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M13-26

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M13-26  [#permalink]

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New post 15 Sep 2014, 23:49
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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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Re M13-26  [#permalink]

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New post 15 Sep 2014, 23:50
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.


Answer: B
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Re: M13-26  [#permalink]

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New post 09 Oct 2014, 14:07
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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.
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Re: M13-26  [#permalink]

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New post 22 Oct 2014, 18:06
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
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Re M13-26  [#permalink]

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New post 31 Aug 2016, 03:46
I think this is a high-quality question and I agree with explanation.
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Re: M13-26  [#permalink]

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New post 23 Oct 2016, 05:19
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.


Answer: B


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.
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Re: M13-26  [#permalink]

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New post 23 Oct 2016, 05:40
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.


Answer: B


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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Re: M13-26  [#permalink]

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New post 16 Jun 2017, 20:52
what if
s =( 2,3,40) &
t= (41,41,41)
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Re: M13-26  [#permalink]

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New post 17 Jun 2017, 03:30
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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Re M13-26  [#permalink]

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New post 25 Mar 2018, 14:57
I think this is a high-quality question and I agree with explanation.
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Re: M13-26  [#permalink]

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New post 17 Apr 2018, 07:18
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?
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Re: M13-26  [#permalink]

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New post 17 Apr 2018, 12:52
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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Re: M13-26  [#permalink]

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New post 10 Jun 2018, 23:24
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
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Re: M13-26  [#permalink]

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New post 11 Jun 2018, 04:39
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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Re: M13-26  [#permalink]

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New post 01 Aug 2018, 06:41
What if it is a single set elements . It didnt talk about number of elements .
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Re: M13-26  [#permalink]

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New post 21 Aug 2018, 02:56
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 :)
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Re: M13-26  [#permalink]

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New post 21 Aug 2018, 03:47
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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