If a, b, c are distinct positive integers less than 25, what is the ma

Question

If a, b, c are distinct positive integers less than 25, what is the maximum possible value of  |a – b| + |b – c| – |c – a| ?

A. 46
B. 44
C. 42
D. 23
E. 21

ArunSharma12 · Answer

|a – b| + |b – c| – |c – a| 
to get maximum value we need to minimize the third term and also as c appears in second term so maximize the second term.
c = 1 and b = 24, maximizes the second term.
when c= 1, a = 2 minimizes the third term.
also when a = 2, first term has the maximum possible difference.
 |a – b| + |b – c| – |c – a| = |2-24|+|24-1|+|1-2| = 22 + 23 - 1 = 44 
Ans: B

TestPrepUnlimited · Answer

Let A, B, C be three points on the number line. We can interpret the question as "What is the maximum possible value of  AB + BC - CA ?"

We can start by making  CA = 1  as we want to minimize what we are subtracting. Afterwards we need to maximize  AB + BC , we can do this by putting A and C on the far left and B on the far right (the other way around also works!). Then set  A = 1, C = 2, B = 24 . This will maximize  AB + BC  and minimize  CA , so the value would be  23 + 22 - 1 = 44.

Ans: B

LeenaPG · Answer

Solution:

Here it could be thought in terms of number line:

|a-b| + |b-c| = |c-a|

so the distance from a to b and then from b to c is same as that of from c to a.

maximum value that b can take is 23.

a=1 and c=1

now substitute this in the equation:
|a–b|+|b–c|–|c–a|
|1-23|+|23-1| - 0= 22+22 = 44
So Ans = B

ScottTargetTestPrep · Answer

Since a, b and c are distinct, we can have 6 possible orders: 1) a > b > c, 2) a > c > b, 3) b > a > c, 4) b > c > a, 5) c > a > b and 6) c > b > a.

Let’s examine each case.

1) If a > b > c, the given expression can be simplified as: a - b + b - c + c - a = 0.

2) If a > c > b, the given expression can be simplified as: a - b + c - b + c - a = 2c - 2b.

3) If b > a > c, the given expression can be simplified as: b - a + b - c + c - a = 2b - 2a.

4) If b > c > a, the given expression can be simplified as: b - a + b - c - (c - a) = 2b - 2c.

5) If c > a > b, the given expression can be simplified as: a - b + c - b - (c - a) = 2a - 2b.

6) If c > b > a, the given expression can be simplified as: b - a + c - b - (c - a) = 0.

Except for the first and last cases, we see that the other four will yield a nonzero value. Let’s examine them, keeping in mind that we want to determine the maximum value of the expression.

2) We can let a = 24, c = 23 and b = 1. So 2c - 2b = 46 - 2 = 44.

3) We can let b = 24, a = 2 and c = 1. So 2b - 2a = 48 - 4 = 44.

4) We can let b = 24, c = 2 and a = 1. So 2b - 2c = 48 - 4 = 44.

5) We can let c = 24, a = 23 and b = 1. So 2a - 2b = 46 - 2 = 44.

We see all the of the 4 nonzero cases yield a maximum value of 44.

Answer: B

Onyedelmagnifico · Answer

For this to be maximum, with the condition that a,b,c<25 
|c-a| has to be small.
Hence |c-a|=1 and |a-b|, |b-c| must have their greatest value at the above condition. 
Hence 
a=1, b=24 and c=2
Hence 
|a-b| + |b-c|- |c-a|= 
23+22-1 = 44 Hence option B

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Anki1405 · Answer

Take a =23, b=1 and c= 24 
We need to minimise the third term |c-a|
Since each term is distinct  let c= 24 and a =23  
therefore,   c-a =1 
to maximise (a-b )and (b-c) take b=1
Then 
 |a – b| + |b – c| – |c – a| 
= |23 – 1| + |1 – 24| – |24 – 23|
= 22+23-1 
= 45-1 
= 44

Correct Choice (B)

StacyArko · Answer

I used a=1, b=13, c=24 and got 46. What am I missing pls

Bunuel · Answer

If a = 1, b = 13, and c = 24, then:

|a – b| + |b – c|  –  |c – a|

= 12 + 11 – 23

= 0

Looks like you may have mistaken the minus sign for a plus.

If a, b, c are distinct positive integers less than 25, what is the ma

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If a, b, c are distinct positive integers less than 25, what is the ma

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