Find all School-related info fast with the new School-Specific MBA Forum

It is currently 25 Oct 2014, 11:54

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Imagine two circles one of radius "r" another "2r" . Let the

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Intern
Intern
avatar
Joined: 15 Jul 2003
Posts: 20
Followers: 0

Kudos [?]: 0 [0], given: 0

Imagine two circles one of radius "r" another "2r" . Let the [#permalink] New post 23 Jul 2003, 04:51
Imagine two circles one of radius "r" another "2r" . Let the bigger circle be stationay and the smaller circle moves on the bigger circle with out " SLIP". In one complete revolution of the smaller circle on the larger one :

How many rotations does the smaller circle go through around its center in completing the revolution.
_________________

abhay pathania
GE US

Manager
Manager
User avatar
Joined: 24 Jun 2003
Posts: 147
Location: India
Followers: 1

Kudos [?]: 1 [0], given: 0

geometry.( some circles in life!!) [#permalink] New post 23 Jul 2003, 05:28
Brainless wrote:
4


Explain...

I think it should be 2
Manager
Manager
avatar
Joined: 10 Jun 2003
Posts: 213
Location: Maryland
Followers: 2

Kudos [?]: 4 [0], given: 0

[#permalink] New post 23 Jul 2003, 06:11
The answer is 2. Their circumferences are in the ratio of 1:2.
SVP
SVP
User avatar
Joined: 03 Feb 2003
Posts: 1613
Followers: 6

Kudos [?]: 57 [0], given: 0

 [#permalink] New post 23 Jul 2003, 07:12
C/c=pi*4*r/pi*2*r=2

2 revolutions
Intern
Intern
avatar
Joined: 15 Jul 2003
Posts: 20
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: geometry.( some circles in life!!) [#permalink] New post 24 Jul 2003, 08:57
abhaypathania wrote:
Imagine two circles one of radius "r" another "2r" . Let the bigger circle be stationay and the smaller circle moves on the bigger circle with out " SLIP". In one complete revolution of the smaller circle on the larger one :

How many rotations does the smaller circle go through around its center in completing the revolution.


2 is correct : But this it becomes tricky when I would ask how many revolutions around the CENTER of the BIGGER circle.
_________________

abhay pathania
GE US

SVP
SVP
User avatar
Joined: 03 Feb 2003
Posts: 1613
Followers: 6

Kudos [?]: 57 [0], given: 0

 [#permalink] New post 24 Jul 2003, 21:43
it is given -- one revolution
GMAT Instructor
User avatar
Joined: 07 Jul 2003
Posts: 771
Location: New York NY 10024
Schools: Haas, MFE; Anderson, MBA; USC, MSEE
Followers: 11

Kudos [?]: 33 [0], given: 0

Re: geometry.( some circles in life!!) [#permalink] New post 25 Jul 2003, 00:13
abhaypathania wrote:
Imagine two circles one of radius "r" another "2r" . Let the bigger circle be stationay and the smaller circle moves on the bigger circle with out " SLIP". In one complete revolution of the smaller circle on the larger one :

How many rotations does the smaller circle go through around its center in completing the revolution.


Be careful gang! This problem is a LOT trickier than it looks!!!!

I submit that the answer is THREE! <evil grin>

Any takers?

:twisted:
_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

GMAT Instructor
User avatar
Joined: 07 Jul 2003
Posts: 771
Location: New York NY 10024
Schools: Haas, MFE; Anderson, MBA; USC, MSEE
Followers: 11

Kudos [?]: 33 [0], given: 0

Re: geometry.( some circles in life!!) [#permalink] New post 25 Jul 2003, 00:20
abhaypathania wrote:
abhaypathania wrote:
Imagine two circles one of radius "r" another "2r" . Let the bigger circle be stationay and the smaller circle moves on the bigger circle with out " SLIP". In one complete revolution of the smaller circle on the larger one :

How many rotations does the smaller circle go through around its center in completing the revolution.


2 is correct : But this it becomes tricky when I would ask how many revolutions around the CENTER of the BIGGER circle.


I respectfully disagree with ALL of you. The correct answer is THREE! Think about it....
_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

SVP
SVP
User avatar
Joined: 03 Feb 2003
Posts: 1613
Followers: 6

Kudos [?]: 57 [0], given: 0

 [#permalink] New post 25 Jul 2003, 00:53
The center of the smaller circle draws a new circle, its radius being 3r.
If we imagine a new circle as a line (its lenght is 6r*pi), and if we fix a point on the smaller circle, then the number of the point's revolutions will be proportional to the circumference of the smaller circle.

Cnew=2*3*pi*r=6r*pi
Csmall=2r*pi

Thus, 3?

