A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order. If \(L_1\) is the list consisting of the first \(n_1\) numbers in L and \(L_2\) is the list consisting of the last \(n_2\) numbers in L, is 17 a mode for L ?

Say the list is a, b, c, c, d, e, f, h, h, h, k, l..
\(L_1\) could be just a, and \(L_2\) could be h,h,k,l OR
\(L_1\) could be a,b,c,c,d and \(L_2\) could be c,c,d..h,h,k,l

(1) 17 is a mode for \(L_1\) and 17 is a mode for \(L_2\).
Since the list is in increasing order and mode is 17 in both cases, the list L1 and L2 either have all numbers within themselves OR only some other 17 are left.
For example 1,3,5,17,17,17,17,17,20,25....L1 could be 1,3,5,17,17 and L2 could be 3,5,17,17,17,17,17,20,25
OR L1 could be 1,3,5,17,17 and L2 could be 17,17,20,25
So the mode of L will be 17 in each case.
Suff

(2) \(n_1 + n_2 = n\)
We do not know the numbers.
Insuff

A
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Hola amigos

I disagree with David from examPAL.

Mode - is the number that occurs most often. If no number in the list is repeated, then there is no mode for the list.

1. \(17\) is a mode for \(L_1\) and \(17\) is a mode for \(L_2\).
Most important thing to note is the numbers are in increasing order. That means that there are only \(2\) possible cases:
Case 1: \(L_1\) and \(L_2\) togther include all the numbers \(L\) has. E.g. \(L\) = {1, 1, 3, 17, 17, 17, 19, 21,}, \(L_1\) = {1, 1, 3 , 17, 17, 17, 19} and \(L_2\) = {3, 17, 17, 17, 19, 21}
Case 2: The numbers of \(L\) not included in \(L_1\) and \(L_2\) are all \(17\). E.g. \(L\) = {1, 1, 3, 17, 17, 17, 17, 17, 17, 17, 19, 21,}, \(L_1\) = {1, 1, 3, 17, 17, 17} and \(L_2\) = {17, 17, 19, 21} Those two reds are leftover.
In both cases \(17\) is the most repeated number and there is no other possible case, thus the mode of \(L\) is \(17\).
Sufficient

2. \(n_1+n_2=n\)
The given information is not repeated in other words here. We already have seen that some \(17\) can be left. Though no info regarding numbers here.
Insufficient

The answer is A

The Logical approach to this question will use the definition of 'mode' - the most frequent value of a set of data.
In statement (1),if the number 17 appears more than all other numbers in both of the list, it must also appear more than any other number in the combined list as well. It is like saying that if a > x and b > y, then a + b > x + y: if 17 appears more than the second frequent number in both lists, it will appear more than the second greatest number in the combined list as well. This statement is enough on its own.
Statement (2) does not tell us which numbers are in the set, so is insufficient.
The correct answer is (A).

Posted from my mobile device

EDIT: *kudos to Shukhrat below for pointing out an error in my explanation of stmt (2) (which did not change the final answer)
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Three cases are possible about the mode of a list of numbers. If the numbers are all distinct, then the list has no mode. If one number's frequency is greater than the frequencies of all other numbers, then the list has one mode. However, it's also possible that a list of not all distinct numbers has multiple modes if two or more numbers have the greatest frequency simultaneously.

We know that \(L\) is a list of numbers, not necessarily distinct, that are arranged in increasing order. The original question: Is 17 a mode for \(L\)?

1) We can infer that in the list no number under 17 has greater frequency than that of 17. We can also infer that in the list no number above 17 has greater frequency than that of 17. Using that no number in the list has greater frequency than that of 17 and that 17 has a frequency of at least 2, we can answer the original question with a definite Yes. \(\implies\) Sufficient

2) No information is given about the actual numbers. If all numbers in the list are 17, then the answer to the original question is Yes. However, if 17 is not even in the list, then the answer to the original question is No. Thus, we can't get a definite answer to the original question. \(\implies\) Insufficient

Answer: A
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Hi All,

We're told that List L contains a total of N numbers, NOT necessarily DISTINCT, that are arranged in INCREASING order and L1 is the list consisting of the first N1 numbers in L and L2 is the list consisting of the last N2 numbers in L. We're asked if 17 is a MODE for L. It's worth noting that the two sub-lists (L1 and L2 could potentially "overlap", meaning that some numbers of List L appear in BOTH groups). Every number in List L is accounted for though (again though, it's possible that some values appear in both sub-lists).

