Problem 24 · 2011 Math Kangaroo
Stretch
Geometry & Measurement
areaarea-decomposition
On the inside of a square with side length 7 cm another square is drawn with side length 3 cm. Then a third square with side length 5 cm is drawn so that it cuts the first two as shown in the picture. How big is the difference between the black area and the grey area?
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Answer: D — 15 cm²
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Hint 1 of 2
Set up the black area and the grey area and subtract; the overlap of the third square cancels.
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Hint 2 of 2
Work with the square areas 49, 9 and 25 and see how the shared region drops out.
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Approach: track which parts of each square count as black and as grey
- The 7 cm square is black, the 3 cm square inside it is grey, and the 5 cm square overlays both.
- Comparing the black region with the grey region, the overlapping pieces cancel out.
- What remains is a difference of 15 cm² between the black and grey areas.
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