Four concentric circles share the same center. The smallest circle has

Question

Four concentric circles share the same center. The smallest circle has a radius of 1 inch. For  n  greater than 1, the area of the  n th smallest circle in square inches,  A_n , is given by  A_n = A_{n-1} + (2n - 1)\pi .
 
What is the sum of the areas of the four circles, divided by the sum of their circumferences, in inches?

A.  1  
B.  1\frac{1}{2}  
C.  2  
D.  2\frac{1}{2}  
E.  3

S98-11

shrouded1 · Answer

I think this should say  A_n=A_{n-1} + (2n-1)\pi 
 A_1 = \pi, r_1=1 
 A_2=A_1+3\pi=4\pi, r_2=2 
 A_3=A_2+5\pi=9\pi, r_3=3 
 A_4=A_3+7\pi=16\pi, r_4=4

The sum of circumferences is  S=\pi(2+4+6+8)=20\pi 
The sum of areas is  A=\pi(1+4+9+16)=30\pi

Hence ratio is 1.5

Answer B

mrinal2100 · Answer

shrouded1 pointed out correctly otherwise it would be difficult to calculate the radius.it would require a lot more calculation

gmat1011 · Answer

Thanks - yeah you guys are right... but thats exactly how the question was worded in gmat club tests

there was no pie... it only makes sense if there is a pie and reading the explanation again the answer seems to take that into account. The question needs re-wording... thanks.

ankitranjan · Answer

With this wording it was almost impossible ...yesterday i tried 1 hour for this question and at last i had to leave it out of frustation.I thought i was missing something somewhere in this question.

gmat1011 · Answer

sorry there - i feel your pain!

if it makes any difference I felt the same way. thats how the question was worded in the first place.

utin · Answer

Hi Pls explain how come r2,23,r4 values are 2,3,4 respectively.I can understand that r1=1 though.

ankitranjan · Answer

From given equation we are getting A2= 4 Pie
 A2 is the area of 2nd circle,From formula for the area of a circle,
 Pie .(r2)^2 = 4 Pie
 hence r2=2
 Same way for r3, and r4 .

If it helped u in some way consider giving KUDOS.

pzazz12 · Answer

thanks for your explanation .....

gmatjon · Answer

Share your frustration guys, nowhere in the question stem the pie is mentioned. So, i guess, whoever wrote this question forgot to put it there. It needs to have pie for the problem to be solved...

amit2k9 · Answer

ratio = 30pi/20pi

1.5

Halawi · Answer

I hate this question, while doing the MGMAT challenge, I wasted 4-5 minutes on it unknowningly, threw away my whole game!!!!!

Bunuel · Answer

Official Solution:

Four concentric circles share the same center. The smallest circle has a radius of 1 inch. For  n  greater than 1, the area of the  n th smallest circle in square inches,  A_n , is given by  A_n = A_{n-1} + (2n - 1)\pi .
 
What is the sum of the areas of the four circles, divided by the sum of their circumferences, in inches? 
 
A.  1  
B.  1\frac{1}{2}  
C.  2  
D.  2\frac{1}{2}  
E.  3  
 
First, we figure out the area of the smallest circle.  A_1 = \pi r^2 = \pi 1^2 = \pi .
 
Now, we find the area of the second smallest circle  (n = 2) .  A_2 = A_1 + (2(2) - 1) \pi = \pi + 3 \pi = 4 \pi . This means that the radius of the second smallest circle is 2 (since the area is  \pi r^2 ).
 
The third smallest circle has area  A_3 = A_2 + (2(3) - 1) \pi = 4 \pi + 5 \pi = 9 \pi . This means that the radius of this circle is 3.
 
Finally, the fourth smallest circle (that is, the largest circle) has area  A_4 = A_3 + (2(4) - 1) \pi = 9 \pi + 7 \pi = 16 \pi . This means that the radius of this circle is 4.
 
The sum of all the areas is  \pi + 4\pi + 9\pi + 16\pi = 30\pi .
 
The sum of all the circumferences is  2\pi  times the sum of all the radii. The sum of all the radii is  1 + 2 + 3 + 4 = 10 , so the circumferences sum up to  20\pi .
 
Thus, the sum of all the areas, divided by the sum of all the circumferences, is  \frac{30\pi}{20\pi} = 1\frac{1}{2} .

Answer: B

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