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GATE 2022 AE – Question 10

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

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 by 2 array and a 3 by 3 array of tangent circles inscribed in equal squares.
  1. 2 : 3
  2. 1 : 1
  3. 3 : 2
  4. 2 : 1

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Correct answer: (B) 1 : 1

Explanation

Take the side of each square as 1 cm.

**Case M (2 × 2 circles).** The circles are tangent to each other and the sides, so each diameter is $1/2$ cm and the radius is $1/4$ cm. Shaded area:
$$4\cdot\pi\left(\frac14\right)^2=\frac{\pi}{4}.$$

**Case N (3 × 3 circles).** Each diameter is $1/3$ cm and the radius is $1/6$ cm. Shaded area:
$$9\cdot\pi\left(\frac16\right)^2=\frac{9\pi}{36}=\frac{\pi}{4}.$$

**Step 3: unshaded areas.** Both squares have area 1, so the unshaded area is $1-\pi/4$ in each case.

Ratio M : N $=\mathbf{1:1}$.