The 8 spokes of a custom circular bicycle wheel radiate from the central axle of the wheel and are arranged such that

[BTGmoderatorDC]

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


OA B

Source: Manhattan Prep

[Scott@TargetTestPrep]

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


OA B

Solution:

We can let the central angle of the smallest sector = x and the common difference = d. So we have:

x, x + d, x + 2d, x + 3d, x + 4d, x + 5d, x + 6d and x + 7d

as the measure of the central angles of all 8 sectors.

The sum of the measure of these 8 central angles is 360 degrees, so we have:

8x + 28d = 360

We are also given that the central angle of the largest sector is 80 degrees, so we have:

x + 7d = 80

Multiplying x + 7d = 80 by 4, we have 4x + 28d = 320. Subtracting this from 8x + 28d = 360, we have:

4x = 40

x = 10

Therefore, the smallest sector is 10/360 = 1/36 of the area of the wheel.

Answer: B