Question

What is the percentage of votes polled in total by all the candidates who lost their security deposits while contesting for constituency A?

• A. 1,40,006
• B. less than 2,00,010
• C. 1,40,010
• D. between 1,40,005 and 1,40,010
• A. 1,75,000
• B. 1,50,000
• C. 1,25,000
• D. 62,500
• A. B, D, C, A
• B. D, B, C, A
• C. B, C, D, A
• D. D, C, B, A
• A. 23.91%
• B. 23.54%
• C. 32.00%
• D. 38.25%

Answer 1

The Correct Option is A

Total valid votes in \(A = 5,00,000\) 

Minimum no of valid votes required to save the security deposits = \(\frac{1}{6}×500000 = 83334\)

As per \(1^{st}\) additional information ⇒ the gap between \(1^{st}\) and \(2^{nd}\) runners up is \(10,000\) 

So the table for \(A⇒ total ⇒ 5,00,000\)

Winner \(→ 275000\)

\(1^{st}\) runner up \(→ 55000\)

\(2^{nd}\) runner up \(→ 85000\)

Total \(→ 455000\)

The valid votes got by the other \(7\) candidates = \(5,00,000 - 4,55,000\)

= \(45000\)

\(∴\)  the % of votes by the candidates who lost the security deposit is = \(\frac{45000}{500000}×100 = 9\%\)

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.

This table provides a clear and simple overview of the vote distribution and calculations for each constituency based on the given information.
So the correct answer is option (B): 9%

Answer 3

Total valid votes in \(A=3,25,000\)

Minimum no of valid votes required to save the security deposits = \(\frac{1}{6}×325,000 = 54167\)

We can see the winner himself/herself has got \(48,750\) no of votes which is less than \(\frac{1}{6}\)th of the total votes. 

\(∴\) all of the candidates got less than \(\frac{1}{6}\)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 = \(12-1 = 11\)).

Answer 4

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.

Answer 5

The correct answer is (A): \(1,40,006\)

As the additional information in condition (2)

The minimum difference in no of votes between any pair of candidates = \(10,000\) 

As there are \(5\) candidates in c, the possible distribution of valid is as follows.

Candidates	\(C_1\)	\(C_2\)	\(C_3\)	\(C_4\)	\(C_5\)	\(Total\)
	1,40,000	1,30,000	1,20,000	1,10,000	1,00,000	6,00,000

Additional \(30\) votes can be distributed among them to maintain the gap at least \(10,000\). 

So the possible answer is \(1,40,006\).

Answer 6

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	-	-	-	-	-	-

Number of votes polled to winning candidate must be 140006.

Answer 7

The correct answer is (A): \(1,75,000\)

Let the no of valid votes = \(100x\)

The table looks like the following:

Total votes → \(100x\)

Winner → \(39375\)

\(1^{st}\) → \(37,500\)

\(2^{nd}\) → \(30,000\)

\(3^{rd}\) → \(10x\)

Total → \(1,06,875\)

As the winner got \(51\) more than \(1^{st}\) runner up (as per \(3\))

Total no of votes has to be more than

= \(\frac{39375 + 37,500 + 30,000}{0.9}\)

= \(1,10,208\)

∴ option \((62,500)\) is ruled out.

To eliminate two more options, lets consider \(1,50,000\) to be the correct one [as one option is less than the other one is the greater than this option]

\(∴ 100x = 1,50,000\)

Total → \(1,50,000\)

Winner → \(45,000\)

\(1^{st}\) → \(37500\)

\(2^{nd}\) → \(30,000\)

\(3^{rd}\) → \(15,000\)

Total → \(1,06,875\)

Minimum rates to save security deposits = \(\frac{1}{6}×150000 = 25000\)

So we can see except 3 candidates all others have lost their security deposits.

Candidates who saved their security deposits = \((100-35)\% = 65\%\) [ as per \(3^{rd}\) additional inform]

\(∴ 65\% \;of\; 150000 = 97500\)

→ which is less than \(1,12,500\)

\(∴\) answer will be more than \(1,50,000\)

Answer 8

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	-	-	-	-	-	-

In constituency B, the requirement for not losing the security deposit is to get at least 16.67% of the votes. Therefore, the third runner-up certainly did not keep their deposit.

Candidates who kept their security deposit collectively received 65% of the votes.