To start, it's worth noting that a "mode" is a number that shows up the most often in a group of numbers (for example, the group 1, 2, 2, 3, 4 has a mode of 2). There can actually be more than one mode in a group though (for example, the group 1, 1, 2, 2, 3, 4 has two modes: 1 and 2).

(1) 17 is a mode for L1 and 17 is a mode for L2.

Fact1 tells us that 17 is a mode for both sub-lists. Note that the prompt did tell us that the list is arranged in INCREASING order...
IF....
-there is NO 'overlap', then some of the 17s appear in L1 and some appear in L2. Totaling all of those individual 17s would make the mode even larger than it is in the two individual sub-lists, meaning that no other value could appear more often in List L.
-there IS an 'overlap' - meaning one (or more) of the 17s appears in BOTH sub-lists, then that total number of 17s will still be a mode. There might be other values that are also modes, but those values could not show up more often than the 17s do, so none of them could "out-number" the 17s.
Fact 1 is SUFFICIENT

(2) N1 + N2 = N

Fact 2 tells us that the two sub-lists have no terms in common (meaning that there is no 'overlap'), but it does not tell us anything about the individual terms or what the mode(s) might be.
Fact 2 is INSUFFICIENT

Final Answer:

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

Steps 1 & 2: Understand Question and Draw Inferences
In this question, we are given

• A certain list, L, contains a total of n numbers, not necessarily distinct, that are arranged in increasing order.
• L1 is the list consisting of the first n1 numbers in L.
• L2 is the list consisting of the last n2 numbers in L.

We need to determine

• Whether 17 is a mode for L or not.

Step 3: Analyse Statement 1
As per the information given in statement 1, 17 is a mode for L1 and 17 is a mode for L2.
Therefore, we can infer that

• 17 must occur in L1, either same or a greater number of times as any other number in L1.
• 17 must occur in L1, either same or a greater number of times as any other number in L2.

As all elements in L are in ascending order, we can also conclude that

• Each number between last occurrence of 17 in L1 and the first occurrence of 17 in L2 must be equal to 17 only.
• Therefore, 17 occurs either same or greater number of times as any other number in L.

o Thus, 17 is a mode for L.

Hence, statement 1 is sufficient to answer the question.

Step 4: Analyse Statement 2
As per the information given in statement 2, n1 + n2 = n.

• However, from this statement, we cannot conclude anything about the mode of L1, L2, or L.

Hence, statement 2 is not sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)
Since we can determine the answer from statement 1 individually, this step is not required.

Hence, the correct answer choice is option A.


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Statement 1 is sufficient you can come up with such lists in which 17 is the mode.

Statement 2: You can just try and see if you need \(n_1+n_2=n\) or not.
L1={1,2,17,17,17} Mode = 17
L2={17,17,17} Mode = 17
L={L1+L2} or L={1,2,17,17,17}

Now does it really matter how many 17s are in L as long as the mode is 17. No.
Hence A

Thank you!
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Bunuel why are we assuming that n1+n2=n in the first equation. Isn't that a necessary condition to say A is sufficient? I would say option C seems appropriate. Can you help?



Posted from my mobile device

Hi legendinthewomb,

We can't assume that N1 and N2 are unique sub-sets with no 'overlap' - and I don't think that any of the explanations assumes that. L1 and L2 could potentially "overlap", meaning that some numbers of List L appear in BOTH groups.

(1) 17 is a mode for L1 and 17 is a mode for L2.

Fact1 tells us that 17 is a mode for both sub-lists. Note that the prompt did tell us that the list is arranged in INCREASING order...
IF....
-there is NO 'overlap', then some of the 17s appear in L1 and some appear in L2. Totaling all of those individual 17s would make the mode even larger than it is in the two individual sub-lists, meaning that no other value could appear more often in List L.
-there IS an 'overlap' - meaning one (or more) of the 17s appears in BOTH sub-lists, then that total number of 17s will still be a mode. There might be other values that are also modes, but those values could not show up more often than the 17s do, so none of them could "out-number" the 17s.

