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.
| 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.
Question 1:
What is the percentage of votes polled in total by all the candidates who lost their security deposits while contesting for constituency A?
EXPLANATION
ANS 9
Let’s try to fill the table first. Partially applying points 1 and 3 we can get the votes for the 2nd runner up in A and the winning candidate in D.
To begin answering this question, we need to first see what is the% of total votes between the candidates we already know about: Winner + 1st + 2nd Runner Up total: 455,000 (91%)
So the vote share among the remaining 7 candidates was only 9%
To lose security deposit, one must get less than or equal to 1/6th i.e. lesser than 16.67% of the total votes. Clearly that is indeed the case with the remaining candidates in A.
Question 2:
How many candidates who contested in constituency B lost their security deposit?
EXPLANATION
The winning candidate in B had a 15% vote share. Meaning everyone else (the other 11) had a less than 16.67% vote share as well so they all will lose their security deposits. Hence, the answer is, ” 11″
Question 3:
What BEST can be concluded about the number of votes polled by the winning candidate in constituency C?
- 1,40,006
- less than 2,00,010
- 1,40,010
- between 1,40,005 and 1,40,010
EXPLANATION
It is mentioned that none in C lost their security deposit and the delta in votes was at least 10,000
The least possible vote share is slightly more than 1/6th, meaning the lowest possible value here is: 600,030 / 6 +1 = 100,005 + 1. If noone lost the security deposit, then the least possible value is 100,006
Assuming a 100,00 vote delta just for the time being we get the following as our possible values
Clearly given the constraints, there isn’t too much we can do to flex the numbers here.This is the only possible way to satisfy the constraints. So this must be the answer.
Question 4:
What was the number of valid votes polled in constituency D?
- 1,75,000
- 1,50,000
- 1,25,000
- 62,500
EXPLANATION
This is how the table looks for D as of now:
Check using the options. Clearly it cannot be 62,500 (Option C) because the total of 2, 3 adds up to a number greater than that
A: 175,000
The table looks like the following:
In this option, clearly the top 3 each have more than 16.67% and the rest have 35%. This option satisfies the constraints and hence is the correct answer is, “1,75,000”.
Question 5:
The winning margin of a constituency is defined as the difference of votes polled by the winner and that of the first runner up. Which of the following CANNOT be the list of constituencies, in increasing order of winning margin?
- B, D, C, A
- D, B, C, A
- B, C, D, A
- D, C, B, A
EXPLANATION
A clearly has the highest margin
With B, margin could be as low as possible since everyone lost their deposit. Highest could be as follows: assume that the first 11 people got 1 – 11 votes only (since there are no ties, we are increasing the numbers by 1 for each position). So the winning margin can be as high as ~48,000
C it is 10,000 as calculated earlier
D is less than 10,000 for sure since it is 5% of the valid votes
So D cannot be more than C. The only option which has D more than C is the last one
The answer is, “B, C, D, A
Question 6:
For all the four constituencies taken together, what was the approximate number of votes polled by all the candidates who lost their security deposit expressed as a percentage of the total valid votes from these four constituencies?
- 23.91%
- 23.54%
- 32.00%
- 38.25%
EXPLANATION
Just calc using available data
For A: Votes that count towards a deposit loss=> 500,000 – (275,000 – 95,000 – 85,000) = 45,000
For B: All except the winner lost, i.e. =>325,000 – 48,750 = 276,250
For C: Noone lost
For D: 35% of the total i.e. 35% of 175,000 => 61,250
Answer: (45,000 + 276,250 + 61,250) / (500,000 + 325,000 + 600,003 + 175,000)
