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# x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y

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MBA Section Director
Affiliations: GMATClub
Joined: 22 May 2017
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WE: Engineering (Computer Software)
x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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28 Sep 2018, 21:13
00:00

Difficulty:

95% (hard)

Question Stats:

27% (03:04) correct 73% (02:46) wrong based on 77 sessions

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x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y. What is the positive difference between the minimum possible value of x and the minimum value of y?

A. -6

B. 0

C. 1

D. 4

E. 6

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Joined: 24 Nov 2017
Posts: 60
Location: India
GMAT 1: 720 Q51 V36
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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29 Sep 2018, 00:50
Given conditions: x + 2y > 20 and 3x - 30 < -y; x and y are positive integers
Objective: To find the difference between the minimum possible value of x and the minimum value of y

3x - 30 < -y can be written as 3x + y < 30

Please refer to the following 3 graphs.

The area that satisfies the first inequality x + 2y > 20 is the region above the red line.

The area that satisfies the second inequality 3x + y < 30 is the region below the green line.

Because x and y are positive integers, the area of interest is restricted to I quadrant and the area enclosed in the triangle ABC are values of x and y that satisfy both the inequalities.

Let us find the coordinates of point C
Solve the two equations x + 2y = 20 and 3x + y = 30
3x + 6y = 60
3x + y = 30
-----------
5y = 30
---------
Or y = 6. Substituting y = 6 in equation (1), we get x = 8.
So, coordinates of point C are (8, 6)

It is evident from the graph that lowest point among the three from y coordinates is C, So, the minimum value of y enclosed in the triangle is an integer greater than 6. So, it has to be 7.
From the graph, we can also deduce that the minimum value of x enclosed in the triangle is an integer greater than 0. So, it has to be 1.

The positive difference between the minimum value of x and minimum value of y is 6.
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Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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29 Sep 2018, 05:01
seems 1 as 0 is not possible which question limits
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Joined: 04 Jan 2015
Posts: 2944
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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29 Sep 2018, 05:19
2

Solution

Given:
• x and y are positive integers
• x + 2y > 20 and 3x – 30 < -y

To find:
• The positive difference between minimum value of x and minimum value of y

Approach and Working:
• x + 2y > 20
• 3x – 30 < -y
Or, 3x + y < 30
Or, -3x – y > -30
Or, -6x – 2y > -60

Adding the above two inequalities, we get,
• x – 6x + 2y – 2y > 20 – 60
Or, -5x > -40
Or, 5x < 40
Or, x < 8

As x is positive integer, minimum possible value of x is 1
• also, if x < 8, we can say from x + 2y > 20,
2y > 20 – x
Or, 2y > 12
Or, y > 6

As y is positive integer, minimum possible value of y is 7

Therefore, the positive difference = 7 – 1 = 6

Hence, the correct answer is option E.

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Joined: 12 Nov 2016
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Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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29 Sep 2018, 05:39
2y > 20 – x
Or, 2y > 12

As X<8, how can we consider x=8

Posted from my mobile device
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2944
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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29 Sep 2018, 05:46
1
Siddharthachepuri@gmail.com wrote:
2y > 20 – x
Or, 2y > 12

As X<8, how can we consider x=8

Posted from my mobile device

Hi,
we did not consider x = 8. As x is less than 8, we used this to conclude that y is greater than 12.
(For example, if x can be maximum 7, then y should be minimum 7 (we wrote it as y > 6).
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Joined: 02 Aug 2009
Posts: 7763
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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12 Dec 2018, 20:49
workout wrote:
x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y. What is the positive difference between the minimum possible value of x and the minimum value of y?

A. -6

B. 0

C. 1

D. 4

E. 6

Let us get the range of one variable first and for that we will have to cancel out the other variable.
So x+2y>20....(I)
3x-30<-y..... multiply by 2, so 6x-60<-2y...(II)
x+2y-2y>20+6x-60.......5x<40.....x<8
Now since x is positive integer and x<8, the minimum possible value of x is 1..

Now take the equation for getting the range of the values of y..
x+2y>20...2y>20-x.... higher the X, lower the y
So take max possible value of x, so 2y>20-7....y>6.5
3x-30<-y....y<30-3x....if you add max value of X, we will get least possible value of y
So y<30-3*7.....y<9
So least possible value is less than 9 but greater than 6.5..
So minimum value of y is 7..

Difference is 7-1=6
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Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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12 Dec 2018, 21:53
EgmatQuantExpert wrote:

Solution

Given:
• x and y are positive integers
• x + 2y > 20 and 3x – 30 < -y

To find:
• The positive difference between minimum value of x and minimum value of y

Approach and Working:
• x + 2y > 20
• 3x – 30 < -y
Or, 3x + y < 30
Or, -3x – y > -30
Or, -6x – 2y > -60

Adding the above two inequalities, we get,
• x – 6x + 2y – 2y > 20 – 60
Or, -5x > -40
Or, 5x < 40
Or, x < 8

As x is positive integer, minimum possible value of x is 1
• also, if x < 8, we can say from x + 2y > 20,
2y > 20 – x
Or, 2y > 12
Or, y > 6

As y is positive integer, minimum possible value of y is 7

Therefore, the positive difference = 7 – 1 = 6

Hence, the correct answer is option E.

Once we calculate value for x, while calculating value for y using first equation we get y>6 and with second equation, we get y<6. How to know which one to take?
Math Expert
Joined: 02 Aug 2009
Posts: 7763
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y  [#permalink]

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12 Dec 2018, 22:40
ritu1009 wrote:
EgmatQuantExpert wrote:

Solution

Given:
• x and y are positive integers
• x + 2y > 20 and 3x – 30 < -y

To find:
• The positive difference between minimum value of x and minimum value of y

Approach and Working:
• x + 2y > 20
• 3x – 30 < -y
Or, 3x + y < 30
Or, -3x – y > -30
Or, -6x – 2y > -60

Adding the above two inequalities, we get,
• x – 6x + 2y – 2y > 20 – 60
Or, -5x > -40
Or, 5x < 40
Or, x < 8

As x is positive integer, minimum possible value of x is 1
• also, if x < 8, we can say from x + 2y > 20,
2y > 20 – x
Or, 2y > 12
Or, y > 6

As y is positive integer, minimum possible value of y is 7

Therefore, the positive difference = 7 – 1 = 6

Hence, the correct answer is option E.

Once we calculate value for x, while calculating value for y using first equation we get y>6 and with second equation, we get y<6. How to know which one to take?

second equation $$y<30-3x$$... This will give us the upper limit as we are getting y is less than something.
So this should give you the maximum value of y..
and you will get maximum value for y when x is minimum in y<30-3x, so y<30-3*1....y<27
so range of values of y is 6<y<27.....
Minimum value is 7 and max value is 26

Hope it helps
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Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y   [#permalink] 12 Dec 2018, 22:40
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