Problem 16 · 2001 AMC 8
Hard
Geometry & Measurement
foldingperimeter
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
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Answer: E — 5/6.
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Hint 1 of 2
Don't picture it abstractly — sketch the paper at each step and just track length × width. The fold turns 4×4 into a 4×2 stack.
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Hint 2 of 2
The cut goes through both folded layers parallel to the fold; unfolding gives one large rectangle and two small ones. Find their dimensions, then compare perimeters.
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Approach: find each rectangle's dimensions, then compare perimeters
- Fold the 4×4 square in half vertically → a 4-tall by 2-wide stack (two layers). Cut that stack in half parallel to the fold: the half nearest the fold stays joined when unfolded (a 4×2 piece), while the outer half is two loose layers (two 4×1 pieces).
- So one large 4×2 rectangle and two small 4×1 rectangles. Their perimeters: small = 2(4 + 1) = 10, large = 2(4 + 2) = 12.
- Ratio = 10 : 12 = 5/6. The reusable habit: in fold-and-cut problems, forget the picture's drama and just bookkeep each piece's length and width.
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