Problem 14 · 2024 Math Kangaroo
Hard
Geometry & Measurement
areaarea-decomposition
Sanjay has three differently coloured circles. First he stacks them on top of one another, as in Figure 1. Then he moves them so that they touch one another pairwise, as in Figure 2. In Figure 1 the visible black area is seven times the area of the white circle. What is the ratio of the visible black area in Figure 1 to that in Figure 2?
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Answer: D — 7 : 6
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Hint 1 of 2
Give the white circle area 1; then the Figure 1 black region is 7, so you immediately know the black circle's area.
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Hint 2 of 2
In Figure 2 the circles only touch (no overlap), so figure out how much of the black is newly hidden compared with Figure 1.
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Approach: scale every area to the white circle, then compare the two pictures
- Let the white circle have area 1; since the visible black in Figure 1 is 7 times that, the visible black in Figure 1 is 7 units.
- The total black circle has area 8 units (the 7 that show plus the 1 unit the smaller circle covers when stacked).
- Re-measuring the black that stays visible once the circles are slid apart to touch pairwise in Figure 2 gives 6 units, so the ratio of visible black in Figure 1 to Figure 2 is \(7:6\), answer D.
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