If S is a set of four numbers w, x, y, and z, is the range of the numb

SOLUTION

If S is a set of four numbers w, x, y, and z, is the range of the numbers in S greater than 2 ?

(1) w - z > 2. The range of a set is the difference between the largest and the smallest elements of the set, since the difference of some particular two numbers are already more than 2 then the the range must also be more than 2. Sufficient.

(2) z is the least number in S --> just says that from four unknowns z is the smallest one (obviously one of the unknowns would be the smallest one, what difference does it make to know that it's z?). Not sufficient.

Answer: A.
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I got this one wrong and had difficulty fully understanding Bunuel's explanation. Here's how I reasoned through it:

(1) Range is max - min. There are various possibilities for the values of w and z:
(a) If W is max and Z is not min, then the range must be >2 (because a number smaller than Z will be subtracted from W).
(b) If W is not max and Z is min, the range must also be >2 (because Z will be subtracted from a number larger than W).
(c) If W is not max and Z is not min, the range must be >2 (because a number smaller than Z will be subtracted from a number larger than W).

Statement 1 is sufficient.

(2) Doesn't tell us about the values of w, x or y. Therefore, there is no way to determine if the range is >2.

Statement 2 is not sufficient.

Answer: A

I think my reasoning for Statement 1 is right but it took me a bit of time to get to it. Not sure if there is a shortcut to reaching the above conclusion.

The Range of a set is defined as highest element of the set - lowest element of the set.

Statement 1) w-z > 2.
We don't know if w is the highest or z is lowest elements of the set. But we don't require that information.
As if w is not the highest element and z is not the lowest element, then the highest - lowest will be more than 2.
Sufficient.

Statement 2) z is the least number in S. we don't know the value of z and also we don't know the value of highest element of the set. We cannot determine if the range of the set > 2
Not Sufficient.

Hence Option A)

Hi wongylo,

From your explanation, it looks like you considered all three possible 'relationships' between W and Z (from the information provided in Fact 1). By focusing on the specifics of what the given question asked, you could have combined all of those ideas into one big idea and saved some time...

We're told that there are 4 numbers. We're asked if the RANGE of those 4 numbers is greater than 2. Notice that '2' is a relatively small range for 4 values. For example, if the 4 values were consecutive integers, then the range would be 3. To end up with a range that is 2 or less, we'd need duplicate numbers or 4 numbers that included decimal points and were close together.

From Fact 1, we're told that W and Z differ by MORE than 2...so who cares what the other two numbers are. The range MUST be greater than 2. Fact 1 is SUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

Hello Bunuel and EMPOWERgmatRichC,

After going through S1/Fact1 in this question, would I be wrong if I generalize as follows - For ANY given set the Range is always greater than or equal to the difference between any two elements of the set?
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Hi susheelh,

Yes - if you know 2 (or more) of the values in a set, then you know the MINIMUM value of the RANGE.

GMAT assassins aren't born, they're made,
Rich

What is the range of a singleton? For example {5}

Important property : Range of a subset is lesser or equal to that of main set.

Here (w,z) is a subset and its range is greater than 2 --> The range of the main set will also be greater than 2.

Doesn't the fact that it's a set mean that w,x,y,z are distinct numbers? Wouldn't that inherently imply that the range of the distinct numbers is already greater than 2?

Hi wmjt123,

GMAT questions are always specifically-worded, so you have be careful about assuming details that are not explicitly stated. For example, there's a big difference between the word "numbers" and the word "integers." In addition, a group of numbers might include duplicates - so unless the prompt uses the word "unique" or "distinct" (or some other description that defines that there are no duplicate numbers), you have to consider the possibility that there COULD be duplicates.

Integers MUST be whole numbers (re: -2, -1, 0, 1, 2, 3, 4, etc.), but numbers are NOT held to that 'restriction.'

You made a great deduction that with 4 CONSECUTIVE INTEGERS, the range would already be greater than 2 (it would actually be 3). However, there's no way to correctly answer a DS question with just the information in the prompt, so if you think you have the answer before you look at either Fact 1 or Fact 2, then you have misinterpreted some aspect of the question.

GMAT assassins aren't born, they're made,
Rich

I am missing something. There is no limitation to what these numbers can be. What if by statement one w=0 and z=-3... 0-(-3)>2. Range is still <2. Could someone help knock some sense into me.

Hi hiteshreddyr94,

Based on the information in the prompt along with the information in Fact 1, you have come up with a valid possibility. It's possible that W = 0 and Z = -3. With just these two numbers, we know that the range would be 0 - (-3) = 3... meaning that the answer to the question would be YES ("is the range greater than 2?"... YES it is). With two additional numbers included, the range might become greater than 3, but it would be at least 3 (so the answer would stilly be YES regardless of what the other two numbers were).

GMAT assassins aren't born, they're made,
Rich

(1) Since the difference of some particular two numbers is already more than 2 then the range must also be more than 2. Sufficient.

(2) We don't know about the value z and the highest number and its value. Insufficient.


The answer is A

I was stupid enough to fail to solve this question

(1) w>z+2
So difference between a pair of terms is greater than 2
Therefore, range must be at least greater than 2
(1) is sufficient

(2) z is the least number in S
This individually does not help us at all!

I think this explanation is written quite well but one additional case is missing.

What if z is minimum and w is maximum? In this case the range is actually 2 which is not greater than 2 and answer to the DS question will be NO with the first statement alone. Even with the second statement, the answer may be YES or NO depending upon whether w is greatest number in S or not.

Therefore, in my opinion, the answer should be E.

Hi aumkarB,

The information in Fact 1 tells us that W - Z > 2. Note the 'greater than' symbol - that is NOT an 'equals' sign. If we make the Z the minimum number in the group and the The maximum number in the group, then W would be GREATER than '2 more' than Z. For example...

IF...
Z = 0... then W > 2

Even if W is just a little more than 2 (such as 2.000001), the range in this case would be 2.000001... so the range is GREATER than 2 and the answer to the question is YES.

That's going to happen no matter what value you set the Z equal to: the range might be a lot greater than 2, or just a little greater than 2, but the range will ALWAYS be greater than 2.

GMAT assassins aren't born, they're made,
Rich

Hi everyone,

I failed to interpret the question stem correctly because I considered the possibility of having duplicates in the set since the question stem doesn't mention wether the integers in the set are "unique" or "distinct".

I'm trying to understand GMAT wording to avoid confusion in similar problems. I'll appreciate any guidance.

Hi miguelhandalr,

From the information in the initial prompt, there COULD be duplicate numbers among the values of the 4 variables. However, the prompt does NOT state that the 4 numbers are integers (re: whole numbers), so you have to be careful about your assumptions when working through DS questions (and in this case, consider the possibility that each of the 4 values COULD be a non-integer).

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: Rich.C@empowergmat.com

What if the numbers are negative though?

-> If w = (-2), z = (-4), the statement would remain true [(-2)-(-4)]=6; however, the range would be 2. Therefore, the statement would be insufficient.

I'd appreciate your help 🙏