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# If P, Q and R are the centers of circles P, Q, and R and the

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Joined: 21 Oct 2013
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If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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22 Jul 2014, 07:34
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78% (02:20) correct 22% (02:21) wrong based on 322 sessions

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If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8

Hi, I want to know why we don't get same answer using PQ = 2?
OE
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π

Attachment:

Eye of the Tiger.jpg [ 11.43 KiB | Viewed 15189 times ]
##### Most Helpful Expert Reply
Math Expert
Joined: 02 Sep 2009
Posts: 59685
If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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22 Jul 2014, 07:51
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3
goodyear2013 wrote:

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8

Hi, I want to know why we don't get same answer using PR = 2?
OE
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π

The radius of P is twice the radius of Q, which is twice the radius of R.

Say the radius of R is 1, so the radius of Q is 2 and the radius of P is 4.

The area of the shaded region = area of Q - area of R = $$4\pi-\pi=3\pi$$.

The area of P = $$16\pi$$.

The ratio = 3/16.

Answer: A.

As for your question PR is not the radius of any circle, so assuming a value for it is not a good idea. PR = radius of Q + radius of R = 2x + x = 2 --> radius of R = x = 2/3 --> the radius of Q is 4/3 --> the radius of P is 8/3.

The area of the shaded region = area of Q - area of R = $$\frac{16}{9}\pi-\frac{4}{9}\pi=\frac{12}{9}\pi$$.

The area of P = $$\frac{64}{9}\pi$$.

The ratio = 12/64 = 3/16. Thew same answer.
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##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 59685
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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22 Jul 2014, 08:09
goodyear2013 wrote:
Hi, I want to know why we don't get same answer using PQ = 2?

PQ is the radius of the middle circle, Q. In my solution above I assumed exactly that PQ = QT = 2.
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Posts: 978
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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03 Jul 2016, 05:25
1
goodyear2013 wrote:
Attachment:
Eye of the Tiger.jpg

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8

Hi, I want to know why we don't get same answer using PQ = 2?
OE
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π

Shaded region = area of circle Q-area of circle R
=pi*(QT)^2-pi*(RT)^2
but QT=2RT
pi*(QT^2-QT^2/4)------>pi*3/4QT^2
but again PT=2QT
substituting----->3/4*pi*(PT/2)^2
3/16*pi*PT^2------>3/16*area of circle P
Ans A
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Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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01 Jun 2017, 16:57
goodyear2013 wrote:
Attachment:
Eye of the Tiger.jpg

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8

Hi, I want to know why we don't get same answer using PQ = 2?
OE
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π

Bunuel why can't RT be 1/2? The largest square in this diagram is P, the second biggest, Q and the smallest R - so if the radius of r is 1/2 then the radius of Q is 1 and the radius of P is 2- but I don't understand why I can't get 3/8 using this method.
Math Expert
Joined: 02 Sep 2009
Posts: 59685
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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01 Jun 2017, 20:52
Nunuboy1994 wrote:
Bunuel why can't RT be 1/2? The largest square in this diagram is P, the second biggest, Q and the smallest R - so if the radius of r is 1/2 then the radius of Q is 1 and the radius of P is 2- but I don't understand why I can't get 3/8 using this method.

You can do that way too. This is a ratio question so if you do math correctly you should get the same answer no matter what numbers you use. But how can I tell what you did incorrectly if you don't show your work?
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Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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05 Jun 2017, 10:49
We know, QT = 2 RT and PT = 2 QT or PT = 4 RT.

Let RT = 1. So, QT = 2 and PT = 4

Now, area of circle P, Q and R will be 16π, 4π and π respectively.

From this we get that area of circle R is 1/4 area of circle Q and 1/16 of circle P. Also, area of circle Q is 1/4 area of circle P.

So area of shaded region = area of circle P - area of circle Q + area of circle R = 16π - 4π + π = 13π

Therefore, the portion of circle P which is shaded = (16π - 13π)/16π = 3/16.
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05 Jun 2017, 23:16
Let the circle with center P radius be x. Area P--Pi*(x2)
Then radius of circle with center Q is x/2. Area Q--Pi(x2/4)
With same method area of circle R--Pi*(x2/16)
Area of shaded region-- Pi (x2/4-x2/16)= Pi *(3x2/16)

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Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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05 Jun 2017, 23:55
1
goodyear2013 wrote:

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8

Hi, I want to know why we don't get same answer using PQ = 2?
OE
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π

Attachment:
Eye of the Tiger.jpg

Let the radius of circle R = 1 unit
So Radius of circle Q = 2 unit
Radius of circle P = 4 unit

Area of shaded region : pi(2^2) - pi(1^1) = 3pi
Area of circle P = pi(4^2 )= 16pi

So ratio of shaded region to Circle P = 3pi/16pi = 3/16

Answer A
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Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

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06 May 2019, 20:11
goodyear2013 wrote:

If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?

A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8

We can let the radius of circles P, Q, and R be 4, 2, and 1, respectively. Therefore, the area of circle P is π(4)^2 = 16π, that of circle Q is π(2)^2 = 4π, and that of circle R is π(1)^2 = π, The area of the shaded region is the difference between the areas of circle Q and circle R. Therefore the area of the shaded region is 4π - π = 3π, and hence 3π/(16π) = 3/16 of circle P is shaded.

Answer: A
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Re: If P, Q and R are the centers of circles P, Q, and R and the   [#permalink] 06 May 2019, 20:11
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