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# Mary divides a circle into 12 sectors. The central angles of these sec

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Math Expert
Joined: 02 Sep 2009
Posts: 59590
Mary divides a circle into 12 sectors. The central angles of these sec  [#permalink]

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18 Apr 2019, 00:25
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Difficulty:

65% (hard)

Question Stats:

50% (02:34) correct 50% (02:22) wrong based on 32 sessions

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Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

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Re: Mary divides a circle into 12 sectors. The central angles of these sec  [#permalink]

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20 Apr 2019, 22:23
1
As 12 angles are in AP, we can assume their values
a-11d,a-9d, a-7d,a-5d,a-3d,a-d,a+d,a+3d,a+5d,a+7d,a+9d,a+11d
Their sum is equal to 360
a-11d+a-9d+a-7d+a-5d+a-3d+a-d+a+d+a+3d+a+5d+a+7d+a+9d+a+11d=360
12a=360
a=30
Let smallest angle is x
x=a-11d
a-x=11d
30-x should be multiple of 11
Only option C gives value of '30-x' a multiple of 11
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Re: Mary divides a circle into 12 sectors. The central angles of these sec  [#permalink]

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18 Apr 2019, 07:42
Since the central angles form a A.P , whose sum will be 360 ( sum of all central angles of a circle)
Number of central angles/sectors = 12 = n
common difference = d

So, 360 = n/2 { 2a + (n-1)d}
substitutingn = 12

360 = 6{ 2a + 11d }
60 - 2a = 11d
substituting option values ( trial method)
option C , a = 8 then 60 - 16 = 11d , d = 44/11 = 4
d has to be a integer and only option C satisfy that condition.
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Hope it helps,

Thanks,
Mofe Bhatia
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Re: Mary divides a circle into 12 sectors. The central angles of these sec  [#permalink]

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20 Apr 2019, 12:19
Is there another explanation? I don't understand the question.
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Re: Mary divides a circle into 12 sectors. The central angles of these sec  [#permalink]

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23 Apr 2019, 19:47
Bunuel wrote:
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Let x = the angle measure of the smallest sector and d = the common difference, so we have:

x + (x + d) + (x + 2d) + … + (x + 11d) = 360

12x + (1 + 2 + .. + 11)d = 360

12x + 66d = 360

2x + 11d = 60

2x = 60 - 11d

x = (60 - 11d)/2

We see that d is at most 4 since x is a positive integer, and if d is 4, then x is the smallest positive integer it can be. So x = (60 - 44)/2 = 8.

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Re: Mary divides a circle into 12 sectors. The central angles of these sec   [#permalink] 23 Apr 2019, 19:47
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