Equal sized circular regions are shaded in a square sheet of paper of 1 cm side length. Two cases, case M and case N, ar...

Question

Equal sized circular regions are shaded in a square sheet of paper of 1 cm side length. Two cases, case M and case N, are considered as shown in the figures below. In the case M, four circles are shaded in the square sheet and in the case N, nine circles are shaded in the square sheet as shown. What is the ratio of the areas of the unshaded regions of case M to that of case N?

A. 2 : 3
B. 1 : 1
C. 3 : 2
D. 2 : 1

Accepted Answer

Ratio of unshaded area (M) to unshaded area (N) = 1 : 1 A unit square (1 cm x 1 cm) contains circles that fit perfectly inside it arranged in a grid. In case M, a 2x2 grid of equal circles is packed in the square; in case N, a 3x3 grid of equal (but smaller) circles. The key insight is that in both cases the total shaded (circular) area is the same fraction of the square, so the unshaded areas are equal.