Last visit was: 21 May 2024, 19:09 It is currently 21 May 2024, 19:09
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.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Difficulty: 505-555 Level,    Algebra,    Min-Max Problems,                         
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 93373
Own Kudos [?]: 625654 [161]
Given Kudos: 81918
Send PM
Most Helpful Reply
Verbal Forum Moderator
Joined: 08 Dec 2013
Status:Greatness begins beyond your comfort zone
Posts: 2099
Own Kudos [?]: 8863 [25]
Given Kudos: 171
Location: India
Concentration: General Management, Strategy
GPA: 3.2
WE:Information Technology (Consulting)
Send PM
Retired Moderator
Joined: 25 Feb 2013
Posts: 894
Own Kudos [?]: 1531 [11]
Given Kudos: 54
Location: India
GPA: 3.82
Send PM
General Discussion
avatar
Intern
Intern
Joined: 15 Jun 2016
Posts: 1
Own Kudos [?]: 4 [4]
Given Kudos: 1
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
4
Kudos
I tackled this question by looking to make the denominators the same.
So Y*(1/X) - X*(X/Y)=? >> (Y-X^2)/XY=?
Now looking at the numerator, I see that as X gets larger, the smaller the overall value of the numerator will be.
You're going to want the Y value to be the largest it can be since X^2 will be subtracted from it. This means that the value of X will need to be the smallest it can be.
Therefore, Y=9 and X=3 >> (9-(3)^2)/(9*3)= 0/27 or 0.
Answer is B
Retired Moderator
Joined: 04 Aug 2016
Posts: 391
Own Kudos [?]: 339 [0]
Given Kudos: 144
Location: India
Concentration: Leadership, Strategy
GPA: 4
WE:Engineering (Telecommunications)
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
why can't x take value as 2 and y as 8?
Math Expert
Joined: 02 Sep 2009
Posts: 93373
Own Kudos [?]: 625654 [0]
Given Kudos: 81918
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
Expert Reply
warriorguy wrote:
why can't x take value as 2 and y as 8?


x cannot be 2 because 2 < x ≤ 8.

But y can be 8 (2 < y ≤ 9).
Retired Moderator
Joined: 05 Jul 2006
Posts: 849
Own Kudos [?]: 1565 [1]
Given Kudos: 49
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
1
Bookmarks
Bunuel wrote:
If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the maximum value of (1/x - x/y)?

A. \(-3\frac{1}{8}\)
B. 0
C. 1/5
D. 5/18
E. 2


1/x - x/y = y-x^2/xy , x^2 is key , x^2 value in the range 9<=x<= 64 , min value of x^2 is 9 that is the max value y can reach , thus to max y-x^2 , 0 is the max
IIM School Moderator
Joined: 04 Sep 2016
Posts: 1260
Own Kudos [?]: 1253 [0]
Given Kudos: 1207
Location: India
WE:Engineering (Other)
Send PM
If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
Abhishek009 niks18 pushpitkc gmatbusters Skywalker18

I did not understand below quote:
Quote:
max (1/x - x/y)

the value of the above expression will be maximum when value of x is least and y is maximum

How do I break the question stem knowing that x and y are positive integers?

1.What can I say about 1/x
2. What can I say about x/y
What can I say about difference between (1) and (2)
IIM School Moderator
Joined: 04 Sep 2016
Posts: 1260
Own Kudos [?]: 1253 [0]
Given Kudos: 1207
Location: India
WE:Engineering (Other)
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
Thanks niks18 for your two cents. I had an additional query.


Quote:
There are two parts to the equation. we have 1/x and then you are subtracting x/y from 1/x. so essentially the value of the resulting equation will be less than 1/x. But the question asks us to maximize 1/x-x/y, so 1/x has to attain the maximum value possible.
1/x will be maximum when x has least value because in that case its reciprocal i.e 1/x will be max. For eg. if you are given x=3 & x=4, then out of these two values max 1/x will be when x=3 because 1/3=0.333 & 1/4=0.25. So now you know that we need the least value of x for the first part.


The highlighted text is an universal fact which holds true even when 1/x reduces to an integer.
E.g. x = 1 (although I do realize that with additional constraints in Q stem, x can not take value of 1,
but can you validate my reasoning.)
Retired Moderator
Joined: 25 Feb 2013
Posts: 894
Own Kudos [?]: 1531 [0]
Given Kudos: 54
Location: India
GPA: 3.82
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
adkikani wrote:
Thanks niks18 for your two cents. I had an additional query.


