Officials

[avenus]

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

[aj5105]

What's the source?

[avenus]

Mahattan, I think. why??

[vineetbatra]

I think answer is E, what is the OA?

Low level official should be a multiple of 2 & 3, and there are several multiples of 2 & 3 under 60.

[mehravikas]

IMO - E

[ketkoag]

E for me too.....

[nervesofsteel]

yes should be E

[beat_gmat_09]

I came across this ques so digged this thread up..
The OA for this question is C, source - Manhattan.
Can anyone pls explain how to evaluate statement B here ?

Thanks

[diebeatsthegmat]

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

do yo have its explanation?

[Dani@MasterGMAT]

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

do yo have its explanation?

Hold on - is it 2x "two times x", or x^2 "x squared"?
If the answer is C, then I believe "x2" and "x3" are in fact "x squared" and "x cubed", in which case x itself will be limited to x=1. If x is greater than 1, for example, x=2, then:
we have 2 high level officials
each supervises 2^2=4 mid level officials - total of 2*4=8 mid level officials.
Each of the 8 mid levels supervises 2^3 = 8 low level officials - for a total of 8*8=64 low levels, which is already over 60 low levels in stat. (1).

Technically, stat. (1) seems to be sufficient on its own in this case, (limits x to 1) but the phrasing of the question apparently allows for low level officials to be supervised by more than one person. Thus, x=2 and we have two high levels, the two high levels could theoretically supervise the same four mid levels (since the question doesn't state that each high level supervises x^2 "different" mid levels), in which case x could equal more than 1 and still satisfy stat. (1). Thus, we need stat. (2) to clarify that each official oversees a different group, in which case any value of x greater than 1 will already push us over the 60 limit.

[beat_gmat_09]

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

do yo have its explanation?

Hold on - is it 2x "two times x", or x^2 "x squared"?
If the answer is C, then I believe "x2" and "x3" are in fact "x squared" and "x cubed", in which case x itself will be limited to x=1. If x is greater than 1, for example, x=2, then:
we have 2 high level officials
each supervises 2^2=4 mid level officials - total of 2*4=8 mid level officials.
Each of the 8 mid levels supervises 2^3 = 8 low level officials - for a total of 8*8=64 low levels, which is already over 60 low levels in stat. (1).

Technically, stat. (1) seems to be sufficient on its own in this case, (limits x to 1) but the phrasing of the question apparently allows for low level officials to be supervised by more than one person. Thus, x=2 and we have two high levels, the two high levels could theoretically supervise the same four mid levels (since the question doesn't state that each high level supervises x^2 "different" mid levels), in which case x could equal more than 1 and still satisfy stat. (1). Thus, we need stat. (2) to clarify that each official oversees a different group, in which case any value of x greater than 1 will already push us over the 60 limit.

I didn't get the explanation for statement 2. Can you pls elaborate.

Thanks.

[diebeatsthegmat]

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

do yo have its explanation?

Hold on - is it 2x "two times x", or x^2 "x squared"?
If the answer is C, then I believe "x2" and "x3" are in fact "x squared" and "x cubed", in which case x itself will be limited to x=1. If x is greater than 1, for example, x=2, then:
we have 2 high level officials
each supervises 2^2=4 mid level officials - total of 2*4=8 mid level officials.
Each of the 8 mid levels supervises 2^3 = 8 low level officials - for a total of 8*8=64 low levels, which is already over 60 low levels in stat. (1).

Technically, stat. (1) seems to be sufficient on its own in this case, (limits x to 1) but the phrasing of the question apparently allows for low level officials to be supervised by more than one person. Thus, x=2 and we have two high levels, the two high levels could theoretically supervise the same four mid levels (since the question doesn't state that each high level supervises x^2 "different" mid levels), in which case x could equal more than 1 and still satisfy stat. (1). Thus, we need stat. (2) to clarify that each official oversees a different group, in which case any value of x greater than 1 will already push us over the 60 limit.

I didn't get the explanation for statement 2. Can you pls elaborate.

Thanks.

welll the problem is that i dont know if it is x^2 and x^3 or 2x and 3x, thus i am stuck at this point

[beat_gmat_09]

it is x^2 and x^3.

[Geva@EconomistGMAT]

I didn't get the explanation for statement 2. Can you pls elaborate.

Let's read the question stem again:
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?

the phrasing here is misleading. Note that the question doesn't explicitly state that the officials supervised by each high level officials are a different set. Theoretically, the sentence "each high level official supervises x^2 mid level officials" still allows some mid level officials to be supervised by both high level officials. If the high levels are A and B, and If there are only 4 mid level officials (H, I, J, K), and all 4 are supervised by both high level officials, this would still be allowed by the question stem: A and B could both supervise the same H, I, J, K, and thus each still supervises 4 mid levels - just not exclusively. Without this exclusivity, we COULD have values of x greater than 1, and still keep the overall number of low level employees less than 60: as long as some employees are counted twice, once as supervised by official A, and once by official B.
This could've been avoided if the question stem read something like "Each high level official exclusively supervises x^2 mid-level officials". Without this crucial word, we need Stat. (2) to come and eliminate this possibility by stating that each employee can only be supervised by one official: so if x=2, then the two high level employees must indeed supervise two separate groups of 4 mid levels, who in turn really supervise 4 separate groups of 8 low levels, bringing the number of low levels to 8*8=64 - which is impossible, according to stat.(1).

[francoimps]

For a shorter explanation:

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.

Solution:

Assuming that no official is supervised by more than one person:

# of HL officials = x (from the given)
# of ML officials = x(x^2) = x^3 (by the fundamental counting principle or, by conversion (x^2 ML per HL)(no. of HL) = total ML )
# of LL officials: = x^3(x^3) = x^6 (by the fundamental counting principle or, by conversion (x^3 LL per ML)(no. of ML) = total HL )


(1) We do not know whether one official can supervise more than 1 person. If at least 1 official can supervise more than 1 person, our equations cannot hold because the total number of HL, ML, or LL will be lower and we cannot determine this by with the current information given.

(2) The statement is useless by itself.

(1) and (2). Since no official is supervised by more than one person, we can use our equations, particularly, the one for the # of LL:
# of LL = x^6 < 60

Since x is an integer, the only x that can make the equation true is when x = 1.

Therefore, the answer is C.