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Math Expert V
Joined: 02 Sep 2009
Posts: 60647
If O is the center of the circle above, what fraction of the circular  [#permalink]

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Difficulty:   5% (low)

Question Stats: 90% (00:40) correct 10% (01:02) wrong based on 1196 sessions

### HideShow timer Statistics If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

Practice Questions
Question: 36
Page: 157
Difficulty: 550

Attachment: Circle.png [ 7.58 KiB | Viewed 40800 times ]

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Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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SOLUTION If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

150°+150°=300° of the circle is not shaded, hence 360°-300°=60° of the circle is shaded, which is $$\frac{60}{360}=\frac{1}{6}$$ of the circular region.

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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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1
Bunuel wrote:

Attachment:
Circle.png
If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

Total angle in a circle = 360 degree
One of Non-shaded angle = 150 degree
Other Non-shaded angle = 150 degree (?) because it is opposite angle
Total Non-shaded angle = 150 + 150 = 300 degree
So, shaded angle = 360 - 300 = 60 degree

--------------- = ------- = 1/6
Total angle 360

Hence, C
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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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1
I would say (C), but it seems such an easy answer. Anyways, here is the explanation: 360-2*150=60
60 degrees is the total of shaded regions, so 60/360 = 1/6.
So, please correct me if I went awry.

p.s.
where is trick here?
those shaded regions opposite to each other do not have equal angles?
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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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1
Bunuel wrote:

Attachment:
Circle.png
If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

Since both Lines pass through centers they are alternate angles. Hence shaded degree measure is 360- (150x2)=60

So fraction of circle shaded = 60/360 = 1/6.

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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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1
Fraction of Shaded region = Angle made at the center by shaded region/ Angle at center of a circle
= (360-150*2)/360 = 60/360 = 1/6
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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SOLUTION If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

150° + 150° = 300° of the circle is NOT shaded, hence 360° - 300° = 60° of the circle IS shaded, which is $$\frac{60}{360}=\frac{1}{6}$$ of the circular region.

Kudos points given to everyone with correct solution. Let me know if I missed someone.
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Posts: 5
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Required fraction is :60/360=1/6
Manager  Joined: 07 Apr 2014
Posts: 98
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Bunuel wrote:
Attachment:
Circle.png
If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

Practice Questions
Question: 36
Page: 157
Difficulty: 550

so shaded has to be 360-300= 60..

60/360 = 1/6.
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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Attached is a visual that should help.
Attachments Screen Shot 2016-05-10 at 5.50.02 PM.png [ 67.54 KiB | Viewed 23918 times ]

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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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2
Bunuel wrote:
Attachment:
Circle.png
If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

In solving this problem there are three angle rules that we must consider.

1) A circle measures a total of 360 degrees.

2) A straight angle measures 180 degrees.

3) Vertical angles, angles that are directly opposite one another, are always equal.

To solve, we can draw out a quick diagram of just the angles: Based on the angles we created, we know two important pieces of information:

1) angle A and 150 degrees sum to 180 degrees because they form a straight angle. Thus, angle A = 30 degrees.

2) angle A = angle B because they are vertical angles. Thus angle B = 30 degrees.

With this information we see that the shaded region represents 60 out of 360 degrees of the circle, or 60/360 = 1/6.

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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Bunuel wrote:
Attachment:
Circle.png
If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

Practice Questions
Question: 36
Page: 157
Difficulty: 550

Given: The angle opposite the 150 degree angle has the same measure.

Call x = each of of the shaded regions

2(150) + 2x = 360
300 + 2x = 360
2x = 60
x = 30

Sum of the angles of the shaded regions = 60 degrees

60/360 = 1/6

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Joined: 26 Jun 2014
Posts: 75
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Non shaded angle = 300 degree.

Shaded angle = 360-300 = 60 degree.
Fraction = 60/360 = 1/6.

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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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We can also say area of each shaded portion is $$\frac{30*\pi*r^{2}}{360}$$ = $$\frac{\pi*r^{2}}{12}$$
Area of 2 shaded portions = 2 * $$\frac{\pi*r^{2}}{12}$$ = $$\frac{\pi*r^{2}}{6}$$

Ratio is $$\frac{Area of shaded portions}{Area of circle}$$ = $$\frac{1}{6}$$
Director  G
Joined: 02 Sep 2016
Posts: 636
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Bunuel I solved this question using the sector formula. I found this angle method really simple. In what all cases can we use it?
VP  D
Joined: 09 Mar 2016
Posts: 1223
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Bunuel wrote:
SOLUTION If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

150°+150°=300° of the circle is not shaded, hence 360°-300°=60° of the circle is shaded, which is $$\frac{60}{360}=\frac{1}{6}$$ of the circular region.

Kudos points given to everyone with correct solution. Let me know if I missed someone.

Hello Bunuel

why didnt my simple proporttion method work here 360 = 100%
300 = x
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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1
dave13 wrote:
Bunuel wrote:
SOLUTION If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

150°+150°=300° of the circle is NOT shaded, hence 360°-300°=60° of the circle IS shaded, which is $$\frac{60}{360}=\frac{1}{6}$$ of the circular region.

Kudos points given to everyone with correct solution. Let me know if I missed someone.

Hello Bunuel

why didnt my simple proporttion method work here 360 = 100%
300 = x

The shaded region is 60 degrees not 300.
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Manager  S
Joined: 23 Sep 2016
Posts: 230
Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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Bunuel wrote:
Attachment:
Circle.png
If O is the center of the circle above, what fraction of the circular region is shaded?

(A) 1/12
(B) 1/9
(C) 1/6
(D) 1/4
(E) 1/3

Practice Questions
Question: 36
Page: 157
Difficulty: 550

IMO C whole circle has 360 degree and as opposite angles are equal then opposite angle to 150 is also 150 and unshaded region is 300
shaded region is 60 so area of sector = 60/360pie r^2 which is 1/6pie r^2 and total are is pie r^2 so 1/6pie r^2/ pie r^2 = 1/6
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Re: If O is the center of the circle above, what fraction of the circular  [#permalink]

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