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

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Manager
Joined: 11 Jul 2010
Posts: 193
Four concentric circles share the same center. The smallest circle has  [#permalink]

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Updated on: 24 Oct 2014, 01:40
5
00:00

Difficulty:

45% (medium)

Question Stats:

73% (02:12) correct 27% (02:23) wrong based on 97 sessions

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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

Originally posted by gmat1011 on 16 Oct 2010, 06:13.
Last edited by Bunuel on 24 Oct 2014, 01:40, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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16 Oct 2010, 15:03
2
gmat1011 wrote:
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)

What is the sum of the areas of the four circles, divided by the sum of their circumferences, in inches?

(C) 2008 GMAT Club / MGMAT challenge

* 1
* 1 1/2
* 2
* 2 1/2
* 3

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

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Posts: 95
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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16 Oct 2010, 20:59
shrouded1 pointed out correctly otherwise it would be difficult to calculate the radius.it would require a lot more calculation
Manager
Joined: 11 Jul 2010
Posts: 193
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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16 Oct 2010, 21:20
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.
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Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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17 Oct 2010, 07:29
mrinal2100 wrote:
shrouded1 pointed out correctly otherwise it would be difficult to calculate the radius.it would require a lot more calculation

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.
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Joined: 11 Jul 2010
Posts: 193
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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17 Oct 2010, 09:02
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.
Manager
Joined: 26 Mar 2010
Posts: 94
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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18 Oct 2010, 19:56
shrouded1 wrote:
gmat1011 wrote:
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)

What is the sum of the areas of the four circles, divided by the sum of their circumferences, in inches?

(C) 2008 GMAT Club / MGMAT challenge

* 1
* 1 1/2
* 2
* 2 1/2
* 3

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

Hi Pls explain how come r2,23,r4 values are 2,3,4 respectively.I can understand that r1=1 though.
Manager
Joined: 08 Sep 2010
Posts: 174
Location: India
WE 1: 6 Year, Telecom(GSM)
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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18 Oct 2010, 20:55
utin wrote:
shrouded1 wrote:
gmat1011 wrote:
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)

What is the sum of the areas of the four circles, divided by the sum of their circumferences, in inches?

(C) 2008 GMAT Club / MGMAT challenge

* 1
* 1 1/2
* 2
* 2 1/2
* 3

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

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

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.
Manager
Joined: 22 Sep 2010
Posts: 77
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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19 Oct 2010, 04:42
Senior Manager
Joined: 18 Aug 2009
Posts: 358
Schools: UT at Austin, Indiana State University, UC at Berkeley
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WE 2: 5.5
WE 3: 6.0
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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12 May 2011, 03:08
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...
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Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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12 May 2011, 03:24
ratio = 30pi/20pi

1.5
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Joined: 06 Dec 2010
Posts: 16
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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29 Dec 2011, 10:10
I hate this question, while doing the MGMAT challenge, I wasted 4-5 minutes on it unknowningly, threw away my whole game!!!!!
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Joined: 02 Sep 2009
Posts: 51121
Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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24 Oct 2014, 01:41
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2
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 $$\frac{30\pi}{20\pi} = 1\frac{1}{2}$$.

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Re: Four concentric circles share the same center. The smallest circle has  [#permalink]

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28 Nov 2018, 23:25
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Re: Four concentric circles share the same center. The smallest circle has &nbs [#permalink] 28 Nov 2018, 23:25
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