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# Region R is a square in the x-y plane with vertices

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Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
Region R is a square in the x-y plane with vertices  [#permalink]

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08 Feb 2013, 10:45
5
15
00:00

Difficulty:

65% (hard)

Question Stats:

65% (03:20) correct 35% (03:06) wrong based on 334 sessions

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Region R is a square in the x-y plane with vertices J = (–1, –2), K = (–1, 4), L = (5, 4), and M = (5, –2). What is the probability that a randomly selected point in region R lies below the line 3x – 5y = 10?
(A) $$\frac{5}{12}$$
(B) $$\frac{5}{18}$$
(C) $$\frac{5}{24}$$
(D) $$\frac{5}{36}$$
(E) $$\frac{5}{72}$$

For thoughts on geometric probability, as well as a complete solution with diagram to this particular problem, see:
http://magoosh.com/gmat/2013/geometric- ... -the-gmat/

Experts: are there any words of wisdom you would like to share on the topic of Geometric Probability?

Mike

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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Re: Region R is a square in the x-y plane with vertices  [#permalink]

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08 Feb 2013, 11:03
1
mikemcgarry wrote:
Region R is a square in the x-y plane with vertices J = (–1, –2), K = (–1, 4), L = (5, 4), and M = (5, –2). What is the probability that a randomly selected point in region R lies below the line 3x – 5y = 10?
(A) $$\frac{5}{12}$$
(B) $$\frac{5}{18}$$
(C) $$\frac{5}{24}$$
(D) $$\frac{5}{36}$$
(E) $$\frac{5}{72}$$

For thoughts on geometric probability, as well as a complete solution with diagram to this particular problem, see:
http://magoosh.com/gmat/2013/geometric- ... -the-gmat/

Experts: are there any words of wisdom you would like to share on the topic of Geometric Probability?

Mike

Basically the square is formed with 4 lines x=-1, x=5, y=-2,y=4
Area of square =36
Now line is given as 5y=3x-10
We can see that the points on which this line intersects 4 lines forming square are (0,-2) and (5,1)
Thus area of triangle formed under the line =15/2
Probability of a point in this area=(15/2)/36 =5/24
Ans C it is!
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Re: Region R is a square in the x-y plane with vertices  [#permalink]

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09 Feb 2013, 17:25
1
The square is formed by x=5 & x=-1 ; y=4 & y=-2

-1 to 5 or -2 to 4 = 6 which is the length of the square (if it was an rectangle we would have to find the length and breadth)

So the area of the square is 36.

To find the points where the line 3x-5y=10 cuts across the square, insert the two values of x or y into the equation

x=5
=> 3*5-5y=10
=> y=1. So (5,1) is first point of intersection of the line with the square.

x=-1
=> 3*(-1)-5y=10
=> y=-(7/5). This point (-1, -7/5) is not on any of the boundary of the square so this is not relevant. Let's try inserting the values of y to find where is the second point of intersection is

y=-2
=> 3x-5*(-2)=10
=> x=0. So (0, -2) is the second point of intersection of the line with the square.

So the line cuts the square at two adjacent sides of the square to form a triangle with base 5 and height 3. Area of this triangle is 15/2

Probability of point lying in this triangle = (15/2)/36 = 5/24
Manager
Joined: 06 Jun 2010
Posts: 154
Re: Region R is a square in the x-y plane with vertices  [#permalink]

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05 Mar 2013, 10:26
1 doubt i have here is that if we try to find x intercept of that line,we get x =10/3 but as per the discussion here,the line intercepts x axis at 5,1.

Thanks,
Shreeraj
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Joined: 28 Jul 2011
Posts: 181
Re: Region R is a square in the x-y plane with vertices  [#permalink]

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15 Mar 2013, 23:21
will vote for C

Area of square = 36

line passes thr' = (0,-2) & (0.3.3)

Line intersect the square @ = (0,-2) & (5,1) - this forms a triangle with base=5 and height = 3

Area of triangle = 15/2

probability = 15/2 * 1/36 = 5 / 24
Intern
Joined: 19 Jul 2010
Posts: 11
Re: Region R is a square in the x-y plane with vertices  [#permalink]

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03 May 2013, 22:32
mikemcgarry wrote:
Region R is a square in the x-y plane with vertices J = (–1, –2), K = (–1, 4), L = (5, 4), and M = (5, –2). What is the probability that a randomly selected point in region R lies below the line 3x – 5y = 10?
(A) $$\frac{5}{12}$$
(B) $$\frac{5}{18}$$
(C) $$\frac{5}{24}$$
(D) $$\frac{5}{36}$$
(E) $$\frac{5}{72}$$

For thoughts on geometric probability, as well as a complete solution with diagram to this particular problem, see:
http://magoosh.com/gmat/2013/geometric- ... -the-gmat/

Experts: are there any words of wisdom you would like to share on the topic of Geometric Probability?

Mike

Hi,

i might be wrong in here, please correct me.

prob = area of triangle / total square area.
but in the question it is What is the probability that a randomly selected point in region R lies below the line 3x – 5y = 10?

so shouldn't the answer be 1 - (5/24)

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Posts: 6001
City: Pune
Re: Region R is a square in the x-y plane with vertices  [#permalink]

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04 May 2013, 07:26
2
tarunjagtap wrote:
mikemcgarry wrote:
Region R is a square in the x-y plane with vertices J = (–1, –2), K = (–1, 4), L = (5, 4), and M = (5, –2). What is the probability that a randomly selected point in region R lies below the line 3x – 5y = 10?
(A) $$\frac{5}{12}$$
(B) $$\frac{5}{18}$$
(C) $$\frac{5}{24}$$
(D) $$\frac{5}{36}$$
(E) $$\frac{5}{72}$$

For thoughts on geometric probability, as well as a complete solution with diagram to this particular problem, see:
http://magoosh.com/gmat/2013/geometric- ... -the-gmat/

Experts: are there any words of wisdom you would like to share on the topic of Geometric Probability?

Mike

Hi,

i might be wrong in here, please correct me.

prob = area of triangle / total square area.
but in the question it is What is the probability that a randomly selected point in region R lies below the line 3x – 5y = 10?

so shouldn't the answer be 1 - (5/24)

Hi Tarun,

Go thru the attached figure. then you will come to know what are you missing exactly.

Narenn
Attachments

cord prob.png [ 22.27 KiB | Viewed 5269 times ]

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Joined: 19 Jul 2010
Posts: 11
Re: Region R is a square in the x-y plane with vertices  [#permalink]

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04 May 2013, 07:47
Narenn wrote:

Hi Tarun,

Go thru the attached figure. then you will come to know what are you missing exactly.

Narenn

oh ok .. got it.. thanks Narenn.. I dont know what was going through my mind..

actually i plotted the line wrongly.. hence the stupid question.
Intern
Joined: 28 Dec 2015
Posts: 39
Re: Region R is a square in the x-y plane with vertices  [#permalink]

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12 Jul 2016, 02:46
X=5 is the equation of the line LM
3x-5y=10 intersects y axis at -2 and x=5 at (5,1).

So,we get a triangle whose area=1/2*5*3=15/2
Now Area of square=6^2=36

So,probability=15/2/36=15/72=5/24
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Re: Region R is a square in the x-y plane with vertices  [#permalink]

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13 May 2018, 11:22
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Re: Region R is a square in the x-y plane with vertices &nbs [#permalink] 13 May 2018, 11:22
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