Official Solution:


(1) The mean of set \(S\) is equal to \(a + b\). Given that \(mean=\frac{(a - b)+(b - a)+(a + b)}{3}=a+b\), which leads to \(a+b=0\). Now, if \(a+b=0\), then \(a-b\) and \(b-a\) are either both zeros (if \(a=b=0\)) or have different signs (if \(a \ne b\)). In any case the median of \(S\) is \(a+b=0\). Sufficient.

(2) The range of set \(S\) is equal to \(2b\). If \(a=b=0\), then \(median=0\) but if \(a=0\) and \(b=1\), then \(median=1\). Not sufficient.


Answer: A
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In option 2, The range (largest - Least )= 2b; how would be arrive at 2b from the set of the three numbers viz : a+b, b-a, a-b ?

It's explained in the solution above:

If a = b = 0, then S = {0, 0, 0} --> the range = 0 - 0 = 0 = 2b.
If a = 0 and b = 1, then S = {-1, 1, 1} --> the range = 1 - (-1) = 2 = 2b.
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a+b - (a-b) = 2b, which is true for a=b=0 as well. Hence that makes b-a as the median. From first option I got a+b as the median. What am I missing here?

Option (1) states mean is a+b, which can be inferred from the question stem.

How does this lead a+b to 0? Could someone please clarify?

By simplifying this: \(mean=\frac{(a - b)+(b - a)+(a + b)}{3}=a+b\)
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doesnt simlifying leads to:

a+B/3=a+b ????????????

Yes, but we can simplify further:

(a + b)/3 = a + b;

a + b = 3(a + b);

2(a + b) = 0;

a + b = 0.
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I worked the second statement as following:

Range= Highest Number - Lowest Number
R=2b
In order for R to be 2b, the only possible combination is (b+a)-(a-b).
Thus, (a-b)≤(b-a)<(b+a) or (a-b)<(b-a)≤(b+a)
So b-a is the median.

Where am I wrong?

(2) The range of set \(S\) is equal to \(2b\). If \(a=b=0\), then \(median=0\) but if \(a=0\) and \(b=1\), then \(median=1\). Not sufficient.
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Judging by this question, you can't assume that the set provided to you is in ascending order? I'd feel more comfortable removing that assumption if anyone can provide an OG question which has a set of unknown variables not provided in ascending order.

A set, by definition, is a collection of elements without any order. (While, a sequence, by definition, is an ordered list of terms.)
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hello expert, I get your explanation for option B, but aren't we worried about finding the mean in terms of a and b.

as explained even if a=b=0 or a=1, b=0, mean is b-a.

The point is that in data sufficiency problems that ask for the value of a quantity, the data given in the statements are sufficient only when it is possible to determine exactly one numerical value for the quantity.
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+1 for A. My take :

Statement 1 : Given that mean is a+b. (a+b)/3=a+b ; i.e a+b=0. This means that the set is -2b,0,2b. The median in this case becomes zero. We get a definite answer.

Statement 2 : Range is 2b. The middle value is b-a. This is not a definite value. Hence not sufficient.

The answer is option A.
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chetan2u,, Bunuel

WRT option-A, if A+B=0, then how is median =0.

Do we assume that mean = median

kindly explain

Thanks

Got it

overlooked the given set.

Thanks

I think this is a high-quality question and I don't agree with the explanation. I think that if (2) is true then the median must be b-a. The examples you give in the solution of a=b=0, median=0; and a=0, b=1, median=1, fit this solution.

Have you checked this post: https://gmatclub.com/forum/m21-184256.html#p1910671 Hope it helps.
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