Case 1:

If the top three candidates kept their security deposit, then the vote distribution is:

• Winner: 37,500 votes + 5x
• 1st runner-up: 37,500 votes
• 2nd runner-up: 30,000 votes
• Total votes: 65x

Case 2:

If the top two candidates kept their security deposit, then the vote distribution is:

• Winner: 37,500 votes + 5x
• 1st runner-up: 37,500 votes
• Total votes: 65x
• This case is invalid because the second runner-up would need to have gotten the deposit, conflicting with the initial condition.

Since the correct allocation of votes does not align with the increasing order of A and D, which should follow the condition but does not in the first option, the first option is the correct answer.

Answer 9

The correct answer is (C): B,C,D,A

Margins

A → \(2,75,000 - 95000 = 180000\)

B →

C → \(1,50,000 - 1, 40,000 = 10,000\)

D → \(39,375 - 37,500 = 1,875\)

Clearly the margin in C is more than that of D. 

So clearly the sequence B , C , D , A is not possible.

Answer 10

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	-	-	-	-	-	-

As calculated previously, candidate D received 175,000 votes. The winner received 5% more votes than the first runner-up, which means the winner got 8,750 more votes than the first runner-up (5% of 175,000 = 8,750). Thus, the winning margin in constituency D is 8,750 votes.

Additionally, the margin in constituency C is at least 10,000 votes.

Therefore, in the increasing order of winning margins, C should always come after D. Since this order is not followed in the third option, the third option is the correct answer.

Answer 11

The correct answer is (A): \(23.91\%\)

The no. of votes got by the candidates who has lost their security deposits is as follows

In \(A →\) \(45000\)

In \(B → 325000 - 48750 = 276250\)

In \(C → 0\)

In \(D → 175000 - 46250 - 37500 - 30000 = 61250\)

\(Req\% = \frac{382500}{500000+325000+600000+175000}×100\)

= \(\frac{382500}{1600000}×100 = 23.91\%\)

Answer 12

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	-	-	-	-	-	-

Understanding the Security Deposit Condition:

• Candidates who didn't lose the deposit must have gotten more than 16.67% of the votes.
• Therefore, the 3rd runner-up must not have kept their deposit.
• Candidates who kept their security deposit collectively received 65% of the total votes.

Case Analysis for Security Deposits:

Case 1:

• If the top three candidates kept their security deposit:

37,500+5𝑥+37,500+30,000=65𝑥37,500+5x+37,500+30,000=65x

• Simplifying:

105,000=65𝑥105,000=65x

• Solving for 𝑥x:

𝑥=1,750x=1,750

• Therefore:

100𝑥=175,000100x=175,000

Case 2:

• If only the top two candidates kept their security deposit:

37,500+5𝑥+37,500=65𝑥37,500+5x+37,500=65x

• Simplifying:

75,000=60𝑥75,000=60x

• Solving for 𝑥x:

𝑥=1,250x=1,250

• But 16.67% of 125,000 = 20,834, which means the 2nd runner-up must have gotten the security deposit. Therefore, this case is not valid.

1. Analyzing Constituencies for Candidates Losing Their Security Deposit:

Constituency A: Total votes = 500,000

• Votes not losing deposit: 275,000 + 95,000 + 85,000 = 455,000
• Votes losing deposit: 500,000 - 455,000 = 45,000

Constituency B: Total votes = 325,000

• Votes not losing deposit: 48,750 (the winner got less than 54,167)
• Votes losing deposit: 325,000 - 48,750 = 276,250

Constituency C: Total votes = 600,030

• None of the candidates lost their deposit, so losing deposit votes = 0

Constituency D: Total votes = 175,000

• Breakdown:
  • Winner = 37,500
  • 1st runner-up = 5x (x = 1,750)
  • Total votes: 37,500 + 8,750 + 30,000 = 76,250
  • Votes not losing deposit: 175,000 - 76,250 = 98,750
  • Votes losing deposit: 175,000 - 76,250 = 98,750

1. Calculating Total and Percentage:

Total votes losing deposit: 45,000 (A) + 276,250 (B) + 0 (C) + 98,750 (D) = 420,000

Total votes across all constituencies: 500,000 (A) + 325,000 (B) + 600,030 (C) + 175,000 (D) = 1,600,030

Percentage of votes losing deposit:

420,0001,600,030×100=26.25%1,600,030420,000​×100=26.25%

So, the percentage of votes polled by candidates who lost their security deposit across all constituencies is approximately 26.25%.