Problem 10 · 2023 Math Kangaroo
Medium
Geometry & Measurement
area-fractionarea-decomposition
A square with area 84 is split into four squares. The upper left square is coloured in black. The lower right square is again split into four squares and so on. The process is repeated infinitely many times. How big is the area coloured in black?
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Answer: B — 28
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Hint 1 of 2
Each step blackens one quarter of the square that is still being divided.
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Hint 2 of 2
The black areas form a geometric series with ratio 1/4.
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Approach: sum the geometric series of black quarters
- At each stage the black piece is 1/4 of the current square, and the next stage works on another 1/4.
- Black fraction = 1/4 + (1/4)(1/4) + ... = (1/4)/(1−1/4) = 1/3 of the whole.
- So the black area is 84·(1/3) = 28.
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