Quote:
There are two parts to the equation. we have 1/x and then you are subtracting x/y from 1/x. so essentially the value of the resulting equation will be less than 1/x. But the question asks us to maximize 1/x-x/y, so 1/x has to attain the maximum value possible.
1/x will be maximum when x has least value because in that case its reciprocal i.e 1/x will be max. For eg. if you are given x=3 & x=4, then out of these two values max 1/x will be when x=3 because 1/3=0.333 & 1/4=0.25. So now you know that we need the least value of x for the first part.


The highlighted text is an universal fact which holds true even when 1/x reduces to an integer.
E.g. x = 1 (although I do realize that with additional constraints in Q stem, x can not take value of 1,
but can you validate my reasoning.)


Hi adkikani

I am not sure why are you calling the highlighted statement as a "Universal fact"? can you clarify more on your query?
IIM School Moderator
Joined: 04 Sep 2016
Posts: 1260
Own Kudos [?]: 1253 [0]
Given Kudos: 1207
Location: India
WE:Engineering (Other)
Send PM
If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
Hi niks18

By universal fact, I meant: if basic conditions are satisfied, the rule holds true for any value of a variable.
Eg. If I am given that x raised to any unknown power yields an even integer, I know for sure:
x is even.

Similarly highlighted text would hold true if (1)the ratio yielded a decimal (x: any value other than 1) or
(2) an integer (I would have to take x to be 1, just to make sure I understood the logic of subtraction clearly).

I wanted to ask you if highlighted text holds true for both conditions above.
In short, for A - B, I need to maximize A and minimize B irrespective of A and B being decimals or integers.
Let me know if we are on same page.
Retired Moderator
Joined: 25 Feb 2013
Posts: 894
Own Kudos [?]: 1531 [1]
Given Kudos: 54
Location: India
GPA: 3.82
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
1
Kudos
adkikani wrote:
Hi niks18

By universal fact, I meant: if basic conditions are satisfied, the rule holds true for any value of a variable.
Eg. If I am given that x raised to any unknown power yields an even integer, I know for sure:
x is even.

Similarly highlighted text would hold true if (1)the ratio yielded a decimal (x: any value other than 1) or
(2) an integer (I would have to take x to be 1, just to make sure I understood the logic of subtraction clearly).

I wanted to ask you if highlighted text holds true for both conditions above.
In short, for A - B, I need to maximize A and minimize B irrespective of A and B being decimals or integers.
Let me know if we are on same page.


Hi adkikani

if we need to maximize A-B, then irrespective of nature of A & B, we will have to maximize A and minimize B. If this is your query, then we are on the same page ;)
VP
VP
Joined: 11 Aug 2020
Posts: 1258
Own Kudos [?]: 203 [1]
Given Kudos: 332
Send PM
Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
1
Bookmarks
Neat question!

If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the maximum value of (1/x - x/y)?

A. −3/18
B. 0
C. 1/5
D. 5/18
E. 2

Start by analyzing 1/x - x/y

How do you maximize this? You want the largest possible value for 1/x and smallest value for x/y. How do you get that? Well to make 1/x as LARGE as possible, you want x to be as SMALL as possible. To make x/y as SMALL as possible, then you want y to be as LARGE as possible.

1/3 - 3/9 = 1/3 - 1/3

B.
Director
Director
Joined: 04 Jun 2020
Posts: 550
Own Kudos [?]: 69 [0]
Given Kudos: 625
Send PM
Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
Bunuel wrote:
If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the maximum value of (1/x - x/y)?

A. \(-3\frac{1}{8}\)
B. 0
C. 1/5
D. 5/18
E. 2


egmat
Is the right approach here to first simplify the question stem?
E.g., I did (y-x^2)/xy

Then, I saw that y needed to be maximized and x needed to be minimized this way.
e-GMAT Representative
Joined: 02 Nov 2011
Posts: 4384
Own Kudos [?]: 30923 [1]
Given Kudos: 639
GMAT Date: 08-19-2020
Send PM
Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
1
Kudos
Expert Reply
woohoo921 wrote:
egmat
Is the right approach here to first simplify the question stem?
E.g., I did (y-x^2)/xy

Then, I saw that y needed to be maximized and x needed to be minimized this way.


Hey woohoo921

Let me ask you a question first:

    o Which one of the following two expressions looks simpler to you?
      • (1/x) – (x/y) OR
      • (y – x2)/xy
    o Thinking again should help you realize that the second expression above is not really a simplification of the given expression. Rather, it is just the result of a manipulation on the given expression.

In my opinion, you saw the opportunity of some processing on the given expression and did it without really thinking what processing to do. Here’s what your thought process should have been - “Okay, I have lower and upper limits of the variables, and I have an expression that I need to maximize and that directly uses these variables. Let me try to draw inferences about the terms in this expression using what I know about the variables themselves.”


