If O is the center of the circle above, what fraction of the circular

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.

Answer: C.
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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


Shaded angle 60
--------------- = ------- = 1/6
Total angle 360

Hence, C

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?

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.

Hence Answer Choice C.

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
Answer C

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.

Answer: C.

Kudos points given to everyone with correct solution. Let me know if I missed someone.
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Part of circle not shaded=150+150=300. So, shaded part is 360-300=60.
Required fraction is :60/360=1/6

not shaded is 150+150 =300.

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

60/360 = 1/6.

Attached is a visual that should help.

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.

The answer is C.
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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

Answer choice
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Non shaded angle = 300 degree.

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

Answer C

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 \(\frac{Area of shaded portions}{Area of circle}\) = \(\frac{1}{6}\)

Bunuel I solved this question using the sector formula. I found this angle method really simple. In what all cases can we use it?

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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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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