Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 61283

If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
11 Sep 2015, 01:01
Question Stats:
49% (01:57) correct 51% (01:45) wrong based on 278 sessions
HideShow timer Statistics
If x and y are both positive, is x greater than 4? (1) x > y (2) In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origin. Kudos for a correct solution.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________




Math Expert
Joined: 02 Sep 2009
Posts: 61283

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
14 Sep 2015, 06:33
Bunuel wrote: If x and y are both positive, is x greater than 4?
(1) x > y (2) In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origin.
Kudos for a correct solution. MANHATTAN GMAT OFFICIAL SOLUTION:We are told that both x and y are positive, and we are asked whether x is greater than 4. There’s not much rephrasing we can do to this question, so let’s move on to the statements. Statement (1): INSUFFICIENT. All we know here is that x is larger than y. It is still possible for x to be any positive number (y just has to be a smaller positive number). Statement (2): INSUFFICIENT. If the point (x, y) lies outside a circle of radius 5 centered at the origin, then that point lies at a distance of more than 5 units away from (0, 0). In the coordinate plane, distance can be computed with the Pythagorean Theorem. We can rephrase this statement to say that x^2 + y^2 > 25. However, x can still be as small or as large a positive number as we wish. Statement (1) and (2) together: INSUFFICIENT. We can definitely pick a very large value for x to satisfy the statements and answer the question with a “Yes.” Just make y smaller to make the first statement true, and if x is bigger than 5, then we’ll definitely have statement (2) true as well. The trick is that we can pick a value of x that is not greater than 4 and still satisfy all the conditions. Let’s try picking x equal to 4 (this would give us an answer of “No” to the question). So we need to see whether there are any values of y that satisfy the following 3 conditions: 1. y is positive (from the stem). 2. y is less than 4 (that is, less than x). 3. x^2 + y^2 > 25. Let’s plug 4 in for x in the last inequality. We get 16 + y^2 > 25 y^2 > 9 y > 3 (since y must be positive, we don’t have to worry about the negative possibilities) The conditions become these: y is greater than 3 and less than 4. Any number between 3 and 4 satisfies the conditions. Notice that y is not restricted to integer values; nothing in the problem indicates that it should be. Thus, we still cannot definitively answer the question of whether x is greater than 4. There are other values of x less than 4 that will work; one tricky part of this problem is that those values of x are greater than 3. The correct answer is E.
_________________




Intern
Joined: 24 Feb 2015
Posts: 44
Schools: Booth PT '18, Anderson '18, LBS '17, INSEAD Jan '17, HEC Dec '17, HKUST '17, ISB '17, Rotman '17, Ivey '17, Schulich Jan '18, NUS '17, IE April '17, Desautels '18, Bocconi '17, Mannheim"17

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
11 Sep 2015, 03:33
St 1 => no info about the value of y => Insufficient
St 2 => (x, y) must satisfy the inequality : x^2 + y^2 > 25 Both points (1,6) & (6, 1) will satisfy this inequality i.e. x can be greater than or even less than 4 => hence, insufficient
Combining both statements => (x, y) such that x > y & x^2 + y^2 > 25 We need to test only with points from Quadrant I since both x & y are positive Both points (6,1) & (3.9, 3.5) will satisfy both the given conditions We still do not get a definite YES/NO answer.
So, correct answer is E .......(Most Probably)



Manager
Joined: 02 Jul 2015
Posts: 100

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
12 Sep 2015, 04:45
Answer as per me is C. only first quadrant is to be considered. As x>y, area between x=0 and x=y is to be considered. thus, C.
Bunuel, is the approach right?



CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 3153
Location: India
GMAT: INSIGHT
WE: Education (Education)

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
12 Sep 2015, 09:05
Bunuel wrote: If x and y are both positive, is x greater than 4?
(1) x > y (2) In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origin.
Kudos for a correct solution. Given: x and y are both POSITIVEQuestion : Is x>4?Statement 1: x>yCase 1: y = 1, x=3 i.e. x < 4 Case 2: y = 1, x=5 i.e. x > 4 NOT SUFFICIENT Statement 2: In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origini.e. \(x^2 + y^2 > 5^2\) i.e. \(x^2 + y^2 > 25\) Case 1: y = 4, x=3.5 i.e. x < 4 Case 2: y = 4, x=4.5 i.e. x > 4 NOT SUFFICIENT Combining the two statements:\(x^2 + y^2 > 25\) and \(x>y\) Case 1: \(y = 5/\sqrt{2} = 3.54\), \(x=3.6\) i.e. x < 4 Case 2: y = 4, x=4.5 i.e. x > 4 Answer: Option E
_________________
Prosper!!!GMATinsightBhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhihttp://www.GMATinsight.com/testimonials.htmlClick here for Our VERBAL & QUANT private tutoring package detailsACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Intern
Joined: 16 Mar 2015
Posts: 5

