Problem 27 · 2016 Math Kangaroo
Stretch
Spatial & Visual Reasoning
paper-cuttingarea
A rectangular piece of paper ABCD is 5 cm wide and 50 cm long. The paper is white on one side and grey on the other. Christina folds the strip as shown so that the vertex B coincides with M, the midpoint of the edge CD. Then she folds it so that the vertex D coincides with N, the midpoint of the edge AB. How big is the area of the visible white part in the diagram?
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Answer: B — 60 cm²
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Hint 1 of 3
Each fold flips a corner flap, so its grey back shows and it hides the white strip underneath it.
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Hint 2 of 3
Find the visible white as the leftover middle strip minus the white that the two folded grey flaps cover.
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Hint 3 of 3
Each flap, once folded inward, lands as a triangle whose base is 13 and height 5 over the white middle.
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Approach: subtract the white hidden by the two folded-in grey flaps
- The strip has area \(5 \times 50 = 250\); each fold turns over an end flap of area 62.5, leaving a white middle band of area \(250 - 2 \times 62.5 = 125\).
- Folding B onto M (and D onto N) lays each grey flap back onto that middle band, where it covers a triangle of base 13 and height 5, area \(\tfrac12 \times 13 \times 5 = 32.5\).
- The two covered triangles sit in opposite halves and do not overlap, so they hide \(2 \times 32.5 = 65\) of white.
- Visible white \(= 125 - 65 = 60\,\text{cm}^2\), answer B.
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