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

Total valid votes in A=3,25,000
Minimum no of valid votes required to save the security deposits = 1/6×325,000 = 54167
We can see the winner himself/herself has got 48,750 no of votes which is less than 1/6 th of the total votes. 
∴ all of the candidates got less than 1/6 th of the valid votes.
∴ security deposits will be forfeited for all of the candidates as winner is exempted from this condition. (total no of such candidates = 12-1 = 11).

So, the correct option is (D): 11

Answer 2

For Constituency A

Total Valid Votes:

• Total valid votes in constituency A = 500,000.

Minimum Votes to Save Security Deposit:

• Minimum number of valid votes required to save the security deposit = 16×500,000=83,33461​×500,000=83,334.

Votes Distribution:

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

Remaining Votes:

• Total valid votes got by the remaining candidates = 500,000 - 455,000 = 45,000.

Candidates Losing Security Deposit:

• All the remaining candidates together received 45,000 votes, which is less than 83,334 votes each, so they lost their security deposits.

Percentage Calculation:

• The percentage of votes received by the candidates who lost their security deposit = 45,000500,000×100=9%500,00045,000​×100=9%.

So, the correct answer for constituency A is:

• The percentage of votes received by candidates who lost their security deposit is 9%.

For Constituency C

Information:

• None of the candidates in constituency C lost their security deposit.
• The difference in votes polled by any pair of candidates is at least 10,000.

Vote Distribution:

• The 5th highest vote-getter must have more than 600,0306=100,0066600,030​=100,006 votes.
• Given the difference between votes is at least 10,000, the only possible case is:
  • 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.
• These votes sum up to exactly 600,030.

For Constituency D

1. Information:
   • The total number of votes in constituency D is represented as 100x.
   • The winning candidate must have received 15𝑥+37,50015x+37,500 votes.

Updated Table Summary

Here's a summary table based on the information provided:

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	-	-	-	-	-	-

Conclusion:

• For constituency A, the percentage of votes received by candidates who lost their security deposit is 9%.
• For constituency C, the vote distribution is given in the table, ensuring all candidates kept their security deposits.
• For constituency D, further information is needed to complete the details.

In constituency B, the threshold for keeping the security deposit is 16×325,00061​×325,000, which equals 54,167 votes. However, since the winner received fewer than 54,167 votes, it means all the other candidates also lost their security deposits. Therefore, the correct answer is that 11 candidates lost their security deposits.