Directions (Q.42-Q.46): Two students, Amiya and Ramya are the only candidates in an election for the position of class representative. Students will vote based on the intensity level of Amiya’s and Ramya’s campaigns and the type of campaigns they run. Each campaign is said to have a level of 1 if it is a staid campaign and a level of 2 if it is a vigorous campaign. Campaigns can be of two types, they can either focus on issues, or on attacking the other candidate.

If Amiya and Ramya both run campaigns focusing on issues, the

• The percentage of students voting in the election will be 20 times the sum of the levels of campaigning of the two students. For example, if Amiya and Ramya both run vigorous campaigns, then 20 × (2+2)%, that is, 80% of the students will vote in the election.
• Among voting students, the percentage of votes for each candidate will be proportional to the levels of their campaigns. For example, if Amiya runs a staid (i.e., level 1) campaign while Ramya runs a vigorous (i.e., level 2) campaign, then Amiya will receive 1/3 of the votes cast, and Ramya will receive the other 2/3.
• The above-mentioned percentages change as follows if at least one of them runs a campaign attacking their opponent.
• If Amiya runs a campaign attacking Ramya and Ramya runs a campaign focusing on issues, then 10% of the students who would have otherwise voted for Amiya will vote for Ramya, and another 10% who would have otherwise voted for Amiya, will not vote at all.
• If Ramya runs a campaign attacking Amiya and Amiya runs a campaign focusing on issues, then 20% of the students who would have otherwise voted for Ramya will vote for Amiya, and another 5% who would have otherwise voted for Ramya, will not vote at all.
• If both run campaigns attacking each other, then 10% of the students who would have otherwise voted for them had they run campaigns focusing on issues, will not vote at all.

What is the minimum percentage of students who will vote in the election?

The correct option is 36%.

To find the minimum percentage of students who will vote in the election, we need to analyze the different combinations of campaign types and levels for the two candidates, Amiya ($A$) and Ramya ($R$).

1. Campaign Parameters:
Let the campaign level of Amiya be $L_A$ and Ramya be $L_R$.
Each campaign level can be:

• Staid: Level = 1
• Vigorous: Level = 2

Thus, the minimum sum of campaign levels is:
$L_A+L_R=1+1=2$
And the maximum sum of campaign levels is:
$L_A+L_R=2+2=4$

2. Base Voter Turnout (When both focus on issues):
The percentage of voting students is given as 20 times the sum of their campaign levels:
$\text{Base Turnout}=20\times (L_A+L_R)\%$
Also, the base vote share for Amiya is proportional to $L_A$ and for Ramya is proportional to $L_R$:

• Base votes for Amiya ($V_A$) = $20\times L_A\%$ of total students
• Base votes for Ramya ($V_R$) = $20\times L_R\%$ of total students

3. Analyzing Different Campaign Scenarios for Voter Turnout:

Scenario A: Both run issue-focused campaigns
The voter turnout depends only on the campaign levels:

• If both run staid campaigns ($L_A=1,L_R=1$):
  $\text{Turnout}=20\times (1+1)=40\%$
  
• If one is staid and the other is vigorous ($L_A+L_R=3$):
  $\text{Turnout}=20\times 3=60\%$
  
• If both run vigorous campaigns ($L_A=2,L_R=2$):
  $\text{Turnout}=20\times (2+2)=80\%$
  

The minimum turnout in this scenario is 40%.

Scenario B: Amiya attacks and Ramya focuses on issues
Here, 10% of those who would have voted for Amiya do not vote at all:
$\text{Non-voters}=10\%\text{of}{V}_A=0.10\times 20L_A\%=2L_A\%$
Thus, the new turnout is:
$\text{Turnout}=20(L_A+L_R)-2L_A$
For minimum turnout, we set both campaign levels to the minimum ($L_A=1,L_R=1$):
$\text{Turnout}=20(1+1)-2\left(1\right)=40-2=38\%$

Scenario C: Ramya attacks and Amiya focuses on issues
Here, 5% of those who would have voted for Ramya do not vote at all:
$\text{Non-voters}=5\%\text{of}{V}_R=0.05\times 20L_R\%=1L_R\%$
Thus, the new turnout is:
$\text{Turnout}=20(L_A+L_R)-1L_R$
For minimum turnout, we set both campaign levels to the minimum ($L_A=1,L_R=1$):
$\text{Turnout}=20(1+1)-1\left(1\right)=40-1=39\%$

Scenario D: Both candidates run campaigns attacking each other
In this case, 10% of the students who would have otherwise voted for either candidate do not vote at all. This means 10% of the total base voter turnout is lost:
$\text{Turnout}=90\%\text{of Base Turnout}=0.90\times 20(L_A+L_R)\%=18(L_A+L_R)\%$
To minimize this turnout, we choose the lowest possible campaign levels ($L_A=1,L_R=1$):
$\text{Turnout}=18\times (1+1)\%=18\times 2\%=36\%$

Conclusion:
Comparing the minimum turnout values across all scenarios:

• Scenario A: 40%
• Scenario B: 38%
• Scenario C: 39%
• Scenario D: 36%

The absolute minimum percentage of students who will vote in the election is 36% (occurring when both run staid campaigns and attack each other).