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|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M

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nick1816
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|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   29 Aug 2019, 17:29
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\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3
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C
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Kinshook
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|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   Updated on: 30 Aug 2019, 20:06
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nick1816 wrote:
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3


\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

Minimum value of the expression is 1 when a=-b ; b=-c; a=-b=c

Maximum value of expression is when
a,b & c have same sign
The maximum value of the expression =3

Difference = 3-1=2

IMO C

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Originally posted by Kinshook on 29 Aug 2019, 18:41.
Last edited by Kinshook on 30 Aug 2019, 20:06, edited 1 time in total.
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nick1816
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   30 Aug 2019, 17:39
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Highlighted part is not correct. It's true only when a=b=c=0 (this case is not possible in this question)
if a=-b and b= -c
then a must be equal to c

At a=-b=c, M has the minimum value that is 1.

Kinshook wrote:
nick1816 wrote:
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3


\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

Minimum value of the expression is 0 when a=-b ; b=-c; c= -a

Maximum value of expression is when
a,b & c have same sign
The maximum value of the expression =3

Difference = 3

IMO E

Posted from my mobile device
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Apo_orv
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   30 Aug 2019, 19:42
Hmmmmm, why can't the expresion value be equal to 0?
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Kinshook
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   30 Aug 2019, 20:07
Corrected. Thanks

nick1816 wrote:
Highlighted part is not correct. It's true only when a=b=c=0 (this case is not possible in this question)
if a=-b and b= -c
then a must be equal to c

At a=-b=c, M has the minimum value that is 1.

Kinshook wrote:
nick1816 wrote:
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3


\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

Minimum value of the expression is 0 when a=-b ; b=-c; c= -a

Maximum value of expression is when
a,b & c have same sign
The maximum value of the expression =3

Difference = 3

IMO E

Posted from my mobile device
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|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   11 Sep 2019, 20:38
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No restrictions are given for values of a, b or c. so to simplify this, lets assume the value to be |a| =|b|=|c| = 1

we can see that, |a| + |b| = |b| + |c| = |c| + |a| = 2.... Hence all bases are same. The max. and min value totally depends on numerators now.

Maximum value can be obtained when sign of all numbers is same (we get all additions inside modulus and hence largest value). when all a, b and c all are positive or negative we will get 6 in the numerator. So the largest possible value of M would be 6/2 = 3

Smallest possible value can be obtained when one of a, b and c is having different sign while other two have same sign. (By this way there would be 2 subtractions inside modulus and 1 addition) By considering any one of a, b or c to be negative, we get 2 in the numerator. So the smallest number than can be obtained is 2/2 = 1

So the difference between largest and the smallest value is 3 -1 = 2, Hence IMO (C)
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   25 Sep 2019, 23:22
Apo_orv wrote:
Hmmmmm, why can't the expresion value be equal to 0?



Hi Apo_orv

Please let me know for which values of a,b or c , the expression will be 0?????

Thanks
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   28 Sep 2019, 03:55
Can somebody please help me?

I am stuck how M can be 1 at minimum. given a = -b = c, the first two terms are euqal to 0, right?
-(a+b)/a+b = -a-b/a+b = -a+a/a+b = -a +a = 0 hence 0,

The 3th term is different :
-(c+a)/c+a => -c-a/c+a = -2a/2a = -1. So how do you get 1 as a result?
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   28 Sep 2019, 04:44
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StressTest wrote:
Can somebody please help me?

I am stuck how M can be 1 at minimum. given a = -b = c, the first two terms are euqal to 0, right?
-(a+b)/a+b = -a-b/a+b = -a+a/a+b = -a +a = 0 hence 0,

The 3th term is different :
-(c+a)/c+a => -c-a/c+a = -2a/2a = -1. So how do you get 1 as a result?



Well

yur resoning is incorrect for above highlighted part

i would rather suggest to try some real numbers ...

Hope it helps
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   28 Sep 2019, 04:59
Rohit2015 wrote:
StressTest wrote:
Can somebody please help me?

I am stuck how M can be 1 at minimum. given a = -b = c, the first two terms are euqal to 0, right?
-(a+b)/a+b = -a-b/a+b = -a+a/a+b = -a +a = 0 hence 0,

The 3th term is different :
-(c+a)/c+a => -c-a/c+a = -2a/2a = -1. So how do you get 1 as a result?



Well

yur resoning is incorrect for above highlighted part

i would rather suggest to try some real numbers ...

Hope it helps


Thank you very much for the reply, but im stuck. When i set in real numbres, how would i choose them?
We are saying that a = -b = c
So a =1, then b =-1 and c =1?
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   28 Sep 2019, 05:02
StressTest wrote:
Rohit2015 wrote:
StressTest wrote:
Can somebody please help me?

I am stuck how M can be 1 at minimum. given a = -b = c, the first two terms are euqal to 0, right?
-(a+b)/a+b = -a-b/a+b = -a+a/a+b = -a +a = 0 hence 0,

The 3th term is different :
-(c+a)/c+a => -c-a/c+a = -2a/2a = -1. So how do you get 1 as a result?



Well

yur resoning is incorrect for above highlighted part

i would rather suggest to try some real numbers ...

Hope it helps


Thank you very much for the reply, but im stuck. When i set in real numbres, how would i choose them?
We are saying that a = -b = c
So a =1, then b =-1 and c =1?


a =1, then b =-1 and c =1?

Yes ...u can take any... :)
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|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   02 Dec 2021, 04:14
I would like to ask:
1. why we will get a 6 as the numerator when the signs are all the same for the maximum part?
2. why do we assume the value to be 1 in the highlighted part? Why is it specifically 1?

omkartadsare wrote:
No restrictions are given for values of a, b or c. so to simplify this, lets assume the value to be |a| =|b|=|c| = 1

we can see that, |a| + |b| = |b| + |c| = |c| + |a| = 2.... Hence all bases are same. The max. and min value totally depends on numerators now.

Maximum value can be obtained when sign of all numbers is same (we get all additions inside modulus and hence largest value). when all a, b and c all are positive or negative we will get 6 in the numerator. So the largest possible value of M would be 6/2 = 3

Smallest possible value can be obtained when one of a, b and c is having different sign while other two have same sign. (By this way there would be 2 subtractions inside modulus and 1 addition) By considering any one of a, b or c to be negative, we get 2 in the numerator. So the smallest number than can be obtained is 2/2 = 1

So the difference between largest and the smallest value is 3 -1 = 2, Hence IMO (C)
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink] New post   06 Feb 2023, 05:49
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Re: |a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M [#permalink]
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