Question of the Week - 41 (If x and y are non-zero integers and...)

Question

e-GMAT          Question of the Week #41

If x and y are non-zero integers and |x – 3| ≤ 2 and -1 ≤ y ≤ 8 then the least value of  \frac{x}{y}  lies in which of the following range?

A.  Between -5.5 and -4.5
 B.  Between -4.5 and -3.5
 C.  Between -3.5 and -2.5
 D.  Between -2.5 and -1
 E.  Greater than -1

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chetan2u · Answer

If x and y are non-zero integers and |x – 3| ≤ 2 and -1 ≤ y ≤ 8 then the least value of  \frac{x}{y}  lies in which of the following range?

We know the range of y, and now we will work on range of x by solving |x – 3| ≤ 2..
Two cases for |x – 3| ≤ 2 : (I) x-3≤ 2...x≤ 5, and (II) -(x-3)≤ 2...3-x≤ 2......1≤ x
Range of x = > x-3≤ 2........1≤x≤ 5

Now, for the least value of  \frac{x}{y} , we will take y as negative, that is -1. The denominator should have the largest possible value, so 5..
 \frac{x}{y} => \frac{5}{-1}=-5 .

-5 is between -5.5 and -4.5, and so the answer is A

Archit3110 · Answer

solve for |x – 3| ≤ 2 x<=5 and -x<=1 ; x>=1 so 1>=x<=5 and -1 ≤ y ≤ 8 so to get least value of x/y ; x = 5 and y = -1 -5 IMO A

EgmatQuantExpert · Answer

Solution 
  Given:  
In this question, we are given that
 • The numbers x and y are non-zero integers.
• Also, |x – 3| ≤ 2 and -1 ≤ y ≤ 8.

To find:  
We need to determine
 • The least value of  \frac{x}{y}  lies in which of the given ranges.

Approach and Working:  
For the non-zero integer x,
 • |x – 3| ≤ 2
Or, -2 ≤ x – 3 ≤ 2
Or, 1 ≤ x ≤ 5

Therefore, possible values of x = 1, 2, 3, 4, 5

Similarly, for the non-zero integer y,
 • -1 ≤ y ≤ 8

Therefore, possible values of y = -1, 1, 2, 3, 4, 5, 6, 7, 8

Now, for  \frac{x}{y}  to be least, we should take one of them as positive and the other one as negative.
 • As we can see from the derived values of x and y, x can be positive only whereas y can be negative.
• Hence, we should take negative value of y, with maximum possible magnitude, and maximum value of x.

Negative value of y, with maximum magnitude = -1
Maximum value of x = 5
 • Therefore, the minimum value of  \frac{x}{y}  =  \frac{5}{-1}  = -5
• This value lies between -5.5 and -4.5

Hence, the correct answer is option A.

Answer: A

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Question of the Week - 41 (If x and y are non-zero integers and...)

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