Sanjay cuts out three circles from three different pieces of colored cardboard. He places them on top of each other as shown in figure 1. He then moves the circles so that all three circles are tangent to each other as shown in figure 2. In figure 1, the area of the visible black region is 7 times the area of the white circle. What is the ratio between the areas of the visible black regions in the two figures? Let the area of white, black, and gray circles be W, B, and G. Then the visible black area in figure 1 is B minus G, which is equal to 7W, and in figure 2, we know that this area is B minus W minus G, and therefore we find that A2 is 7W minus W, which is 6W, and therefore our ratio is 7 to 6.

geometry

circles

area ratio

tangent circles

cardboard