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.

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

The correct answer is 9%.

Let us systematically work through the data for Constituency A step by step.

Step 1: Identify the known data for Constituency A

• Number of candidates contesting: 10
• Total valid votes polled: 5,00,000
• Votes polled by winning candidate: 2,75,000
• Votes polled by first runner up: 95,000
• Additional fact 1: The first runner up polled 10,000 more votes than the second runner up.

Step 2: Find the votes polled by the second runner up

From Additional Fact 1:
Votes of second runner up = Votes of first runner up − 10,000

$=\mathrm{95,000}-\mathrm{10,000}=\mathrm{85,000}$

Step 3: Find the total votes polled by all remaining candidates (rank 4 through 10 — i.e., 7 candidates)

Total votes accounted for by top 3 (winner + first runner up + second runner up):

$\mathrm{2,75,000}+\mathrm{95,000}+\mathrm{85,000}=\mathrm{4,55,000}$

Votes polled by the remaining 7 candidates:

$\mathrm{5,00,000}-\mathrm{4,55,000}=\mathrm{45,000}$

Step 4: Determine the security deposit threshold

A defeated candidate loses their security deposit only if they fail to secure more than one-sixth of the total valid votes polled.

Security deposit threshold:

$\frac{1}{6}\times \mathrm{5,00,000}=\mathrm{83,333.33}$

So any defeated candidate who polled ≤ 83,333 votes loses their security deposit.

Step 5: Check who loses their security deposit

• Winning candidate (2,75,000 votes): Not a defeated candidate — does NOT lose deposit.
• First runner up (95,000 votes): 95,000 > 83,333 → Does NOT lose deposit.
• Second runner up (85,000 votes): 85,000 > 83,333 → Does NOT lose deposit.
• Remaining 7 candidates together polled only 45,000 votes in total. The maximum any single one of them could have polled is strictly less than 85,000 (since rankings are strict with no ties), and in fact the total for all 7 is only 45,000. Therefore, each of the 7 candidates individually polled far fewer than 83,333 votes. All 7 candidates lose their security deposits.

Step 6: Calculate the percentage of votes polled by all security-deposit-losers

Total votes by deposit-losing candidates = 45,000
Total valid votes in Constituency A = 5,00,000

$\text{Percentage}=\frac{\mathrm{45,000}}{\mathrm{5,00,000}}\times 100=9\%$

Therefore, the percentage of votes polled in total by all the candidates who lost their security deposits while contesting for Constituency A is 9%.