problems which are based on concentric circles.

Question 1 Four concentric circles are drawn with the radii at interval of X units. If the radius of inner circle is X units then the ratio of the area between the 4th and 3rd circles to the area between the 2nd and 1st circles is:
a)1:2 b)2:1 c)4:3 d)7:3
Answer : d)7:3
Solution :
Let X be the radius of inner(1st) circle.
Given that the radii of the circles are at the intervals of X units.
Then X + X = 2X, 2X + X = 3X, 3X + X = 4X are the radii of successive circles respectively.
Now, the area of first circle = pi(X)² = pi x X²
Area of second circle = pi(2X)² = 4 x pi x X²
Area of third circle = pi(3X)² = 9 x pi x X²
Area of fourth circle = pi(4X)² = 16 x pi x X²
Then, the area between 1st and 2nd circles = (4 x pi x X²) - (pi x X²) = 3piX²br />
And the area between 3rd and 4th circles = (16 x pi x X²) - (9 x pi x X²) = 7piX²
Therefore the required ratio = 7piX² / 3piX² = 7/3
Hence the answer is 7:3

Question 2
88 and 66 are the sum and difference of the perimeter of two concentric circles respectively. Then the difference between the area of the circles is:
a)462 b)154 c)none of these d)cannot be determined
Answer : a)462
Solution :
Let R be the radius of the outer circle
And r be the radius of the inner circle.
Then their perimeters are 2 x pi x R and 2 x pi x r.
Now, the sum of their perimeters is given by
2 x pi x R + 2 x pi x r = 2pi(R + r) = 88
(R + r) = 44/pi ....eqn1
Similarly their difference is given by
2pi(R - r) = 66 ....eqn2
(R - r) = 33/pi
Now, we have to find the difference of their area
i.e., pi(R²) - pi(r²)
pi(R² - r²) = pi(R + r)(R - r)
From eqn1 & 2, we have pi(R+r)(R-r) = pi x 44/pi x 33/pi
44 x 33 / pi
44 x 33 x 7/22 = 66 x 7 = 462
Hence the answer is 462.

Question 3
Three concentric circles are given. The radius of inner circle is r, that of middle one is 2r and that of outer one is 3r. Given that the area of ring formed by 1st and 2nd is A and the area of ring formed by 2nd and 3rd is B. Then the relation between A and B is:
a) A = B B) B = 2A c) A < B d) A > B
Answer : c )A < B
Solution :
Given that A is the area of ring formed by 1st and 2nd circles.
i.e., to get the area of the ring, we have to subtract the area of the 1st from 2nd.
Similarly, the subtraction of 3rd and 2nd gives B.
Let r, 2r, 3r be the radius of the concentric circles.
Then, their area will be pi(r²), pi(2r)² = 4pi(r²), pi(3r)² = 9pi(r²).
Therefore, A = 4pi(r²) - pi(r²) = 3pi(r²)
and B = 9pi(r²) - 4pi(r²) = 5pi(r²)
From above, we have A < B.