The 8 spokes of a custom circular bicycle wheel radiate from the centr

Question

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

Kudos for a correct solution.

A. 1/72
B. 1/36
C. 1/18
D. 1/12
E. 1/9

Bunuel · Accepted Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

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.

GMATinsight · Answer

Bunuel : Looks like repeated Question (Posted twice right now)

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 = 10

Smallest Sector = (80-7d) = 80-7*10 = 10
Smallest sector as Fraction of entire circle = 10/360 = 1/36

Answer: option B

anudeep133 · Answer

Solution:  Sum of a a.p is (n/2)(first term + last term) = 360
4(a + 80) = 360 ==> a = 10
Fraction = 10/360 = 1/36

Option B

TheKingInTheNorth · Answer

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

Kudos for a correct solution.

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 ....

Nunuboy1994 · Answer

I think this is the most concise method for this problem- no need to find the minimum of the set- the amount of time that would take is unrealistic for the GMAT

spetznaz · Answer

+1 for option B. Use sum of terms formula for AP. The sum of AP must be 360. Find the value of the first angle, then find the fraction. Answer comes to 1/36

ScottTargetTestPrep · Answer

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

dcummins · Answer

Can form 2 equations:
1. Adding up all the arithmetic series values gives you 360.
2. Largest value: x+7d=80

solve by substitution to give x=10
10/360
= 1/36

ahamd · Answer

8 spokes hence the no of angles in between 8 spokes is 8.

Sum of the angles=360=no. of angles *(1st angle +last angle)/2

360=8/2(1st angle+80)
1st angle =10 degrees.

we know that

Area of a sector /area of a circle=central angle/360

Area of a sector which has 10 degree central angle/area of circle=10/360=1/36 .Option B

bumpbot · Answer

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The 8 spokes of a custom circular bicycle wheel radiate from the centr

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