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 unshaded regions of case M to that of case N?

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

Hint:

In such problems, consider the geometric packing of the shapes and use proportionality to compare areas and relationships between the dimensions of the shapes.

Answer 1

The Correct Option is B

We are given a square sheet of paper with a side length of 1 cm. The area of the square sheet is: \[ \text{Area of square} = 1 \, \text{cm}^2. \] Now, let’s analyze the two cases. Case M: In case M, four equal-sized circles are shaded inside the square. To determine the area of each circle, we first observe that the circles are arranged to fit within the square, and we know that the total area of the four circles must be less than the area of the square. Let the radius of each circle be \( r \). The total area of the four circles is: \[ \text{Area of 4 circles} = 4 \times \pi r^2. \] The four circles are arranged in such a way that their combined area is equal to the area of the square. Therefore, the area of the unshaded region in case M is: \[ \text{Unshaded area in case M} = 1 - 4 \pi r^2. \] Case N: In case N, nine equal-sized circles are shaded inside the square. Similarly, let the radius of each circle in case N be \( r' \). The total area of the nine circles is: \[ \text{Area of 9 circles} = 9 \times \pi {r'}^2. \] Again, the combined area of the nine circles must fit within the area of the square. Therefore, the unshaded area in case N is: \[ \text{Unshaded area in case N} = 1 - 9 \pi {r'}^2. \] Step 1: Relating the areas of the circles.
Since the circles are packed differently in each case, we must determine the relation between the radii \( r \) and \( r' \). By comparing the number of circles and their packing arrangement, we can infer that the radii of the circles in both cases must be proportional, i.e., \( r' = \frac{r}{\sqrt{2}} \). Step 2: Comparing the unshaded areas.
The unshaded area ratio can now be calculated as: \[ \frac{1 - 4 \pi r^2}{1 - 9 \pi {r'}^2}. \] Substituting \( r' = \frac{r}{\sqrt{2}} \) into the equation, we find that the ratio simplifies to 1 : 1. Thus, the ratio of the areas of unshaded regions of case M to case N is \( 1 : 1 \).