PR and QS are perpendicular diameters of a large circle with centre O

Question

https://gmatclub.com/forum/download/file.php?id=54337 PR and QS are perpendicular diameters of a large circle with centre O, while PO, QO, RO and SO are diameters of four identical smaller circles, as shown above. What is the ratio of the yellow shaded region to the area of the red shaded region? A. 1:1 B. 1:2 C. 1:3 D. 3:1 E. 2:1 Untitled.png

Kinshook · Answer

Given:PR and QS are perpendicular diameters of a large circle with centre O, while PO, QO, RO and SO are diameters of four identical smaller circles, as shown above.

Asked: What is the ratio of area of the yellow shaded region to the area of the red shaded region?

Let the radius of smaller circle be 1 cm and radius of larger circle be 2 cm.

Area of smaller circle = pi
Area of larger circle = 4pi

Let the area of red shaded region be x and area of yellow shaded region be y

Area of larger circle - area of 4 smaller circle = 4pi - 4*pi = 0
Area of larger circle = 4pi = (4*pi - x) + y 
x = y

The ratio of area of the yellow shaded region to the area of the red shaded region
= y: x = 1:1

IMO A

AnirudhaS · Answer

Let the bigger circle's radius is R, 
Then smaller circle radius = R/2
Let the total area in red = x
Let the total area in yellow = y

Now,  \pi.R^2 = y + \pi (\frac{R}{2})^2.4 - x 
or,  \pi R^2 = y + \pi (\frac{R^2}{4}).4 - x 
or, x=y

Answer:  A

ScottTargetTestPrep · Answer

Notice that by symmetry, the area of each component of the red shaded regions are equal and similarly for the yellow shaded regions. Let x be the area of a component of the red shaded region and y be the area of a component of the yellow shaded region.

Let r be the radius of one of the smaller circles. Then, the smaller circles have area ?r^2, and the unshaded region within one of these circles is ?r^2 - 2x. Since there are four such unshaded regions, the area of the white region in the figure is 4 * (?r^2 - 2x) = 4?r^2 - 8x.

On the other hand, the area of the white region is the difference between the area of the larger circle and the combined area of the red and yellow shaded regions. The area of the larger region is ?(2r)^2 = 4?r^2 and the combined area of the red and yellow regions is 4x + 4y. Thus, the area of the white region is 4?r^2 - (4x + 4y).

Setting the two expressions for the area of the white region equal, we obtain:

4?r^2 - 8x = 4?r^2 - (4x + 4y)

4?r^2 - 8x = 4?r^2 - 4x - 4y

-8x = -4x - 4y

4y = 4x

Since the area of the red shaded region is 4x and the area of the yellow shaded region is 4y, the ratio of the areas is 4x/4y = 1 or 1 : 1.

Answer: A

PR and QS are perpendicular diameters of a large circle with centre O

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PR and QS are perpendicular diameters of a large circle with centre O

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