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# There are x high-level officials (where x is a positive integer). Each

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There are x high-level officials (where x is a positive integer). Each  [#permalink]

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03 Nov 2014, 08:15
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Tough and Tricky questions: Word Problems.

There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

Kudos for a correct solution.

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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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04 Nov 2014, 09:49
Statement 1 is clearly Insufficient as 1^3, 2^3, 3^3 all are less then 60.

Statement 2 tells that no official is supervised bt more than one person. x=1. Sufficient.
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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04 Nov 2014, 20:38
Clearly, neither of statement 1 and 2 is sufficient.

Combining 1 and 2 :-

From statement 1, the different combinations are :
High officials = 1 , Mid level official = 1 , Low Level officials = 1
High officials = 2 , Mid level official = 4 , Low Level officials = 8
High officials = 3 , Mid level official = 9 , Low Level officials = 27

From statement 2, "No official is supervised by more than one person." , we cannot conclusively say which group we can pick.

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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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04 Nov 2014, 22:05
There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

x----x^2(each of whom)----x^3

1) 2----4----8 (8*4= 32) 32<60 yes
3----9----27 (27*9= 243) 243<60 no

thus 4 high level officials

2) insufficient

thus A.

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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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06 Nov 2014, 13:57
Not sure if i'm on the wrong track here but 1 looks sufficient to me.

2 high (test):
2, each of which supervises 2^2 (so 4). So 2 * 4 = 8 mids.

8 mids

low level?
8, each of which supervises 2^3 (=8), so 8*8 = 64. Too large for statement 1 to hold, therefore 1 high official - sufficient.

statement 2 makes no sense to me so I think its a distractor, therefore A.
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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06 Nov 2014, 23:57
Bunuel When will the OA be updated for these questions??
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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07 Nov 2014, 03:24
Bunuel When will the OA be updated for these questions??

The OA's will be announced on this weekend.
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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07 Nov 2014, 21:53
2
The key is EACH OF which means if there are 2 high level there are 8 mid level and 8*8=64 low level.

So A looks sufficient- BUT, the stem does not say the low level cannot report to more than one mid level managers. Say that 5 low level report to 2 mid level managers, then there can be 1 or 2 high level managers - so not sufficient.

B resolves this- if no one has more than one manager the only solution is to have 1 high level, 1 mid level and 1 low level.

C- combined sufficient.

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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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08 Nov 2014, 13:32
1
If x = 1, then there's 1 high level, 1 med level, and 1 low level
If x = 2, there's 2 high level, each of whom supervise x^2 = 4, for a total of 8 mid level, each of whom supervise x^3 = 8 for 64 low level.

1 alone: tells you fewer than 60 low level, but x could equal 2 if low level employees report to more than one manager.
2 alone: could be any value for x

1 & 2 combined allow you to definitively answer, so answer is C
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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10 Nov 2014, 02:46
Let H,M and L represent High,Medium and Low level officers resp.
By given condition:
if H=1 then M=1 and L=1
if H=2 then M=2*(2^2)=8 and L=8*(2^3)=64

(1) It tells us that L<60, thus only first of the above two situations satisfies. i.e H=1, it won't be true if H>1-->Sufficient
(2) In each of the above two situations, No official is supervised by more than one person as it is given in the question statement itself that each of relatively higher level officers supervise a set of lower level officers--> insufficient

Hence, the correct answer is choice(A)
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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22 Apr 2015, 23:43
Bunuel wrote:

Tough and Tricky questions: Word Problems.

There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

Kudos for a correct solution.

Hi Bunuel,

I marked A. Could you please explain how does B contribute to the solution.

aimtoteach
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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23 Apr 2015, 00:27
Bunuel wrote:

Tough and Tricky questions: Word Problems.

There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

Kudos for a correct solution.

I also marked A, not sure how statement 1 contributes to the answer... My reasoning:

Statement 1: tells us that x^3<60

Statement 2: no official is supervised by more than one person: insinuates that x = x^2 = x^3 ... I thought this could only be true for x=1

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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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23 Apr 2015, 00:34
sabineodf wrote:
Bunuel wrote:

Tough and Tricky questions: Word Problems.

