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What is the maximum and minimum possible value of the expression
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11 Mar 2022, 00:11
11 Mar 2022, 00:11
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What is the maximum and minimum possible value of the expression \(\frac{|x+y|}{|x|+|y|} + \frac{|y+z|}{|y|+|z|} + \frac{|z+x|}{|z|+|x|}\) ?
A. 2 and 1
B. 3 and 0
C. 3 and 1
D. 4 and 0
E. 4 and 1
A. 2 and 1
B. 3 and 0
C. 3 and 1
D. 4 and 0
E. 4 and 1
Retired Moderator
Joined: 16 Apr 2020
Status:Founder & Quant Trainer
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Posts: 1546
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WE:Education (Education)
What is the maximum and minimum possible value of the expression
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25 Mar 2022, 05:00
25 Mar 2022, 05:00
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OA Explanation:
Remember: |a+b| ≤ |a|+|b|
[https://gre.myprepclub.com/forum/gre-quant-absolute-values-modulus-theory-18389.html#p52039]
All three fraction terms separated by + sign can have maximum value as 1, which means the maximum value of the entire expression is 3
When is this possible?
|a+b| = |a|+|b|
Now, what can be the minimum value of each fraction term?
The answer is: It can be equal to 0
When is this possible?
\(x= -y\) or \(y = -z\) or \(x = -z\)
But wait - Among x, y, and z at least two will have the same sign
Here is why?
if \(x= -y\) and \(y = -z\) then \(x ≠ -z\), because \(x\) will become equal to \(z\)
Therfore, out of the three fraction terms - atmost two terms can be zero and atleast one will have to be 1
So, the minimum value of the entire expression would be 1
Hence, option C
Remember: |a+b| ≤ |a|+|b|
[https://gre.myprepclub.com/forum/gre-quant-absolute-values-modulus-theory-18389.html#p52039]
All three fraction terms separated by + sign can have maximum value as 1, which means the maximum value of the entire expression is 3
When is this possible?
|a+b| = |a|+|b|
Now, what can be the minimum value of each fraction term?
The answer is: It can be equal to 0
When is this possible?
\(x= -y\) or \(y = -z\) or \(x = -z\)
But wait - Among x, y, and z at least two will have the same sign
Here is why?
if \(x= -y\) and \(y = -z\) then \(x ≠ -z\), because \(x\) will become equal to \(z\)
Therfore, out of the three fraction terms - atmost two terms can be zero and atleast one will have to be 1
So, the minimum value of the entire expression would be 1
Hence, option C
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What is the maximum and minimum possible value of the expression
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09 Dec 2023, 02:10
09 Dec 2023, 02:10
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Theory: |A + B| ≤ |A| + |B|
=> Maximum value of \(\frac{|x+y|}{|x|+|y|} + \frac{|y+z|}{|y|+|z|} + \frac{|z+x|}{|z|+|x|}\) will be when all x , y and z have the same sign and numerator and denominator will cancel out to give us
Maximum value = 1 + 1 + 1 = 3
Ideally Minimum value should be zero if we had two different variables in each part of the absolute value. But here we have 3 variables shared across three Absolute values. So, two of them will have the same sign and value and other can have a different sign and same value.
Ex: x = 1, y = 1 and z = -1
=> \(\frac{|1+1|}{|1|+|1|} + \frac{|1-1|}{|1|+|-1|} + \frac{|-1+1|}{|-1|+|1|}\)
=> Minimum value = \(\frac{2}{2}\) + 0 + 0 = 1
So, Answer will be C
Hope it helps!
Watch the following video to learn the Basics of Absolute Values
- - As |A| + |B| will always add up, irrespective of the signs of A and B, as after coming out of Absolute value they will become non-negative
- But, |A +B| can get reduced if both have oppositive signs
=> Maximum value of \(\frac{|x+y|}{|x|+|y|} + \frac{|y+z|}{|y|+|z|} + \frac{|z+x|}{|z|+|x|}\) will be when all x , y and z have the same sign and numerator and denominator will cancel out to give us
Maximum value = 1 + 1 + 1 = 3
Ideally Minimum value should be zero if we had two different variables in each part of the absolute value. But here we have 3 variables shared across three Absolute values. So, two of them will have the same sign and value and other can have a different sign and same value.
Ex: x = 1, y = 1 and z = -1
=> \(\frac{|1+1|}{|1|+|1|} + \frac{|1-1|}{|1|+|-1|} + \frac{|-1+1|}{|-1|+|1|}\)
=> Minimum value = \(\frac{2}{2}\) + 0 + 0 = 1
So, Answer will be C
Hope it helps!
Watch the following video to learn the Basics of Absolute Values
