The number line shown contains three points R, S, and T, whose coordin

Say R = -3, S = 1 and T = 2. In this case r = |-3| = 3, s = 1, and t = 2. The average of the coordinates of the points R, S, and T would be (2 + 1 + (-3))/3 so, (s + t - r)/3.
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You certainly are not dumb; your intelligence led you to see that the way absolute value is often taught and the way it shows up here seem incongruent.

I'll avoid the semantics / syntax categorization, except to say that, overly simplified, the issue is what "absolute value" denotes.

In this case, we have a negative variable, \(-r\). It might help to memorize this rule:

If r is negative, i.e.
if \(r < 0,\)
then \(|r| = -r\)

The expression on the RHS (\(-r\)) is not negative. It is positive. It is the negative (opposite) of a negative value:

\(-(-5) = 5\)

Another way to see it; if x is a negative number

|x| =
-x =
(-1)(a negative number) =
a positive number

|-3| =
-(-3) =
(-1)(the negative number -3) =
3

For this problem, I assigned values to the variables and calculated the mean with numbers that had both the intuitively correct signs, and, given the rule, the mathematically correct signs. (-1 + 2 + 5)/3 = mean of 2.
Only Answer E gives that result.

A great post that might help: Bunuel ,Absolute Value, Tips and Tricks,here.

These sites might also help.
Scroll a long way for a discussion of absolute value and negative numbers, here,

and here.

Hope this helps.
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Video solution from Quant Reasoning:
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The arithmetic mean of three numbers x y z is (x+y+z)/3

In this case we know that the numbers are s, t, -r (since r is the absolute value of a negative number ... and s,t are the absolute values of positive numbers)

Hence mean is (s+t-r)/3

Posted from my mobile device
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This question is kind of Deceptive . you can still have (+(-R)+S+T)/3 to make D correct . E could also be rationalized like (S+T-(+R))/3 . This is why I hate the GMAT . We are arguing semantics/Syntax . Unless I am just dumb and don't comprehend.

We can let R, S, and T be the actual values. Thus, we want to determine the value of (R + S + T)/3. Since S and T are positive, S = s and T = t. However, since R is negative, R = -r. Thus, the average becomes:

(-r + s + t)/3

(s + t - r)/3

Answer: E
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Responding to a pm:

Note that point R is to the left of 0 so it will be negative. S and T are to the right of 0 so they are positive.
Say R = -4, S = 2, T = 3

We are given that r, s and t are absolute values of the co-ordinates so r = 4, s = 2 and t = 3 (absolute values are positive)
So R = -r

Average of the co-ordinates of points R, S and T = (R + S + T)/3 = (-r + s + t)/3
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VeritasKarishma, can you please explain as to why did we not consider the absolute value of R to calculate the mean? The question mentions that the points on the number line have absolute values.

Thanks!

Small letters r, s and t are variables. The question defines them as absolute values of the points R, S and T. But the question asks for the average of the actual co-ordinates of R, S and T. It does not ask for the average of r, s and t. That would simply be (r+s+t)/3.
But the actual co-ordinate of R would be -r since it will be negative.
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The number line shown contains three points R, S, and T, whose coordinates have absolute values r, s, and t, respectively. Which of the following equals the average (arithmetic mean) of the coordinates of the points R, S, and T ?

say, the coordinates of the points R, S, and T be -5, 1, 2 .
( R is left side of 0(so negative) whereas S and T are on the right side.(so positive) )

now, absolute values of -5, 1, 2 are 5, 1, 2 which are r, s, and t according to the given data .
so, r= 5 ; s=1 ; t=2

now, the average (arithmetic mean) of the coordinates of the points R, S, and T = (-5+1+2 )/3 =[ (-(5)) +1+2 )] /3
=( -r +s+t )/3 = (s+t-r)/3

correct answer E

Solution:

We totally agree that this question has a deceptive usage of words.
The one way we would see it is as follows:

Let the points be R,S,T and they have coordinates -a, b, c respectively.

r, s, t are the absolute values of the coordinates -a,b,c respectively.

With R being on the negative side of the number line would have,

r=|-a|= -(-a)

=> -a (the coordinate in context)= -r

​s = |b| = b

t = |c| = c

Thus the coordinates -a,b,c in terms of r, s, t are -r, s ,t respectively.

