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If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the

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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
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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.
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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?
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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
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).
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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
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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
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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)
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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]

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.)
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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]

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.)

I am not sure why are you calling the highlighted statement as a "Universal fact"? can you clarify more on your query?
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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.
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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
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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.

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
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Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the [#permalink]
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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.
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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.
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Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
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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)

• 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

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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

Looking forward to hearing from you!
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Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
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

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.

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.
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Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the [#permalink]
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