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The 8 spokes of a custom circular bicycle wheel radiate from the centr

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The 8 spokes of a custom circular bicycle wheel radiate from the centr  [#permalink]

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New post 14 Sep 2015, 22:06
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The 8 spokes of a custom circular bicycle wheel radiate from the central axle of the wheel and are arranged such that the sectors formed by adjacent spokes all have different central angles, which constitute an arithmetic series of numbers (that is, the difference between any angle and the next largest angle is constant). If the largest sector so formed has a central angle of 80°, what fraction of the wheel’s area is represented by the smallest sector?

A. 1/72
B. 1/36
C. 1/18
D. 1/12
E. 1/9


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Re: The 8 spokes of a custom circular bicycle wheel radiate from the centr  [#permalink]

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New post 14 Sep 2015, 22:43
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Bunuel wrote:
The 8 spokes of a custom circular bicycle wheel radiate from the central axle of the wheel and are arranged such that the sectors formed by adjacent spokes all have different central angles, which constitute an arithmetic series of numbers (that is, the difference between any angle and the next largest angle is constant). If the largest sector so formed has a central angle of 80°, what fraction of the wheel’s area is represented by the smallest sector?

A. 1/72
B. 1/36
C. 1/18
D. 1/12
E. 1/9


Kudos for a correct solution.


Largest angle = 80
Let, Difference between any two angles in A.P. = d

i.e. Sum of all angles will be
80 + (80-d) + (80-2d) + (80-3d) + (80-4d) + (80-5d) + (80-6d) + (80-7d) = 640 - 28d

But sum of all central angles in a circle = 360

i.e. 640 - 28d = 360
i.e. d = 280/28 = 10

Smallest Sector = (80-7d) = 80-7*10 = 10
Smallest sector as Fraction of entire circle = 10/360 = 1/36

Answer: option B
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Re: The 8 spokes of a custom circular bicycle wheel radiate from the centr  [#permalink]

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New post 15 Sep 2015, 01:41
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Bunuel wrote:
The 8 spokes of a custom circular bicycle wheel radiate from the central axle of the wheel and are arranged such that the sectors formed by adjacent spokes all have different central angles, which constitute an arithmetic series of numbers (that is, the difference between any angle and the next largest angle is constant). If the largest sector so formed has a central angle of 80°, what fraction of the wheel’s area is represented by the smallest sector?

A. 1/72
B. 1/36
C. 1/18
D. 1/12
E. 1/9


Kudos for a correct solution.




8 spokes, then the circle is broken up into 8 sectors. The central angles of these sectors are 8 different numbers: call them a, b, c, d, e, f, g, and h, with a as the smallest number and h as the biggest number.

All of these angles add up to 360°, the total central angle in a circle.

Since there are 8 angles, the average of all the angles must be 360/8 = 45°.

Since the angle measures are evenly spaced as a series, the average of all the angles must also be the average of the smallest & the largest angles. That is, (a + h)/2 = 45.

Finally, since h = 80, we can figure out a, which equals 10.

A 10° angle is 10/360 = 1/36 of the circle. The sector with this angle occupies just 1/36 of the wheel’s area.

OA must be B.
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Re: The 8 spokes of a custom circular bicycle wheel radiate from the centr  [#permalink]

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New post 15 Sep 2015, 06:13
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Total of all the central angles = 360
Since, there are 8 sectors, average angle = 360/8 = 45
largest angle = 80, smallest angle = x → (80+x)/2 = 45 → x = 10
sector angle & the sector area are in the same proportion → fraction of the total area occupied by the smallest sector = 10/360 = 1/36
Answer: B
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Re: The 8 spokes of a custom circular bicycle wheel radiate from the centr  [#permalink]

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New post 24 Sep 2018, 12:51
Smallest Angle: A1= A
Second Angle: A2= A+i
Third Angle: A3= A+2i
N Angle: AN= A+(N-1)i
Data:
8 angle (a) 80= A+7i
All angles(b) 360= 8A + 28i
multiply (a).-4 and add to (b)
40=4A
A=10

10/360 = 1/36
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Re: The 8 spokes of a custom circular bicycle wheel radiate from the centr &nbs [#permalink] 24 Sep 2018, 12:51
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