Problem 21 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
areaarea-decomposition
Inside the gray square there are three white squares; the number in each shows its area. The white squares have sides parallel to the sides of the gray square. If the area of the gray square is 81, what is the area of the gray region not covered by the white squares?
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Answer: C — 52
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Hint 1 of 2
The gray square has area 81, so its side is 9; find the side of each white square from its area.
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Hint 2 of 2
The middle white square spans what is left across the side after the corner squares, so its side is 9 − 3 − 2.
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Approach: find each square's side, then subtract the white areas
- The gray square has area 81, so its side is 9. The corner white squares have areas 9 and 4, so their sides are 3 and 2.
- The middle white square stretches across the row between them, so its side is 9 − 3 − 2 = 4, giving area 16.
- Gray left over = 81 − 9 − 4 − 16 = 52.
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