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
16 Sep 2015, 10:12
Hi Experts,
I thought a lot on this but couldnt picture a values of x and Y when x>y  Condition on combining both options.
Let us forget a equation x^2+Y^2 > 25 for a moment. Can you help me plotting the points on the graph which satisfies below conditions
X Can be less then 4 X < Y X,Y lie outside the circle of radius 5



Intern
Joined: 28 Aug 2017
Posts: 34

If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
18 Sep 2017, 01:51
Bunuel wrote: If x and y are both positive, is x greater than 4?
(1) x > y (2) In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origin.
Kudos for a correct solution. Given: x > 0, y > 0 x > 4? Statement 1: x > y This statement is insufficient to prove whether x > 4 because there are innumerable pairs of positive numbers where x > y and x < 4 such as (2, 1) and also x > y and x > 4 such as (5, 3), etc. Statement 2: Given: the point (x, y) lies outside a circle of radius 5 centered at the origin. The point (x, y) lies on a circle of radius \(\sqrt{x^2 + y^2}\) centered at the origin. Since the point (x, y) lies outside the circle of radius 5, this implies that the radius of the circle with the point (x, y) is greater than 5. So, \(\sqrt{x^2 + y^2} > 5\) \(x^2 + y ^2 > 25\) But, there are many pairs of numbers which satisfy this inequality without giving a definite answer on whether x > 4 such as (2, 5), (4, 4), (5, 1), etc. Since the individual statements are insufficient individually, on combining them: x > y and \(x^2 + y ^2 > 25\) Since, x > y and x and y are positive \(x^2 > y^2\) \(x^2 + x^2 > y^2 + x^2 > 25\) \(2x^2 > 25\) x > \(5/\sqrt{2}\) x > 3.5 But, combining the statements hasn't concluded if x > 4. Hence, the statements are sufficient neither individually nor combined. Therefore, the answer is E.



Senior Manager
Status: Whatever it takes!
Joined: 10 Oct 2018
Posts: 381
GPA: 4

If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
Updated on: 14 Sep 2019, 23:47
This is how I solved. Hope it helps. Posted from my mobile device
Attachments
IMG_20190915_130727.jpg [ 2.97 MiB  Viewed 339 times ]
Originally posted by EncounterGMAT on 23 Oct 2018, 08:37.
Last edited by EncounterGMAT on 14 Sep 2019, 23:47, edited 1 time in total.



Current Student
Joined: 05 Oct 2018
Posts: 40
Location: United States
GMAT 1: 770 Q49 V47
GPA: 3.95
WE: General Management (Other)

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
23 Oct 2018, 12:07
Bunuel wrote: If x and y are both positive, is x greater than 4?
(1) x > y (2) In the coordinate plane, the point (x, y) lies outside a circle of radius 5 centered at the origin.
Kudos for a correct solution. From the stem we know x > 0 and y > 0 (1) Clearly not sufficient. X = 2, y = 1 works and so does x = 5, y = 4 (although we are not limited to integers as in the choices I made) (2) we now know x^2+y^2 > 25, and x > 0, y > 0 This is still not sufficient. Examples include x = 1, y = 100 and x = 6, y = 2 (+) now in the coordinate plane, we only care about points in quadrant one that are both below the line y=x in and outside the circle y^2+x^2 = 25 It is obvious that we can find values of x that satisfy this >4, so the question is are there any values of 4 or less. Well the minimum value has to be just greater than 2x^2 = 25 (subbing in y=x for y in the equation of the circle). That gives us the square root of 12.5 which is less than the square of 4 (16), so x could be less than or equal to 4 still. Answer E



NonHuman User
Joined: 09 Sep 2013
Posts: 14086

Re: If x and y are both positive, is x greater than 4?
[#permalink]
Show Tags
27 Nov 2019, 04:40
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 x and y are both positive, is x greater than 4?
[#permalink]
27 Nov 2019, 04:40