Is my reasoning correct?
SVP
SVP
User avatar
Joined: 03 Feb 2003
Posts: 1613
Followers: 6

Kudos [?]: 57 [0], given: 0

 [#permalink] New post 25 Jul 2003, 00:55
but in this case, the number of revolutions around the center of the bigger circle is 2.
Manager
Manager
User avatar
Joined: 24 Jun 2003
Posts: 147
Location: India
Followers: 1

Kudos [?]: 1 [0], given: 0

 [#permalink] New post 25 Jul 2003, 01:21
stolyar wrote:
The center of the smaller circle draws a new circle, its radius being 3r.
If we imagine a new circle as a line (its lenght is 6r*pi), and if we fix a point on the smaller circle, then the number of the point's revolutions will be proportional to the circumference of the smaller circle.

Cnew=2*3*pi*r=6r*pi
Csmall=2r*pi

Thus, 3?

Is my reasoning correct?


Stolyar, the smaller circle's centre will draw a new circle with a radius of r and not 3r.
GMAT Instructor
User avatar
Joined: 07 Jul 2003
Posts: 771
Location: New York NY 10024
Schools: Haas, MFE; Anderson, MBA; USC, MSEE
Followers: 11

Kudos [?]: 33 [0], given: 0

 [#permalink] New post 25 Jul 2003, 01:52
prashant wrote:
stolyar wrote:
The center of the smaller circle draws a new circle, its radius being 3r.
If we imagine a new circle as a line (its lenght is 6r*pi), and if we fix a point on the smaller circle, then the number of the point's revolutions will be proportional to the circumference of the smaller circle.

Cnew=2*3*pi*r=6r*pi
Csmall=2r*pi

Thus, 3?

Is my reasoning correct?


Stolyar, the smaller circle's centre will draw a new circle with a radius of r and not 3r.


Try this:

Take two quarters. Both have radius r. Suppose you keep one quarter stationary while the other quarter rotates around it one time (say start at 12 o'clock position, roll clockwise, then end at 12 o'clock position). According to logic seen so far, the outer quarter will only make ONE revolution about its center (that's the question asked, right?). Put the outer quarter in the heads position with George facing left and "LIBERTY" on top. Let's define one revolution of the outer quarter as the most obvious one: when George facing left and "liberty" is on top again. Now roll the outer quarter carefully around the stationary quarter, being careful not to let it slip. Note when the quarter is at about the 6 o'clock position, GEORGE IS FACING LEFT and LIBERTY IS ON TOP. As far as I am concerned, and by any reasonable definition, the outer quarter has completed one full revolution about ITS center. Keep going and teh quarter will make yet another revolution about its center and end up at teh 12 o'clock position with George once again facing left.

HOW DID THIS HAPPEN? Well, the quarter made ONE revolution RELATIVE TO THE SURFACE of the inner quarter, but the in following the surface of the inner quarter, forced it to make an additional revolution.

Thus for the circle of radius r rolling around the circle of radiur 2r. The circle will make 2 complete revolutions RELATIVE TO THE SURFACE OF THE INNER CIRCLE, and just following the path of the surface of the inner circle will take the outer circle through one additional complete revolution, thus it will make a total of 3 revolutions. (Cut 2 circles out of paper, draw a reference arrow, and try it yourself if you are not convinced!!!).
_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

Manager
Manager
User avatar
Joined: 24 Jun 2003
Posts: 147
Location: India
Followers: 1

Kudos [?]: 1 [0], given: 0

geometry.( some circles in life!!) [#permalink] New post 25 Jul 2003, 02:27
Fair try Akamaibrah. I see your point.

But could you please give us a more mathematical explanation rather than physical visualisation based explanation. I'm still struggling to solve it without using coins and paper cuttings :?

Thanks!
Manager
Manager
User avatar
Joined: 24 Jun 2003
Posts: 147
Location: India
Followers: 1

Kudos [?]: 1 [0], given: 0

Re: geometry.( some circles in life!!) [#permalink] New post 25 Jul 2003, 02:36
prashant wrote:
Fair try Akamaibrah. I see your point.

But could you please give us a more mathematical explanation rather than physical visualisation based explanation. I'm still struggling to solve it without using coins and paper cuttings :?

Thanks!


I've got it!

Stolyar - your reasoning is correct.

In the case where the two coins have the same radius r, the centre of the outer coin will draw out a circle of radius 2r as it goes around the circumference of the inner coin. Therefore, the distance travelled by the centre of the outer coin to come back to its starting position is 2*pi*2r which implies that the outer coin would have completed (2*pi*2r)/(2*pi*r) = 2 revolutions

With the same logic, in abhay's this case, the centre of the outer circle with radius r draws a new circle of radius 3r, as Stolyar mentioned. Therefore, the revolutions done by smaller circle, using the same logic, would be 3.

I feel quite dumb....this wasn't so tough..... :oops:

but anyway...c'est la vie
GMAT Instructor
User avatar
Joined: 07 Jul 2003
Posts: 771
Location: New York NY 10024
Schools: Haas, MFE; Anderson, MBA; USC, MSEE
Followers: 11

Kudos [?]: 33 [0], given: 0

Re: geometry.( some circles in life!!) [#permalink] New post 25 Jul 2003, 02:39
prashant wrote:
Fair try Akamaibrah. I see your point.

