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Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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7 00:00

Difficulty:   55% (hard)

Question Stats: 53% (01:10) correct 47% (01:23) wrong based on 138 sessions

### HideShow timer Statistics Is the range of a combined set $$(S, T)$$ is bigger than the sum of ranges of sets $$S$$ and $$T$$ ?

(1) The largest element of $$T$$ is bigger than the largest element of $$S$$.

(2) The smallest element of $$T$$ is bigger than the largest element of $$S$$.

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Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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Official Solution:

The range of a set is the difference between the largest and smallest elements of a set.

$$range_t=t_{max}-t_{min}$$;

$$range_s=s_{max}-s_{min}$$;

Question: $$range_{t \text{ and } s} \gt (t_{max}-t_{min})+(s_{max}-s_{min})$$?

(1) The largest element of $$T$$ is bigger than the largest element of $$S$$. Given: $$t_{max} \gt s_{max}$$, so the largest element of combined set is $$t_{max}$$ but we still don't know which is the smallest element of combined set:

If it's $$t_{min}$$ then the question becomes is $$t_{max}-t_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$0 \gt s_{max}-s_{min}$$ and the answer would be NO;

If it's $$s_{min}$$ then the question becomes is $$t_{max}-s_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$t_{min} \gt s_{max}$$ and the answer would be sometimes NO and sometimes YES. Not sufficient.

(2) The smallest element of $$T$$ is bigger than the largest element of $$S$$. Given: $$t_{min} \gt s_{max}$$, so the largest element of the combined set is $$t_{max}$$ and the smallest element of the combined set is $$s_{min}$$.

So the question becomes is $$t_{max}-s_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$t_{min} \gt s_{max}$$? And that is given to be true, so the answer is YES. Sufficient.

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Manager  Joined: 06 Mar 2014
Posts: 228
Location: India
GMAT Date: 04-30-2015

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1
Is there any way, we can present the above statements with some examples?

Some other way in which we don't have to resort to variables such as tmax and tmin.
Intern  Joined: 22 Jun 2014
Posts: 21
Concentration: General Management, Finance
GMAT 1: 700 Q50 V34 GRE 1: Q800 V600 GPA: 3.68

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imagine the two sets as two line segments in a straight line. the lines can overlap each other, one inside another or be in different place altogether. to answer the question, we need to know if the lines overlap or there is a finite distance between them?
The 1st point says the right end of line t is on the right side of right end of line s - this doesn't answer our question. The 2nd point says left end of line t is on the right side of right end of line s --> there is a finite distance between the lines. We just got our answer
Senior Manager  Joined: 31 Mar 2016
Posts: 376
Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34 GPA: 3.8
WE: Operations (Commercial Banking)

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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 12 Jul 2013
Posts: 7

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Bunuel wrote:
Official Solution:

The range of a set is the difference between the largest and smallest elements of a set.

$$range_t=t_{max}-t_{min}$$;

$$range_s=s_{max}-s_{min}$$;

Question: $$range_{t \text{ and } s} \gt (t_{max}-t_{min})+(s_{max}-s_{min})$$?

(1) The largest element of $$T$$ is bigger than the largest element of $$S$$. Given: $$t_{max} \gt s_{max}$$, so the largest element of combined set is $$t_{max}$$ but we still don't know which is the smallest element of combined set:

If it's $$t_{min}$$ then the question becomes is $$t_{max}-t_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$0 \gt s_{max}-s_{min}$$ and the answer would be NO;

If it's $$s_{min}$$ then the question becomes is $$t_{max}-s_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$t_{min} \gt s_{max}$$ and the answer would be sometimes NO and sometimes YES. Not sufficient.

(2) The smallest element of $$T$$ is bigger than the largest element of $$S$$. Given: $$t_{min} \gt s_{max}$$, so the largest element of the combined set is $$t_{max}$$ and the smallest element of the combined set is $$s_{min}$$.

So the question becomes is $$t_{max}-s_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$t_{min} \gt s_{max}$$? And that is given to be true, so the answer is YES. Sufficient.

Hi Bunuel,

Wont the hypothesis break down in the following screnario

T=[1,2,3,4,5]
S=[0,0,0,0]

so the answer to the question would be "E" cannot be determined.

Kindly let me know if i have misunderstood.
Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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rt1601 wrote:
Bunuel wrote:
Official Solution:

The range of a set is the difference between the largest and smallest elements of a set.

$$range_t=t_{max}-t_{min}$$;

$$range_s=s_{max}-s_{min}$$;

Question: $$range_{t \text{ and } s} \gt (t_{max}-t_{min})+(s_{max}-s_{min})$$?

