GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 10 Dec 2018, 20:38

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### Free lesson on number properties

December 10, 2018

December 10, 2018

10:00 PM PST

11:00 PM PST

Practice the one most important Quant section - Integer properties, and rapidly improve your skills.
• ### Free GMAT Prep Hour

December 11, 2018

December 11, 2018

09:00 PM EST

10:00 PM EST

Strategies and techniques for approaching featured GMAT topics. December 11 at 9 PM EST.

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 21 Oct 2013
Posts: 419
If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

22 Jul 2014, 06:34
4
1
6
00:00

Difficulty:

25% (medium)

Question Stats:

80% (02:19) correct 20% (02:18) wrong based on 367 sessions

### HideShow timer Statistics

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 10601 times ]
Math Expert
Joined: 02 Sep 2009
Posts: 51072
If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

22 Jul 2014, 06:51
1
2
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.

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.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 51072
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

22 Jul 2014, 07:09
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.
_________________
VP
Joined: 05 Mar 2015
Posts: 1004
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

03 Jul 2016, 04:25
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
Director
Joined: 12 Nov 2016
Posts: 734
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37
GRE 1: Q157 V158
GPA: 2.66
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

01 Jun 2017, 15:57
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: 51072
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

01 Jun 2017, 19:52
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?
_________________
Manager
Joined: 09 Dec 2015
Posts: 114
Location: India
Concentration: General Management, Operations
Schools: IIMC (A)
GMAT 1: 700 Q49 V36
GPA: 3.5
WE: Engineering (Consumer Products)
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

05 Jun 2017, 09:49
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.
Intern
Joined: 08 Mar 2017
Posts: 49
Location: India
Concentration: Operations, General Management
Schools: ISB '20, XLRI"20
GMAT 1: 600 Q40 V33
GPA: 2.79
WE: Project Management (Manufacturing)

### Show Tags

05 Jun 2017, 22:16
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)

Posted from my mobile device
Director
Joined: 13 Mar 2017
Posts: 658
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

05 Jun 2017, 22:55
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

_________________

CAT 2017 99th percentiler : VA 97.27 | DI-LR 96.84 | QA 98.04 | OA 98.95
UPSC Aspirants : Get my app UPSC Important News Reader from Play store.

MBA Social Network : WebMaggu

Appreciate by Clicking +1 Kudos ( Lets be more generous friends.)

What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish".

Non-Human User
Joined: 09 Sep 2013
Posts: 9101
Re: If P, Q and R are the centers of circles P, Q, and R and the  [#permalink]

### Show Tags

02 Jul 2018, 10:03
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If P, Q and R are the centers of circles P, Q, and R and the &nbs [#permalink] 02 Jul 2018, 10:03
Display posts from previous: Sort by

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.