There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

Kudos for a correct solution.

I also marked A, not sure how statement 1 contributes to the answer... My reasoning:

Statement 1: tells us that x^3<60

Statement 2: no official is supervised by more than one person: insinuates that x = x^2 = x^3 ... I thought this could only be true for x=1

Reread my reasoning and the question and I retract my statements haha but I am still confused about the questions. I still don't understand how to get the answer though!

But I've realized that statement 1 has more value than I initially thought because it has narrowed the options of x down to 1, 2 or 3...

So what is B giving us that tells us which one it is?
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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23 Apr 2015, 02:11
aimtoteach wrote:
Bunuel wrote:

Tough and Tricky questions: Word Problems.

There are x high-level officials (where x is a positive integer). Each high-level official supervises x^2 mid-level officials, each of whom, in turn, supervises x^3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.
(2) No official is supervised by more than one person.

Kudos for a correct solution.

Hi Bunuel,

I marked A. Could you please explain how does B contribute to the solution.

aimtoteach

Hello aimtoteach and sabineodf

Let's imagine that we have firm with 10 low level workers and 2 managers.
But we know nothing about hierarchy of this firm. Maybe each manager has 5 dedicated subordinates or maybe it's mix hierarchy and each manager has 10 subordinates: each from this employees has two managers and report to each managers for different tasks.

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There are x high-level officials (where x is a positive integer). Each  [#permalink]

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08 Mar 2016, 09:41
There are x high-level officials (where x is a positive integer). Each high-level official supervises x2 mid-level
officials, each of whom, in turn, supervises x3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.

(2) No official is supervised by more than one person.
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Posts: 53020
Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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08 Mar 2016, 11:00
APARNASHAH wrote:
There are x high-level officials (where x is a positive integer). Each high-level official supervises x2 mid-level
officials, each of whom, in turn, supervises x3 low-level officials. How many high-level officials are there?

(1) There are fewer than 60 low-level officials.

(2) No official is supervised by more than one person.

Merging topics. Please search before posting.
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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03 Oct 2016, 06:51
Hi Bunuel,
could you please post explanation why is it not E?
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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15 Jan 2017, 16:40
1
Let's assume that there are 2 high level officials. This means that each of these 2 high level officials supervises 4 (or x^2) mid-level officials, and that each of these 4 mid-level officials supervises 8 (or x^3) low-level officials.

It is possible that the supervisors do not share any subordinates. If this is the case, then, given 2 high level officials, there must be 2(4) = 8 mid-level officials, and 8(8) = 64 low-level officials.

Alternatively, it is possible that the supervisors share all or some subordinates. In other words, given 2 high level officials, it is possible that there are as few as 4 mid-level officials (as each of the 2 high-level officials supervise the same 4 mid-level officials) and as few as 8 low-level officials (as each of the 4 mid-level officials supervise the same 8 low-level officials).

Statement (1) tells us that there are fewer than 60 low-level officials. This alone does not allow us to determine how many high-level officials there are. For example, there might be 2 high level officials, who each supervise the same 4 mid-level officials, who, in turn, each supervise the same 8 low-level officials. Alternatively, there might be 3 high-level officials, who each supervise the same 9 mid-level officials, who, in turn, each supervise the same 27 low-level officials.

Statement (2) tells us that no official is supervised by more than one person, which means that supervisors do not share any subordinates. Alone, this does not tell us anything about the number of high-level officials.

Combining statements 1 and 2, we can test out different possibilities.

If x = 1, there is 1 high-level official, who supervises 1 mid-level official (12 = 1), who, in turn, supervises 1 low-level official (13 = 1).

If x = 2, there are 2 high-level officials, who each supervise a unique group of 4 mid-level officials, yielding 8 mid-level officials in total. Each of these 8 mid-level officials supervise a unique group of 8 low-level officials, yielding 64 low-level officials in total. However, this cannot be the case since we are told that there are fewer than 60 low-level officials.

Therefore, based on both statements taken together, there must be only 1 high-level official. The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
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Re: There are x high-level officials (where x is a positive integer). Each  [#permalink]

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Re: There are x high-level officials (where x is a positive integer). Each   [#permalink] 17 Oct 2018, 14:05
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