Let me now show this to you in concrete steps. In fact, let me do it for both kinds of expressions – you will see how the solution would differ for each of these two starting expressions.

    o Using the given expression:
    • Goal: Maximize (1/x) – (x/y)
      • This is a difference of two numbers, (1/x) and (x/y).
      • The difference between these two numbers will be maximum when (1/x) takes its greatest possible value and (x/y) takes its least possible value.
        • Now, Max (1/x) is when x is minimum. (Here, when x = 3, 1/x = 1/3)
        • And Min (x/y) is when x is minimum, and y is maximum. (Here, x = 3 and y = 9 give x/y = 1/3)
        • Overall, max value of the expression is 1/3 – 1/3 = 0
      • Observe how the value of x is to be minimized for both these steps above. (The same value of x, x = 3, is required)

    o Using your modified expression:
    • Goal: Maximize (y - x2)/xy
      • This is a quotient of two numbers, (y – x2) and (xy).
      • While xy is always positive for our given ranges of x and y, you can say nothing about (y – x2). It may be positive, negative, or 0.
      • Now, the quotient will be maximum when the numerator, (y-x2) takes its greatest possible value and the denominator (xy) takes its least possible value.
        • Now, (y – x2) is itself a difference of two numbers. This difference is maximum when y is maximum and x2 is minimum, or x is minimum since x is positive. (Here, max y = 9 and min x = 3. Thus, max y – x2 = 0)
        • And Min (xy) is when x is minimum, and y is minimum. (Here, x = 3 and y = 3)
      • Observe how the values of y are not consistent in the two steps. This creates more complication, and we need to go back to the drawing board.
      • A second look will show that if max (y-x2) is anyway 0, the denominator does not really matter! So, the max value of your expression is still 0.

I hope you can see how much more complicated the second analysis is compared to the first analysis. 😊

Remember, simplifying should really mean simplifying!

Hope this helps!

Best,

Shweta Koshija

Quant Product Creator, e-GMAT
Senior Manager
Senior Manager
Joined: 13 Jul 2022
Posts: 336
Own Kudos [?]: 571 [0]
Given Kudos: 236
Location: India
Concentration: Finance, Nonprofit
GPA: 3.74
WE:Corporate Finance (Non-Profit and Government)
Send PM
Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
Bunuel wrote:
warriorguy wrote:
why can't x take value as 2 and y as 8?


x cannot be 2 because 2 < x ≤ 8.

But y can be 8 (2 < y ≤ 9).


Hi,

I did get the logic that the value of 1/x has to be maximised and value of x/y minimized
However, I got confused seeing the x in the numerator of x/y. I thought this numerator x also has to be factored, even if it implies reducing the value of 1/x
I am still unclear about the logic or impact of x in the numerator of x/y
Can you please help me out with the logic here and correct my line of thought?

Looking forward to hearing from you!
Math Expert
Joined: 02 Sep 2009
Posts: 93373
Own Kudos [?]: 625654 [0]
Given Kudos: 81918
Send PM
Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
Expert Reply
RenB wrote:
If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the maximum value of (1/x - x/y)?

A. \(-3\frac{1}{8}\)
B. 0
C. 1/5
D. 5/18
E. 2

Hi,

I did get the logic that the value of 1/x has to be maximised and value of x/y minimized
However, I got confused seeing the x in the numerator of x/y. I thought this numerator x also has to be factored, even if it implies reducing the value of 1/x
I am still unclear about the logic or impact of x in the numerator of x/y
Can you please help me out with the logic here and correct my line of thought?

Looking forward to hearing from you!


To maximize (1/x - x/y), we need to maximize 1/x and minimize x/y.

To maximize 1/x, we need to minimize x. The minimum possible value of x is 3. Thus, the maximum value of 1/x is 1/3.

To minimize x/y, we need to minimize x (AGAIN) and maximize y. The minimum possible value of x is 3 and the maximum possible value of y is 9. Thus, the minimum value of x/y is 3/9.

Therefore, the maximum value of (1/x - x/y) is obtained when x = 3 and y = 9, which is 1/3 - 3/9 = 0.

Answer: B.

P.S. To address your doubt, when maximizing (1/x - x/y), the goals of minimizing x for the first term (1/x) and minimizing x for the second term (x/y) are aligned and do not contradict each other. By minimizing x, we are able to maximize the overall expression, as both terms benefit from the same action.
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 33141
Own Kudos [?]: 829 [0]
Given Kudos: 0
Send PM
Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
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.
GMAT Club Bot
Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
Moderator:
Math Expert
93373 posts