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?
Choose the correct answer:
A) 1/72
B) 1/36
C) 1/18
D) 1/12
E) 1/9
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= 
<gadget id="calculator">280/28</gadget>
<output>10</output>
10   Smallest Sector= (80-7d) = 80-7*10= 
<gadget id="calculator">80-7*10</gadget>
<output>10</output>
10   Smallest sector as Fraction of entire circle= 10/360 = 1/36   Answer: option 1/36
<result>B</result>
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?