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27 Jun 2022, 07:15
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C
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Difficulty:
35%
(medium)
Question Stats:
75% (01:31) correct
25%
(02:23)
wrong
based on 16
sessions
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There are two co-centric circles of unequal diameters. The area between the circles is shaded. If the area of the shaded region is 3 times the area of the smaller circle, then the circumference of the larger circle is how many times the circumference of the smaller circle?
(A) 3
(B) 2.5
(C) 2
(D) √3
(E) √2
(A) 3
(B) 2.5
(C) 2
(D) √3
(E) √2
27 Jun 2022, 07:57
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We are dealing with two circles that have the same centre, however, one circle is larger than the other, let us denote the radius of the larger / outer circle as r2 and the radius of the inner / smaller circle as r1
The shaded area is simply the difference between the area of the larger circle and the area of the smaller circle, this can be expressed as:
pi*(r2)^2 - pi*(r1)^2 , which is just the difference in their areas expressed mathematically
We are told that the shaded area is three times the area of the smaller circle
This means that the above expression is 3*pi*(r1)^2 , where r1 is the radius of the smaller circle as explained earlier
Putting this all together, we know that: pi*(r2)^2 - pi*(r1)^2 = 3*pi*(r1)^2
Re-arranging the above equation we get: pi*(r2)^2 = 4*pi*(r1)^2
Divide both sides by pi: (r2)^2 = 4*(r1)^2
Now root both sides: r2 = 2*r1 - this is telling us that the ratio of the radius of the larger circle to the radius of the smaller circle is such that r2/r1 = 2
Therefore the diameter of the larger circle is double that of the smaller circle, and as C=pi*d, the circumference of the larger circle must be twice as large
Answer (C)
The shaded area is simply the difference between the area of the larger circle and the area of the smaller circle, this can be expressed as:
pi*(r2)^2 - pi*(r1)^2 , which is just the difference in their areas expressed mathematically
We are told that the shaded area is three times the area of the smaller circle
This means that the above expression is 3*pi*(r1)^2 , where r1 is the radius of the smaller circle as explained earlier
Putting this all together, we know that: pi*(r2)^2 - pi*(r1)^2 = 3*pi*(r1)^2
Re-arranging the above equation we get: pi*(r2)^2 = 4*pi*(r1)^2
Divide both sides by pi: (r2)^2 = 4*(r1)^2
Now root both sides: r2 = 2*r1 - this is telling us that the ratio of the radius of the larger circle to the radius of the smaller circle is such that r2/r1 = 2
Therefore the diameter of the larger circle is double that of the smaller circle, and as C=pi*d, the circumference of the larger circle must be twice as large
Answer (C)
27 Jun 2022, 09:37
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This is a classic GMAT question where the area of the shaded region is the subtraction of other two areas.
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