Last visit was: 23 Apr 2026, 14:34 It is currently 23 Apr 2026, 14:34
Home
Messages
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
805+ (Hard)|   Absolute Values|   Algebra|      
User avatar
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 12 Mar 2026
Posts: 1,841
Own Kudos:
8,509
 [22]
Given Kudos: 707
Location: India
Posts: 1,841
Kudos: 8,509
 [22]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 29 Aug 2019, 17:29
1
Kudos
Add Kudos
21
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

38% (01:56) correct 62% (01:41) wrong based on 231 sessions

History

Date
Time
Result
Not Attempted Yet
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

A. 1
B. 1.5
C. 2
D. 2.5
E. 3
ShowHide Answer
Official Answer
C
Most Helpful Reply
User avatar
Kinshook
User avatar
Major Poster
Joined: 03 Jun 2019
Last visit: 23 Apr 2026
Posts: 5,986
Own Kudos:
5,858
 [6]
Given Kudos: 163
Location: India
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
My Reviews
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
Posts: 5,986
Kudos: 5,858
 [6]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M Updated on: 30 Aug 2019, 20:06
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.
2
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
nick1816
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

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

\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
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

Posted from my mobile device
General Discussion
User avatar
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 12 Mar 2026
Posts: 1,841
Own Kudos:
8,509
 [1]
Given Kudos: 707
Location: India
Posts: 1,841
Kudos: 8,509
 [1]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 30 Aug 2019, 17:39
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
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
nick1816
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
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
Show more
avatar
Apo_orv
avatar
Joined: 27 Nov 2016
Last visit: 14 Jul 2022
Posts: 4
Own Kudos:
2
Given Kudos: 48
Concentration: Finance, Entrepreneurship
Posts: 4
Kudos: 2
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 30 Aug 2019, 19:42
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hmmmmm, why can't the expresion value be equal to 0?
User avatar
Kinshook
User avatar
Major Poster
Joined: 03 Jun 2019
Last visit: 23 Apr 2026
Posts: 5,986
Own Kudos:
5,858
Given Kudos: 163
Location: India
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
My Reviews
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
Posts: 5,986
Kudos: 5,858
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 30 Aug 2019, 20:07
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Corrected. Thanks

nick1816
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
nick1816
\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
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
Show more
User avatar
omkartadsare
User avatar
Joined: 11 Jun 2019
Last visit: 15 Nov 2021
Posts: 28
Own Kudos:
51
 [2]
Given Kudos: 135
Location: India
Posts: 28
Kudos: 51
 [2]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 11 Sep 2019, 20:38
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
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)
User avatar
Rohit2015
User avatar
Joined: 20 Jul 2015
Last visit: 09 Apr 2020
Posts: 24
Own Kudos:
27
 [1]
Given Kudos: 7
Posts: 24
Kudos: 27
 [1]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 25 Sep 2019, 23:22
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Apo_orv
Hmmmmm, why can't the expresion value be equal to 0?
Show more


Hi Apo_orv

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

Thanks
User avatar
StressTest
User avatar
Joined: 15 Mar 2019
Last visit: 19 Dec 2020
Posts: 17
Own Kudos:
7
Given Kudos: 12
Posts: 17
Kudos: 7
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 28 Sep 2019, 03:55
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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?
User avatar
Rohit2015
User avatar
Joined: 20 Jul 2015
Last visit: 09 Apr 2020
Posts: 24
Own Kudos:
27
 [1]
Given Kudos: 7
Posts: 24
Kudos: 27
 [1]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 28 Sep 2019, 04:44
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
StressTest
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?
Show more


Well

yur resoning is incorrect for above highlighted part

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

Hope it helps
User avatar
StressTest
User avatar
Joined: 15 Mar 2019
Last visit: 19 Dec 2020
Posts: 17
Own Kudos:
7
Given Kudos: 12
Posts: 17
Kudos: 7
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 28 Sep 2019, 04:59
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Rohit2015
StressTest
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
Show more

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?
User avatar
Rohit2015
User avatar
Joined: 20 Jul 2015
Last visit: 09 Apr 2020
Posts: 24
Own Kudos:
27
 [1]
Given Kudos: 7
Posts: 24
Kudos: 27
 [1]
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 28 Sep 2019, 05:02
Kudos
Add Kudos
Bookmarks
Bookmark this Post
StressTest
Rohit2015
StressTest
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?
Show more

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

Yes ...u can take any... :)
avatar
eddo124
avatar
Joined: 02 Dec 2021
Last visit: 30 Apr 2022
Posts: 2
Own Kudos:
1
Given Kudos: 6
Posts: 2
Kudos: 1
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 02 Dec 2021, 04:14
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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
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)
Show more
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,962
Own Kudos:
1,117
Posts: 38,962
Kudos: 1,117
|a+b|/|a|+|b| + |b+c|/|b|+|c| +|c+a|/|c|+|a|=M 06 Jul 2025, 18:22
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.
Moderators:
Bunuel
Math Expert
109785 posts
Krunaal
Tuck School Moderator
853 posts