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# a,b and c are distinct natural numbers less than 20. What is the maxim

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Re: a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
This is a moderately difficult question on maximization and minimization.

In this question, to maximize the value of the expression, we need to maximize the first two parts and minimize the third part, since it is negative.

To maximize |a-b|, the best way is to take the biggest possible value for a, and the smallest possible value for b. This means we take a = 19 and b = 1.
So, maximum value of |a-b| = 18.

The value of b is fixed now. In order to maximize the second part, we need to maximize c, so that the difference is maximized. The maximum possible value for c = 18.
Maximum value of |b-c| = 17.

From these values, |c-a| = 1. This is also the minimum value of |c-a|.

So, the maximum possible value of the given expression = 18 + 17 – 1 = 34.
The correct answer option is B.

Hope this helps!
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a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
To answer this question I use a a number line

---------0--c---b------------a--

If you want to maximize the result of the expression you have to put the number so that the differences between ab and bc are maximised

---------0--b--------------c-a--

---------0--1--------------18-19--
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Re: a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
CrackVerbalGMAT wrote:
This is a moderately difficult question on maximization and minimization.

In this question, to maximize the value of the expression, we need to maximize the first two parts and minimize the third part, since it is negative.

To maximize |a-b|, the best way is to take the biggest possible value for a, and the smallest possible value for b. This means we take a = 19 and b = 1.
So, maximum value of |a-b| = 18.

The value of b is fixed now. In order to maximize the second part, we need to maximize c, so that the difference is maximized. The maximum possible value for c = 18.
Maximum value of |b-c| = 17.

From these values, |c-a| = 1. This is also the minimum value of |c-a|.

So, the maximum possible value of the given expression = 18 + 17 – 1 = 34.
The correct answer option is B.

Hope this helps!

Since, a,b,c are positive natural numbers, the sign of a,b,c will always be positive.
So if we open the brackets,
a-b-b+c-c+a = 2(a-c)

To maximise this we can put a = 19 and c =1 , so the maximum value would come out to be 36.
What is wrong with this solution? Please explain and clarify
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Re: a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
Prerana94 wrote:
Since, a,b,c are positive natural numbers, the sign of a,b,c will always be positive.
So if we open the brackets,
a-b-b+c-c+a = 2(a-c)

To maximise this we can put a = 19 and c =1 , so the maximum value would come out to be 36.
What is wrong with this solution? Please explain and clarify

You can't open a module like that. There are always two signs of module irrespective it contains positive, negative, rational or irrational number.

So, |a-b| + |b-c| - |c-a|
Will be ±(a-b) + ±(b-c) - ±(c-a)
There will be more than just 2(a-c) values.

Posted from my mobile device
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Re: a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
1
Kudos
Always seems to be late at night when I finish off on one of these crazy ones by Nick.....

We are looking to MAXIMIZE the following:

[a - b] + [b - c] - [c - a] = ?

(1st) We want to MINIMIZE the effect that [c - a] has on the total Expression because the (+)Positive Value will be SUBTRACTED from the Other (+)Pos. Values

However, at the same time we want to MAXIMIZE the effect that [a - b] + [b - c] has by giving it the Greatest Value we can

Distance Concept of Absolute Value:

the Modulus Given by, for example: [X - 4] = 3

Tells you that the DISTANCE between X and +4 is exactly 3 UNITS in Either Direction on the Number Line.

X = +7 or +1

Applying the Same Logic:

(1st)

[a - b] + [b - c] ------- We want to MAXIMIZE the DISTANCE between A and B......while simultaneously MAXIMIZING the DISTANCE between B and C on the Number Line

(2nd)
- [c- a] ----- To MINIMIZE the Distance, we want to put C as CLOSE to A as we can on the Number Line

Further, we are told that A, B, and C are all Distinct (+)Positive Numbers Less Than < 20

Step 1: Put C as Close as we can to A on the Number Line:

Case 1: A--C---------------------------

or

Case 2: C--A------------------------

Step 2: on the Number Line, we want A as FAR AWAY as possible from B --- while B is as FAR AWAY as possible from C

Case 1: A--C-------------------------B

or

Case 2: C--A--------------------------B

We could also try to swap A with B in Case 1--- and move C closer to A on the other end of the Scale

Case 3: B-------------------C---A

Let's Test the 3 Cases Spreading the DISTINCT (+)Positive Integers Out as Far as Possible:

Case 1: A--C-----------------------B

A = 1 ; C = 2 ; and B = 19

[A - B] + [B - C] - [C - A] = ?

[1 - 19] + [19 - 2] - [2 -1] = ?

[-18] + [+17] - [+1] = ?

18 + 17 - 1 = 34*****

Let's Try Case 2:

Case 2: C--A----------------------B

C = 1 ; A = 2 ; and B = 19

[2 - 19] + [19 - 1] - [1 - 2] = ?

17 + 18 - 1 = 34******

Finally, let's try Case 3:

Case 3: B---------------------C--A

B = 1 ; C = 18 ; and A = 19

[19 - 1] + [1 - 18] - [18 - 19] = ?

[+18] + [-17] - [-1] = ?

18 + 17 - 1 = 34******

Therefore, as long as we follow the Distance Approach and keep C as close to A as possible and Separate A and B as FAR as we can ------ the MAXIMUM Value will be obtained = 34

-B-

34
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Re: a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
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Re: a,b and c are distinct natural numbers less than 20. What is the maxim [#permalink]
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