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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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Author:  UUBA [ 20 Oct 2021, 11:14 ]
Post subject:  Re: If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the
2 < x <= 8; Min x = 3, Max x = 8 (Because x is an integer, given in question)
2 < y <= 9; Min y = 3, Max Y = 9 (Because Y is an integer, given in question)

To make Max ( 1/x - x/y )
( Max - Min )
( 1/min x - min x/max y )
(1/3 - 3/9) = (1/3 -1/3) = 0
Author:  woohoo921 [ 06 Dec 2022, 16:00 ]
Post subject:  Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the
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.
Author:  egmat [ 11 Dec 2022, 20:23 ]
Post subject:  Re: If x and y are integers such that 2 < x 8 and 2 < y 9, what is the
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
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