anjaneiya wrote:Representation in a local community board is based on the community's population, with 25 board members representing the first 500,000 people, and each additional board member representing an additional 30,000 people. Is the population of the town less than 500,000?
(1) There were 23 members on the community board.
(2) If the population were three times as large, the community board would have had 51 members.
Answer:
D
Beautiful problem, anjaneiya. Where is it from, please?
Let us call P_n the population that corresponds to exactly n people in its community board, ok?
Please check the following affirmations, because they will be crucial in the arguments that follow!
(I) 0 <= P_25 <= 500,000, because till 500,000 people (this number included), we will have exactly 25 board members.
(II) 500,001 = 500,000 + 1 +
0. 30,000 < = P_
26 <= 500,000 + 30,000 (explain to yourself)
(III) 500,000 + 1 +
1. 30,000 <= P_
27 <= 500,000 + 2.30,000 (again)
(IV) 500,000 +1 +
2.30,000 <= P_
28 <= 500,000 + 3.30,000 (again)
Now I guess you will all agree with the following generalization:
Important: please note that (in red): 26 - 0 = 27 - 1 = 28 - n = ....
(V) 500,000 + 1 +
(n-26).30,000 <= P_n <= 500,000 + [(n-26)+1] .30,000 (explain!)
That´s all we need for solving it all, let´s see.
(1) SUFFICIENT.
If statement (1) means
EXACTLY 23 (what is usually the case), it seems to contradict the question stem, because as mentioned in (I), the question stem makes us understand that we have at least 25 board members when the population is sufficiently small. ANYWAY we are sure that sttm (1) is sufficient, because if the population were equal or greater than 500,000 we would need CERTAINLY more board members than 24 (therefore than 23 for sure).
(2) This one is the hard/nice one.
From what I explained, please note that if we call population as P, we know that in the hypothetical 3P population we would have, from (V), that:
500,000 + 1 +
(51-26).30,000 <= P_51 <= 500,000 + [(51-26)+1] .30,000
therefore 3P must be in the interval [500,000 + 1 + 25.30,000 ; 500,000 + 1 + 26.30,000], that is,
P must be in the interval [166,667 + 25.10,000 ; 166,667+26.10,000].
From the fact that 166,667 + 260,000 < 500,000 we are sure P is less than 500,000, therefore sttm(2) is also sufficient.
Regards,
Fabio.
P.S.1: (I), (II), etc are easily build but (V) is NOT
if you don´t do the first ones... therefore please note that while building (I), (II), etc you are SAVING time!
P.S.2: the calculation of the extreme values of the interval is NOT superfluous, because we could have discovered (in other scenario/numbers) that P would be (say) between 490,000 and 510,000 and, in this case, sttm (2) would NOT be sufficient to answer the question asked!
For the interested readers, this is my "final version" of this beautiful problem (the one I will use with my students):
Representation in a local community board is based on the community´s population, with 25 board members representing the first 500,000 people, and each additional board member representing an additional 30,000 people. Is the population of the community less than 500,000 ?
(1) If the population were two times as large, the community board would have had exactly 32 members.
(2) If the population were 2.85 times as large, the community board would have had exactly 56 members.
Answer to this new problem:
A Enjoy!
