Manhattan GMAT Challenge Problem of the Week – 19 Sept 2011:
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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?
Each spoke is a radius of the circular wheel. If there are 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.
The correct answer is B.
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