This question has some clever wording in it. The question mentions that coordinates R, S, T have absolute values r, s, t.

The question then asks for the average of R, S, T but uses r, s, t (i.e. the absolute values) in the answer choices.

Therefore, to take into account the fact that coordinate R is a negative, we must multiply r by -1. As a result, the correct answer choice becomes (s + t - r) / 3. Hope this helps!

Hi longhaul123

Try to think in the easiest manner possible ignoring absolute value or any fancy stuff, I too was confused at the starting.

consider 3 integers: -7(R), 5(S),and 8(T)

If someone asks to find the mean (average) of the 3 numbers above, we can say

\(\frac{-7 + 5 + 8}{3}\) -------> st1

Now, when the question says absolute value they have simply given the positive value of each term i.e: |-7|--> 7(r), |5|--> 5(s), |8|--> 8(t)

Since, to find mean (average), as in st1 above, it is necessary to consider the sign (+/-) of the number the answer is (E)

Here is my interpretation of this question.

Take aside the portion saying absolute values, we know basic average is simply adding all of the numbers and dividing by the number of numbers. Say R is -4, S is 1 and T is 2. We know the average would be (-4 + 1 + 2)/3.

The question itself is a very tricky/deceptive in the wording. It says the number line has the points R, S, and T with the absolute values of r, s, and t. While R, S and T might have absolute values of r, s and t, the question ultimately asks what the average of points R, S and T are, not r, s and t. The key might be in the use of the word "coordinate" which factors in positive/negative values, instead of the absolute value of r, s and t.

It wants the average to EQUAL the coordinates of R, S, and T. So all we need is an equation that equals the average of (-4 + 1 +2)/3. E (s + t - r)/3 is the only option that equals that.

Ultimately r, s, and t are absolute values, so the the average equation would have to incorporate a negative for r. So (|s| + |t| - (|r|))/3

Since absolute values r, s & t > 0
But real values of S & T > 0 But R < 0
Therefore real values of R, S & T are -r, s & t respectively

Average of R, S & T = (s + t -r) /3

IMO 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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This one was quite confusing ... initially. If it helps, use different letters for r, s, t so that you do not mix them up with R, S, T. Here is how I went about it.

Let us suppose that R = -6, S = 1, T = 2. Now, if you just asked for the AVG of R, S and T, and not given (A)-(E), how would you answer it?

(R + S + T)/3
(-6 + 1 + 2)/3 ~~~> AVG = -3/3 i.e. -1

Now looking at (A)-(E), you know that whichever one it is, its got to = -1.

(D) gives you 3 as the average and so is wrong. (E) takes you back to -1.

Very tricky. If you plug in numbers however... you see the trick.

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

Another way to understand absolute value is as the distance from zero. For example, |x| is the distance between x and 0 on a number line.

Same way absolute value of R is given to us as r, so that is distance be r and 0. So if absolute value of r = 4 we can interpret this are r is at distance of 4 units from Zero. As shown in figure- r is on left side of Zero so r is negative.
Value associated with r is -4 while s and t are positive so if s=2 and t =4 we can take s and t as is

Since we have to cal average we take actual values with sign and not just absolute value .

Now cal Average = -4+2+4/3

Same way -r+s+t/3

Ans choice E

Hope this helps

Solution for anyone confused AF about the language used in the problem:

The question is asking for the mean of R, S, and T - and based on the answer choices, it must be in terms of r, s, and t.

Since r, s, and t are absolute values, you need to modify r to -r in the answer, because R=-r

Basically, how do you re-write (R+S+T)/3 in terms of r, s, and t. R=-4. r=4. So you need to add in the subtraction/negative sign.

I chose D originally as well. This is really a dumb question. Being able to do this kind "logic" - or in my mind, interpreting nonsense - is really pointless.

If you read the question once again it says:

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 ?

Part in bold text --> Absolute values of coordinates of R, S, T are r, s, and t. It does not mean that the coordinates themselves are all +ve. So, when you'll take average of the three coordinates, you will have to account for their signs as per the number line representation above.

For example:

As per number line representation,

Coordinate of point R = -r
ABSOLUTE Value of coordinate of R = |-r| = r

Hope this helps.