Voting System Puzzle

Question

Answer Questions 79 to 82 on the basis of the information given below:

The year is 2089. Beijing, London, New York, and Paris are in contention to host the 2096 Olympics. The eventual winner is determined through several rounds of voting by members of the IOC with each member representing a different city. All the four cities in contention are also represendted in IOC.

• In any round of voting, the city receiving the lowest number of votes in that round gets eliminated. The survivor after the last round of voting gets to host the event.
• A member is allowed to cast votes for at most two different cities in all rounds of voting combined. (Hence, a ember becomes ineligible to cast a vote in a given round if both the cities (s)he voted for in earlier rounds are out of contention in that round of voting).
• A member is also ineligible to cast a vote in a round if the city (s)he represents is in contention in that round of voting.
• As long as the member is eligible, (s)he must vote and vote for only one candidate city in any round of voting.
• The following incomplete table shows the information on cities that received the maximum and minimum votes in different rounds, the number of votes cast in their favour, and the total votes that were cast in those rounds.

    Round | Total votes cast |   Maximum votes cast  |     Eliminated
          |                  |  City  | No. of votes |  City  | No. of votes
    ------|------------------|--------|--------------|--------|-------------
       1  |                  | London |      30      |New York|      12
       2  |        83        | Paris  |      32      |Beijing |      21
       3  |        75        |        |              |        |

It is also known that:

• All those who voted for London and Paris in round 1, continued to vote for the same cities in subsequent rounds as long as these cities were in contention. 75% of those who voted for Beijing in round 1, voted for Beijing in round 2 as well.
• Those who voted for New York in round 1, voted either for Beijing or Paris in round 2.
• The difference in votes cast for the two contending cities in the last round was 1.
• 50% of those who voted for Beijing in round 1, voted for Paris in round 3.
  
  

79. What percentage of members from among those who voted for New York in round 1, voted for Beijing in round 2?

(1) 33.33
(2) 50
(3) 66.67
(4) 75

80. What is the number of votes cast for Paris in round 1?

(1) 16
(2) 18
(3) 22
(4) 24

This is an MBA entrance exam puzzle. Please share your approach.

Answer 1

This is just a bunch of bullet points I've written as I worked through it - there were no other notes taken, so apologies if these are not perfectly readable. It will give an insight into how I would solve it though.

• From the rules, all cities (except those still in the running) must vote in round $2$
• This means there are $83$ cities that voted + $3$ cities still in the running ($1$ city removed each round)
• Which means round $1$ had $82$ votes, because there was $1$ more city ineligible to vote than round $2$
• Using the minimum/maximum votes we know:
• Round $1$ was: London, Paris, Beijing and New York
• Round $2$ was: London, Paris and Beijing
• Round $3$ was: London and Paris



• In Round $1$ the makeup of the votes was:
• London: $30$
• New York: $12$
• Paris $+$ Beijing: $40$



• As London and Paris voters were consistent, we know the $30$ for London never changed throughout.
• In Round $2$, we know the exact votes:
• Paris: $32$
• London: ($83-(P+B)=(83-53)=30$
• Bejing: $21$


• $75\%$ of the Beijing Round $1$ voters were Beijing round $2$ voters (but that is NOT $75\%$ of the $21$)


• We know $9$ cities ($84-75$) couldn't vote in Round $3$
• This means $9$ cities changed their vote from New York to Beijing in round $1$ (and couldn't vote again)
• Which gives us the answer for $79$. $9/12$ voters $(75\%)$ changed from New York to Beijing in Round $1$ $\rightarrow$ Round $2$. (Answer $4$)


• As London and Paris voters didn't change between Round $1$ and Round $2$, the number of Beijing carry-on votes was $12$ $(21-9)$
• This means there were originally $(1/0.75 * 12)=16$ votes for Beijing
• Which solves round $1$ voting:
• London: $30$
• New York: $12$
• Beijing: $16$
• Paris: $(82-(16+12+30))=24$

Which gives us the answer for $80$. $24$ voters originally voted for Paris in Round $1$. Which is answer d