CAT 2025 Lesson : Numerical Reasoning - Case 4

Read the following and answer the questions that follow

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 represented in IOC.

(a) 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.
(b) A member is allowed to cast votes for at most two different cities in all rounds of voting combined. (Hence, a member 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.)
(c) 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.
(d) 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.



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.
[CAT 2005]

10) 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

11) What is the number of votes cast for Paris in round 1?

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

12) What percentage of members from among those who voted for Beijing in round 2 and were eligible to vote in round 3, voted for London?

(1) 33.33
(2) 38.10
(3) 50
(4) 66.67

13) Which of the following statements must be true?

I. IOC member from New York must have voted for Paris in round 2.
II. IOC member from Beijing voted for London in round 3.

(1) Only I
(2) Only II
(3) Both I and II
(4) Neither I nor II

Solution

The total votes in Round 2 includes everyone eligible in Round 1 along with the eliminated city which is New York.

Hence the total votes in Round 2 is 82 + 1(New York's vote) = 83.

From this, we can infer that the total votes must be 82 in Round 1.

Going by this logic, the total votes should have been 84 in round 3, but is 75. The only way this could have happened is if 9 of the voters who voted for New York and then Beijing were not allowed to vote.

So Round 3 has 84 – 9 (people who voted for New York and Beijing) = 75 voters

From the case and the questions we can understand that this is a case that requires movement of votes across the rounds. Hence, let us form a structure accordingly.



9 people who voted for New York voted for Beijing in Round 2 (on elimination of New York). Hence 9 votes from New York in Round 1 is moved to Beijing in Round 2.

New York did not participate in Round 2 and 3, hence, the cell related to it can be struck out.

Similarly, Beijing did not participate in Round 3, hence the Beijing cell can be struck out too in round 3.



The total votes of New York in Round 1 are 12, out of which 9 voted for Beijing. The remaining 3 could have voted for either London or Paris in Round 2.

Similarly, the total number of votes for Beijing in Round 2 were 21, for London in Round 1were 30 and for Paris in Round 2 were 32.

The total votes in Round 2 is 83. Total votes of Paris and Beijing in Round 2 add up to 53, when subtracted from 80 gives the total votes of London in Round 2, which is 30.

The 21 votes of Beijing comprises of 9 votes of New York and the remaining 12 votes.
These 12 votes could either be 11 votes from Beijing in Round 1 who continued to vote for Beijing in Round 2 and the 1 New York vote who was eliminated in Round 1.

But it is given that 75% of the voters for Beijing in Round 1 voted for Beijing in Round 2. If 75% is 11 we will not get an integer number of votes. Thus, this possibility is eliminated.

Hence, the New York candidate who was eliminated in Round 1 did not vote for Beijing in Round 2 and the 12 remaining votes were from Beijing voters from Round 1.

If 75% is 12 votes, then 100% is 16 votes. The total voters of Beijing in Round 1 are 16.



The total votes in Round 1 are 82. London, Beijing and New York had a total of 58 votes, hence, Paris had a total vote of 24 in Round 1.

Those who voted for London and Paris in Round 1 continued to vote for these cities in Round 2, so, the 30 and 24 votes of London and Paris respectively were carried to Round 2 also.

London has a total vote of 30 in Round 2. Hence the voters from New York who did not vote for Beijing and the voters from Beijing who did not vote for Beijing in Round 2 cannot have voted for London.

Hence the remaining 3 votes from New York and 4 votes from Beijing was used for Paris in Round 2. Also, the New York candidate who hasn't voted for Beijing will also end up voting for Paris in Round 2.

Paris has a total of 32 votes in Round 2 which comprises of 24 votes from Paris voters from round 1 + 4 votes from Beijing + 3 votes from New York + 1 vote from New York candidate.



The total vote difference between the two cities in Round 3 was just 1. The total votes in Round 3 is 75, which means, one city got 38 votes and another got 37 votes.

Out of the 16 votes of Beijing, 50% vote for Paris, which is 8 votes. Hence, the remaining 8 voted for London in Round 3.

London already carries its 30 votes from Round 1 and 2, so, 30 + 8 votes makes 38 votes.

Hence, the total votes Paris would have got is 37.

These 37 votes consists of the 24 votes of Paris from Round 1 and Round 2 + 8 voted from Beijing + 3 votes from New York + 1 vote from the New York candidate + 1 vote from the Beijing candidate.



10)
9 members out of 12 from New York voted for Beijing in Round 2

=$\dfrac{9}{12} \times 100$= 75%

Answer: (4) 75

11)
From the above table, the number of votes for Paris in round 1 is 24

Answer: (4) 24

12)
Out of 21 members who voted for Beijing in Round 2, only 12 of them are eligible to vote in Round 3, out of the 12 eligible members, 8 voted for London in Round 3.

=$\dfrac{8}{12} \times 100$= 66.67% Answer: (4) 66.67

13)
Statement I is true, as the IOC member from New York must have voted for Paris in Round 2.
Statement II is not true, as the IOC member from Beijing did not vote for Paris in Round 3.

Answer: (1) Only I

Answer:
10) (4) 75
11) (4) 24
12) (4) 66.67
13) (1) Only I