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The 8 spokes

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The 8 spokes

by BTGmoderatorDC » Sun Feb 11, 2018 10:49 pm
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

I'm confused how to set up the formulas here. Can any experts help?

OA B
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by GMATGuruNY » Mon Feb 12, 2018 3:54 am
lheiannie07 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
Since the difference between one central angle and the next is constant, the central angles constitute an EVENLY SPACED SET.
For any evenly spaced set:
Average = (biggest + smallest)/2.

Since there are 8 spokes, the number of central angles = 8.
Since the 8 central angles form a 360° circle, the average angle measurement = (sum of the angles)/(number of angles) = 360/8 = 45°.
Substituting average = 45 and biggest = 80 into the blue equation above, we get:
45 = (80 + smallest)/2
90 = 80 + smallest
Smallest = 10°.

In any circle:
(sector area)/(circle area) = (central angle)/360.
Since the smallest angle = 10°, we get:
(smallest sector area)/(circle area) = 10/360 = 1/36.

The correct answer is B.
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by Scott@TargetTestPrep » Sun Jun 23, 2019 6:22 pm
BTGmoderatorDC 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

I'm confused how to set up the formulas here. Can any experts help?

OA B

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

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Founder and CEO
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by Scott@TargetTestPrep » Sun Jun 23, 2019 6:22 pm
BTGmoderatorDC 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

I'm confused how to set up the formulas here. Can any experts help?

OA B

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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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