But could you please give us a more mathematical explanation rather than physical visualisation based explanation. I'm still struggling to solve it without using coins and paper cuttings :?

Thanks!


If you were to "unravel" the circumference of the inner circle into a straight line, the outer coin would clearly rotate 2 times since the circumference of the inner circle is twice that of the outer circle. However, since the actual path is also ONE complete circle, just following the path adds another circle to the number of rotations (imagine a ant walking along the edge of a coin held upright. Just by following his nose completely around the quarter, he completes one full "somersault" without knowing it).

Hence, the number of complete rotations is

[(CircumferenceInnerCircle)/(CircumferenceOUterCircle)] + 1.

Since Circumference is proportionate to Radius, the equation is also:

R_inner/R_outer + 1

and in this case

2r/r + 1 = 2 + 1 = THREE.

Stolyar's explanation is a little hard to understand (for me anyway), but I think he means this. Let's say you poke a hole in the middle of the outer circle and put in an axle. As the outer circle of radius r goes around the circle of radius 2r, the axle travels a total distance of 2*Pi*3r.

If you took that same "circle" (it is now, i suppose, a wheel") and rolled it along a path of length 2*Pi*3r, it would make 3 revolutions. That really is exactly equivalent to my equation.

The radius of the axle's path is 2*Pi*(R_inner + R_outer). Tne number of rotations is this length divided by the circumference of the outer circle which is 2 * Pi * R_outer. Hence:

Rotations = 2*Pi * (R_inner + R_outer) / (2 * Pi * R_outer)
= (R_inner + R_outer) / R_outer
= R_inner/R_outer + R_outer/R_outer
= R_inner/R_outer + 1 which is the same result that I derived "logically" (I don't trust the math unless I can visualize it in a way that makes sense).

The main thing is that we all agree that the answer is THREE.

HTH!
_________________

Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993


Last edited by AkamaiBrah on 25 Jul 2003, 03:03, edited 2 times in total.
Manager
Manager
User avatar
Joined: 24 Jun 2003
Posts: 147
Location: India
Followers: 1

Kudos [?]: 1 [0], given: 0

Re: geometry.( some circles in life!!) [#permalink] New post 25 Jul 2003, 02:47
AkamaiBrah wrote:
prashant wrote:
Fair try Akamaibrah. I see your point.

But could you please give us a more mathematical explanation rather than physical visualisation based explanation. I'm still struggling to solve it without using coins and paper cuttings :?

Thanks!


If you were to "unravel" the circumference of the inner circle into a straight line, the outer coin would clearly rotate 2 times since the circumference of the inner circle is twice that of the outer circle. However, since the actual path is also ONE complete circle, just following the path adds another circle to the number of rotations (imagine a ant walking along the edge of a coin held upright. Just by following his nose completely around the quarter, he completes one full "somersault" without knowing it).

Hence, the number of complete rotations is

[(CircumferenceInnerCircle)/(CircumferenceOUterCircle)] + 1.

Since Circumference is proportionate to Radius, the equation is also:

R_inner/R_outer + 1

and in this case

2r/r + 1 = 2 + 1 = THREE.



I love your answer - very crisp and clear!
And the ant example works very well !

:)
Intern
Intern
avatar
Joined: 15 Jul 2003
Posts: 20
Followers: 0

Kudos [?]: 0 [0], given: 0

 [#permalink] New post 27 Jul 2003, 21:34
Bingo...

Three it is.

the explaination given is correct too. The other way u could solve it is by making the radius of the inner circle to zero... then the outter cirlce would rotate about a point... ( ONCE ) and still would have travelled zero distance about the point...the other way to resolve the mistry is to make diagrams and then evaluate rotation angle ("thita') about the point of rotation of the outter circle.
Absolute Revolutions are always calculated based on the point of rotation rather then the center of rotation. where the point of rotation in this is the point of contact between the bigger and the smaller circle.
  [#permalink] 27 Jul 2003, 21:34
    Similar topics Author Replies Last post
Similar
Topics:
2 Two circles, one with radius 10 inches and the other with ra AccipiterQ 2 03 Nov 2013, 20:28
63 Experts publish their posts in the topic A rectangle is inscribed in a circle of radius r. If the alex1233 31 16 May 2012, 06:44
6 Experts publish their posts in the topic Consider a quarter of a circle of radius 16. Let r be the JubtaGubar 3 21 Apr 2012, 09:28
Experts publish their posts in the topic A circle of radius r ahs a circumference equal to C. A puma 4 12 Apr 2008, 10:53
Experts publish their posts in the topic A point is arbitrarily selected inside a circle of radius R. bmwhype2 4 20 Nov 2007, 02:36
Display posts from previous: Sort by

Imagine two circles one of radius "r" another "2r" . Let the

  Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

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®.