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

Its an AP question .... it is given clearly in the question .

Let the smallest angle be a

and the circle has 8 sectors and hence 8 angle with a common difference d

hence all the angles can be written in AP form with Cd as d ,

a, a+d, a+2d, a+3d ,a+4d, a+5d, a+6d ,a+7d,

given that a+7d = 80 --------1

also

a + a+d + a+2d + a+3d +

a+4d + a+5d + a+6d + a+7d = 360 ( as sum of all the angle is 360)

which is 8a + 28d = 360 --------2

solving 1 and 2

we get a=10

We are almost done ,

now the question ask what fraction of the wheel’s area is represented by the smallest sector ?

(10/360)( pie r*r)/ (pie r*r) = 10/360= 1/36

B ans ....

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Regards ,