Question

In an election several candidates contested for a constituency. In any constituency, the winning candidate was the one who polled the highest number of votes, the first runner up was the one who polled the second highest number of votes, the second runner up was the one who polled the third highest number of votes, and so on. There were no ties (in terms of number of votes polled by the candidates) in any of the constituencies in this election. 
In an electoral system, a security deposit is the sum of money that a candidate is required to pay to the election commission before he or she is permitted to contest. Only the defeated candidates (i.e., one who is not the winning candidate) who fail to secure more than one sixth of the valid votes polled in the constituency, lose their security deposits. 
The following table provides some incomplete information about votes polled in four constituencies: A, B, C and D, in this election.

	Constituency
	A	B	C	D
No. of candidates contesting	10	12	5	8
Total No. of valid votes polled	5,00,000	3,25,000	6,00,030
No. of votes polled by the winning candidate	2,75,000	48,750
No. of votes polled by the first runner up	95,000			37,500
No. of votes polled by the second runner up				30,000
% of valid votes polled by the third runner up				10%

The following additional facts are known: 
1. The first runner up polled 10,000 more votes than the second runner up in constituency A.
2. None of the candidates who contested in constituency C lost their security deposit. The difference in votes polled by any pair of candidates in this constituency was at least 10,000.
3. The winning candidate in constituency D polled 5% of valid votes more than that of the first runner up. All the candidates who lost their security deposits while contesting for this constituency, put together, polled 35% of the valid votes.
How many candidates who contested in constituency B lost their security deposit? [This Question was asked as TITA]

• A. 9
• B. 13
• C. 8
• D. 11

Answer 1

The Correct Option is D

Constituency A Analysis

Total Votes Cast:

• Constituency A recorded a total of 500,000 valid votes.

Security Deposit Threshold:

• To retain their security deposit, a candidate needed to secure at least 1/6th of the total valid votes, calculated as \( \frac{1}{6} \times 500,000 = 83,334 \) votes.

Vote Distribution Among Top Candidates:

• Winner: 275,000 votes.
• 1st Runner-up: 95,000 votes (10,000 more than the 2nd runner-up).
• 2nd Runner-up: 85,000 votes.
• Combined votes for these three: 275,000 + 95,000 + 85,000 = 455,000 votes.

Votes for Other Candidates:

• The remaining candidates collectively received 500,000 - 455,000 = 45,000 votes.

Candidates Forfeiting Security Deposit:

• As the collective vote count for the remaining candidates (45,000) is less than the individual threshold (83,334), all these candidates forfeited their security deposits.

Vote Percentage Analysis:

• The proportion of votes obtained by candidates who lost their security deposit is \( \frac{45,000}{500,000} \times 100 = 9\% \).

Therefore, for constituency A:

• The percentage of votes secured by candidates who forfeited their security deposit is 9%.

Constituency C Analysis

Key Information:

• No candidate in constituency C lost their security deposit.
• The minimum vote difference between any two candidates was 10,000.

Vote Distribution Pattern:

• The 5th highest vote-getter must have polled more than \( \frac{600,030}{6} = 100,006 \) votes.
• Given the minimum 10,000 vote difference, the vote distribution is as follows:
  • Winner: 140,006 votes.
  • 1st Runner-up: 130,006 votes.
  • 2nd Runner-up: 120,006 votes.
  • 3rd Runner-up: 110,006 votes.
  • 4th Runner-up: 100,006 votes.
• The sum of votes for these top five candidates is 600,030.

Constituency D Information

1. Provided Data:
   • The total votes in constituency D are represented as 100x.
   • The winning candidate's vote count is given by \( 15x + 37,500 \).

Updated Summary Table

The following table summarizes the data:

Constituency	Total Votes	Winner Votes	1st Runner-Up Votes	2nd Runner-Up Votes	3rd Runner-Up Votes	4th Runner-Up Votes	5th Highest Votes	Total of Top 5 Votes
A	500,000	275,000	95,000	85,000	-	-	-	455,000
C	600,030	140,006	130,006	120,006	110,006	100,006	>100,006	600,030
D	100x	15x+37,500	-	-	-	-	-	-

Summary of Findings:

• Constituency A: 9% of the votes were cast for candidates who forfeited their security deposit.
• Constituency C: The table indicates the vote distribution, ensuring all candidates retained their security deposits.
• Constituency D: Insufficient information is available to provide a complete analysis.

In constituency B, the minimum votes required to secure the deposit was \( \frac{1}{6} \times 325,000 = 54,167 \). Since the winner received fewer than 54,167 votes, all other candidates also failed to meet this threshold and lost their security deposits. Consequently, 11 candidates lost their security deposits in constituency B.