As such, Fact 1 is SUFFICIENT

GMAT assassins aren't born, they're made,
Rich
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Very Tricky Official Question.
I did it wrong as C , Missed out the word 'Last'.
Always remember to Read The Word questions Very slowly Taking care of each word.
Any word missed out will result in a wrong answer.
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Bunuel How do we know from the question that L1 and L2 are sub-lists of L and not just lists of random numbers that also consist of N1 and N2 numbers? And how do we know that L1 (and L2) has to go in the ascending order without skipping any numbers?

Hi pruekv,

GMAT questions are always specifically worded, so you have to pay careful attention to everything that you're told (and what you are NOT told) - and the question that is ASKED. This prompt specifically tells us several things:

1) List L is a list of N numbers, NOT necessarily distinct, that are arranged in INCREASING order.

This means that we have an unknown number of N numbers, some of those numbers might appear more than once and the numbers are arranged in order from least to greatest.

2) L1 is a list of the FIRST N1 numbers in List L.

This means that L1 is a group of numbers from list L, starting at the FIRST number and including numbers up to a certain point. In simple terms, you might think of this as a subset of the 'smaller' numbers in List L (up to a certain point in the List).

3) L2 is a list of the LAST N2 numbers in List L.

This means that L2 is also group of numbers from list L, starting at the LAST number and including numbers going 'backwards' up to a certain point. In simple terms, you might think of this as a subset of the 'larger' numbers in List L (down to a certain point in the List).

Some additional things to keep in mind:
-The numbers are not necessarily consecutive integers (since the prompt never states that they are).
-The sub-lists L1 and L2 might include some 'overlap' - meaning the largest numbers in L1 might also be the smallest numbers in L2.

GMAT assassins aren't born, they're made,
Rich
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It's not possible to understand the question within 2 minutes; let alone answering the question

Hi waihoe520,

The idea that you should answer every Quant question in under 2 minutes is NOT good pacing advice (certain 'gettable' questions will require up to 3 minutes of effort on the Official GMAT, so if you're not prepared to put in that time, then you will likely miss out on some points in the Quant section).

This prompt is actually a great 'concept question' - meaning that you don't have to do any math to solve it as long as you recognize the concepts involved. The key to solving this question (relatively) quickly is to recognize that some of the numbers in L1 might also be in L2 (and what that means to the entire group of numbers).

Were you able to correctly answer this question - and how did you go about approaching it overall?

GMAT assassins aren't born, they're made,
Rich
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Would it make a difference if we are not told that numbers are in increasing order?

Hi VeritasKarishma,

Any easy explanation for this?

Thanks in advance

Imagine the list L.
Numbers in ascending order... This is the diagram that came to my mind on reading the question.



L1 is the list with FIRST few numbers, L2 is the list with LAST few numbers. L1 and L2 may not overlap and there may be some extra numbers in L, L1 and L2 may divide the complete list between themselves or L1 and L2 may overlap with some numbers common to both.

Is 17 mode for L? That means does 17 occur maximum number of time in L?

(1) 17 is a mode for \(L_1\) and 17 is a mode for \(L_2\).

17 occurs the most in L1 and 17 occurs the most in L2.

L1 and L2 may not overlap and there may be some extra numbers in L - All these extra numbers will be 17 too since the list is in increasing order
L1 and L2 may divide the complete list between themselves - Most 17s in both list so there will be most 17s in combined list
L1 and L2 may overlap with some numbers common to both - Most 17s in L1 and the same most 17s in L2 so most 17s in combined list. The other numbers on both sides of 17 will appear fewer number of time.

Sufficient alone.

(2) Not sufficient alone since 17 isn't even mentioned.

Answer (A)
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Hi chetan2u , thanks for the explanation

I thought (1) is insufficient, because there is no indication that n_1 and n_2 is adjacent. In other words, I concerned that there is a gap between n_1 and n_2.
If the gap exists, the don't know how big it is or what consists the gap.

Please help me what I missed

Thank you in advance!!
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Hi suminha,

You bring up a good point: what if there is a group of numbers BETWEEN group L1 and group L2? At this point, what additional work did you do when you worked through Fact 1...?

If there is a set of numbers that is not in either L1 or L2, how would it be possible for 17 to be the MODE for BOTH of those groups (since there are numbers 'in between' the two groups, how could 17 be in BOTH groups?)? Remember that the numbers are arranged in order from least to greatest and there could be duplicates.....

There is only one way for that to occur: if ALL of the numbers that are NOT in L1 nor in L2 are all 17s. In this situation, 17 would still be the mode for the entire list and the answer to the question would still be 'YES.'

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