Hence average of the coordinates -a, b, c = -r + s + t /3 (Option e)

Pls note that the question asks you of the average of the coordinates.

Hope this helps
Devmitra Sen(GMAT Expert Quant)
​
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Even though this is an official question, I do not particularly appreciate it. Not only is it deceptively worded, it is confusingly worded and would present difficulties in terms of understanding the question. This would be an issue especially for non-native speakers of English.

The question has given the absolute values of three points on a number line. These three aboslute values represent the positive values of the distances of the three points from the origin point O. Therefore, we first need to translate the meaning.

Given these absolute values of the distances of three points, what are the actual coordinates of the three points? We will first need to determine the actual coordinates in order to calculate the average value of the three distances.

Note that points S and T lie to the positive side of the number line and the point R lies to the negative side of the number line. So, the points S and T have positive coordinates and the point R has negative coordinates.

Actual coordinate of T = +(absolute value of the coordinate of T) = +t

Actual coordinate of S = +(absolute value of the coordinate of S) = +s

Actual coordinate of R = -(absolute value of the coordinate of R) = -r

Therefore, the average value (arithmetic mean) of the three coordinates = (Sum of the actual coordinates)/3 = (s + t - r)/3

ANSWER: (E)

Answer: Option E

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Let R,S and T be -3, 1 and 2. Average of R,S AND T is 1+2+(-3)/3 = 0. Absoloute values of R,S and T are r,s and t i.e 3, 1 and 2. Now option C also gives us the same average 3-1-2/3 = 0. How option C is different from option E ?

Hi BrentGMATPrepNow, to clarify regarding absolute value is |-r| = r right? So not sure why answer choice here is not D but E? Thanks Brent

If r is positive, then |-r| = r
Example, |-3| = 3

However, if r is negative, then |-r| does NOT equal r
Example, |-(-3)| does NOT equal -3

Hi all!
This is a perfect question to slip and be "trapped", if you are not careful.
Before I move to the operation, lets deep think a bit.
r, s, and t are the absolute values of points R,S,T.
For example if S= 1, s=|1|=1

What exactly is Absolute value or the Mod function?
The algebraic definition defines |x| = x if x>0 and -x if x<0. So |-1| = -(-1)=+1.

Okay!
With that thought at place, observe the point R. This is the link to your getting the question right if the GMAT thinking and observation is at place or wrong if you ignore the difference between R, r and the mod function rules(if you chose D)


R is to the left of 0.What kind of number is to the left of 0? -A negative number ! Right?
What is the absolute value of a negative number? We started off with this definition!
|x| =-x when x<0

We are asked => the average (arithmetic mean) of the coordinates of the points R, S, and T ?
This is R+S+T/3
There is no problem to see that S=s and T=t as both of these coordinates are to the right of 0 and positive. |x|=x when x>0
For R, the value of r = -R (Since r= |R| and R<0)
So value of the coordinate R= -r

Hence the average of the coordinates is R+S+T/3= -r+s+t/3 or option E.

Guard yourself against D(observation and gap of analysis) and B(Trapped due to low focus and loss of grip on the question).
If you have marked A or C, please go back to the fundamentals of absolute values. You need to enhance conceptual clarity.
If you got this question right, great
If NOT, what's there to lose? You only gain multiple perspectives from the Official Questions.

Let me know if you have any questions or concerns in the comments by tagging us.

Devmitra Sen
GMAT Mentor
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I don't understand why the answer is not C when you use these numbers: 3-1-2/3=0/3 similar to the average (2+1+-3/3=0)
I understand that R is negative hence R=-3 but isn't the absolute version of R=l-3l then r=3? therefore in your answers you should be looking for an average that matches -3+1+2/3=0. Plugging in the lowercase numbers (3-1-2)/3=0 as well. Here I was using 3 and not -3 because I took the absolute value of uppercase R to get lowercase r. Am i missing something?

From the number line given, it follows that (the coordinate of R) < 0 < (the coordinate of S) < (the coordinate of T). Hence, (the coordinate of R) must be negative, while the coordinates of S and T must be positive.
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