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

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Manager
Joined: 09 Jul 2018
Posts: 63
x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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16 Sep 2018, 21:51
1
15
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Difficulty:

95% (hard)

Question Stats:

44% (02:14) correct 56% (02:58) wrong based on 68 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 minimum value of y?
A. -6
B. 0
C. 1
D.4
E.6
Math Expert
Joined: 02 Aug 2009
Posts: 7971
x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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16 Sep 2018, 23:14
1
1
ritu1009 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 minimum value of y?
A. -6
B. 0
C. 1
D.4
E.6

Let's get both terms in just one variable..
(A) x+2y>20...(I)
3x-30<-y or -y>3x-30.... Multiply this with 2
-2y>6x-60.....(ii)
Add I and ii...
x+2y-2y>20+6x-60........x>6x-40..........5x<40.........x<8
So minimum value of x can be 1, as x is positive integer

(B) Now to get minimum value of y, take x as maximum that is 7 and substitute in (I)
x+2y>20..7+2y>20....y>13/2, ao y can have minimum possible value as 7, the next integer after 13/2
Or get the equations in term of y..
So multiply (I) by 3.........3x+6y>60
Add this to (II) -y-3x>-30
So 3x+6y-y-3x>60-30.......5y>30....y>6.....so min value of y is 7

Difference between the two = 7-1=6

E
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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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17 Sep 2018, 00:05
chetan2u wrote:
ritu1009 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 minimum value of y?
A. -6
B. 0
C. 1
D.4
E.6

Let's get both terms in just one variable..
x+2y>20...(I)
3x-30<-y or -y>3x-30.... Multiply this with 2
-2y>6x-60.....(ii)
Add I and ii...
x+2y-2y>20+6x-60........x>6x-40..........5x<40.........x<8
So minimum value of x can be 1, as x is positive integer

Now to get minimum value of y, take x as maximum that is 7 and substitute in (I)
x+2y>20..7+2y>20....y>13/2, ao y can have minimum possible value as 7, the next integer after 13/2

Difference between the two = 7-1=6

E

Hi chetan2u,

If we substitute the value of x= 1, y = 7.. the equation x + 2y > 20 doesn't hold true. Can you please help if I am missing anything here.
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Math Expert
Joined: 02 Aug 2009
Posts: 7971
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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17 Sep 2018, 02:20
1
rahul16singh28 wrote:
chetan2u wrote:
ritu1009 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 minimum value of y?
A. -6
B. 0
C. 1
D.4
E.6

Let's get both terms in just one variable..
x+2y>20...(I)
3x-30<-y or -y>3x-30.... Multiply this with 2
-2y>6x-60.....(ii)
Add I and ii...
x+2y-2y>20+6x-60........x>6x-40..........5x<40.........x<8
So minimum value of x can be 1, as x is positive integer

Now to get minimum value of y, take x as maximum that is 7 and substitute in (I)
x+2y>20..7+2y>20....y>13/2, ao y can have minimum possible value as 7, the next integer after 13/2

Difference between the two = 7-1=6

E

Hi chetan2u,

If we substitute the value of x= 1, y = 7.. the equation x + 2y > 20 doesn't hold true. Can you please help if I am missing anything here.

When we look for minimum possible value, we find it irrespective of the other variable..
So we find minimum of x just ensuring that y is positive integer.
And similarly we take max value of x to get minimum value of y..
Say x+y=8 and we have to find min possible value of x and y if x and g are positive integers..
So the values will be 1 each for both even if 1+1 is not 8
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Posts: 230
Concentration: Finance, Economics
WE: Corporate Finance (Commercial Banking)
x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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17 Sep 2018, 08:53
Hi chetan2u ,

how can we solve the following two equations in one go.

y>6
and x+2y>20

TIA
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Manager
Joined: 09 Jul 2018
Posts: 63
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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17 Sep 2018, 23:49
Shank18 wrote:
Hi chetan2u ,

how can we solve the following two equations in one go.

y>6
and x+2y>20

TIA

I too have the same issue. once y is calculated , x is only obtained from 3x-30<-y and not x+2y>20.
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Joined: 02 Sep 2018
Posts: 5
Location: India
Schools: ISB '20
GMAT 1: 710 Q49 V38
GPA: 2.9
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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18 Sep 2018, 04:23
3
ritu1009

Think of it this way,

x+2y>20.... (1)
3x+y<30.... (2)

Multiply (1) with 3 and it becomes 3x+6y>60

compare it with equation (2) and you will see that the only difference lies in 5y. So 5y must definitely be
more than 30. thus 5Y>30 and thus Y>6

now multiply equation (2) with 2 and it becomes 6x+2y<60
Again compare with equation (1) and you will see a difference of 5x
So 5x<40 and thus x<8

Now take minimum values for X and Y knowing that they are positive integers,
X=1 and Y=7
Thus difference is 6
Manager
Status: In last prep stage
Joined: 11 Jun 2017
Posts: 156
GMAT 1: 630 Q44 V33
GMAT 2: 680 Q47 V37
GPA: 3.2
Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.  [#permalink]

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18 Sep 2018, 06:49
chetan2u wrote:
ritu1009 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 minimum value of y?
A. -6
B. 0
C. 1
D.4
E.6

Let's get both terms in just one variable..
x+2y>20...(I)
3x-30<-y or -y>3x-30.... Multiply this with 2
-2y>6x-60.....(ii)
Add I and ii...
x+2y-2y>20+6x-60........x>6x-40..........5x<40.........x<8
So minimum value of x can be 1, as x is positive integer

Now to get minimum value of y, take x as maximum that is 7 and substitute in (I)
x+2y>20..7+2y>20....y>13/2, ao y can have minimum possible value as 7, the next integer after 13/2

Difference between the two = 7-1=6

E

I did not understand this statement:Now to get minimum value of y, take x as maximum that is 7 and substitute.
Why should we take ma of x ?
What is we assume that x and y are both 1,won't that satisfy the inequalities and answer is 0.Please let me know ,where i am wrong in this approach.
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Re: x and y are positive integers such that x + 2y > 20 and 3x – 30 < -y.   [#permalink] 18 Sep 2018, 06:49
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