(1) The largest element of $$T$$ is bigger than the largest element of $$S$$. Given: $$t_{max} \gt s_{max}$$, so the largest element of combined set is $$t_{max}$$ but we still don't know which is the smallest element of combined set:

If it's $$t_{min}$$ then the question becomes is $$t_{max}-t_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$0 \gt s_{max}-s_{min}$$ and the answer would be NO;

If it's $$s_{min}$$ then the question becomes is $$t_{max}-s_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$t_{min} \gt s_{max}$$ and the answer would be sometimes NO and sometimes YES. Not sufficient.

(2) The smallest element of $$T$$ is bigger than the largest element of $$S$$. Given: $$t_{min} \gt s_{max}$$, so the largest element of the combined set is $$t_{max}$$ and the smallest element of the combined set is $$s_{min}$$.

So the question becomes is $$t_{max}-s_{min} \gt t_{max}-t_{min}+s_{max}-s_{min}$$. Or: is $$t_{min} \gt s_{max}$$? And that is given to be true, so the answer is YES. Sufficient.

Hi Bunuel,

Wont the hypothesis break down in the following screnario

T=[1,2,3,4,5]
S=[0,0,0,0]

so the answer to the question would be "E" cannot be determined.

Kindly let me know if i have misunderstood.

For this case you still have an YES answer to the question for (2). The range of combined set is 5 and the sum of the ranges of T and S is 4 + 0 = 4.
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Intern  B
Joined: 29 Nov 2016
Posts: 3

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what if
s =( 2,3,40) &
t= (41,41,41)
Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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digvijay99 wrote:
what if
s =( 2,3,40) &
t= (41,41,41)

In this case the range of combined set would be 41 - 2 = 39 and the sum of the ranges of S and T would be 38 + 0 = 38. So, for (2) you'll get the same YES answer to the question, we've got in the solution.

Does this make sense?
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Intern  B
Joined: 28 Mar 2017
Posts: 9
GMAT 1: 550 Q43 V23 ### Show Tags

I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 10 Jan 2018
Posts: 13
Location: United States (NY)
GMAT 1: 710 Q49 V38 GMAT 2: 770 Q51 V44 GPA: 3.36

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What about both S and T are single element set? Let S = {0} and T = {1} . T's smallest element is larger than S's biggest element . S and T can merge into {0,1} with range to be 1. The sum of the range of S and T is 1. So (2) is insufficient. What's wrong with this logic?
Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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Canteenbottle wrote:
What about both S and T are single element set? Let S = {0} and T = {1} . T's smallest element is larger than S's biggest element . S and T can merge into {0,1} with range to be 1. The sum of the range of S and T is 1. So (2) is insufficient. What's wrong with this logic?

The range of a single-element set is 0.
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Intern  B
Joined: 15 May 2018
Posts: 8
GMAT 1: 750 Q51 V40 GRE 1: Q166 V164 GPA: 3.97
WE: Consulting (Consulting)

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Let us take 2 sets:
1) S = {1,2,3}
2) T = {4,6}

This satisfies statement 2.
Range of S = 2
Range of T = 2
Total = 4
Range of S,T = 5

It satisfies, so far so good.

Now take t ={4,5}, with S remaining same. Now Range of S,T is 4 and not greater than the sum. Answer should be (E) I believe
Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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AkarshS wrote:
Let us take 2 sets:
1) S = {1,2,3}
2) T = {4,6}

This satisfies statement 2.
Range of S = 2
Range of T = 2
Total = 4
Range of S,T = 5

It satisfies, so far so good.

Now take t ={4,5}, with S remaining same. Now Range of S,T is 4 and not greater than the sum. Answer should be (E) I believe

In case S = {1, 2, 3} and T = {4, 5}.

The sum of ranges of sets S and T = 2 + 1 = 3, while the range of combined set is 4. So, you'd still have an YES answer. The correct answer to the question is B, as explained above.
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Intern  B
Joined: 09 Jan 2018
Posts: 4

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What if it is a single set elements . It didnt talk about number of elements .
Intern  B
Joined: 03 May 2017
Posts: 4
Location: India
Concentration: General Management, Technology
Schools: HEC Montreal

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Hi Bunuel,

Let's say:
S: 2
T: 4,6

Range of (S,T) will not be greater than individual ranges.

Let;s say:
S:1
T:7,9

Range of (S,T) will be greater than individual ranges.

Please guide Math Expert V
Joined: 02 Sep 2009
Posts: 56272

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Milind27 wrote:
Hi Bunuel,

Let's say:
S: 2
T: 4,6

Range of (S,T) will not be greater than individual ranges.

Let;s say:
S:1
T:7,9

Range of (S,T) will be greater than individual ranges.

Please guide S: 2 --> range = 0
T: 4,6 --> range = 2
Combined {2, 4, 6} --> range = 4.
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