: total room area minus the interior area, all in unit squares. The room is 12×16 = 192 sq ft and the interior is 10×14 = 140 sq ft, so the border is 192 − 140 = 52 sq ft = 52 unit tiles — no corner bookkeeping at all.
- Then add the 35 big tiles for the interior: 52 + 35 = 87.
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