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 maximum possible voting margin with which one of the candidates can win?

The correct answer is 29%.

To find the maximum possible voting margin with which one of the candidates can win, we need to analyze all possible combinations of campaign levels and campaign types for Amiya and Ramya.

Let:
• $L_A$ be Amiya's campaign level ($L_A\in \{1,2\}$, where 1 is staid and 2 is vigorous).
• $L_R$ be Ramya's campaign level ($L_R\in \{1,2\}$, where 1 is staid and 2 is vigorous).
• $T_A$ and $T_R$ be the types of campaigns run by Amiya and Ramya respectively (focusing on issues, or attacking the opponent).

Step 1: Determine Baseline Votes (When both focus on issues)
If both candidates focus on issues, the total voter turnout as a percentage of all class students is:
$V_{\text{base}}=20\times (L_A+L_R)\%$

Among those who vote, the baseline votes received by Amiya ($v_A$) and Ramya ($v_R$) as a percentage of total class students are:
$v_A=\frac{L_A}{L_A+L_R}\times V_{\text{base}}=20\times L_A\%$
$v_R=\frac{L_R}{L_A+L_R}\times V_{\text{base}}=20\times L_R\%$

Thus, the baseline voting percentages depending on campaign levels are:
1. If both run staid campaigns ($L_A=1,L_R=1$):
$v_A=20\%$, $v_R=20\%$
2. If Amiya is vigorous and Ramya is staid ($L_A=2,L_R=1$):
$v_A=40\%$, $v_R=20\%$
3. If Amiya is staid and Ramya is vigorous ($L_A=1,L_R=2$):
$v_A=20\%$, $v_R=40\%$
4. If both run vigorous campaigns ($L_A=2,L_R=2$):
$v_A=40\%$, $v_R=40\%$

Step 2: Analyze Scenarios with Attacking Campaigns
Let us evaluate the final votes ($V_A$ and $V_R$) and the margin under different campaign strategies:

Case A: Ramya attacks Amiya, and Amiya focuses on issues
According to the rules, 20% of Ramya's baseline voters switch to Amiya, and 5% of Ramya's baseline voters do not vote at all.
Final votes for Ramya: $V_R=v_R\times (1-0.20-0.05)=0.75\times v_R$
Final votes for Amiya: $V_A=v_A+0.20\times v_R$
Let's compute the margin for the level configurations:
• If $L_A=2$ (Amiya vigorous) and $L_R=1$ (Ramya staid):
Baseline: $v_A=40\%$ and $v_R=20\%$
Final votes for Ramya: $V_R=0.75\times 20\%=15\%$
Final votes for Amiya: $V_A=40\%+(0.20\times 20\%)=44\%$
Voting Margin for Amiya: $V_A-V_R=44\%-15\%=29\%$

Case B: Amiya attacks Ramya, and Ramya focuses on issues
Here, 10% of Amiya's baseline voters switch to Ramya, and 10% do not vote.
Final votes for Amiya: $V_A=0.80\times v_A$
Final votes for Ramya: $V_R=v_R+0.10\times v_A$
• If $L_A=1$ (Amiya staid) and $L_R=2$ (Ramya vigorous):
Baseline: $v_A=20\%$ and $v_R=40\%$
Final votes for Amiya: $V_A=0.80\times 20\%=16\%$
Final votes for Ramya: $V_R=40\%+(0.10\times 20\%)=42\%$
Voting Margin for Ramya: $V_R-V_A=42\%-16\%=26\%$

Case C: Both run campaigns attacking each other
Here, 10% of the baseline voters for each candidate will not vote at all.
Final votes: $V_A=0.9\times v_A$ and $V_R=0.9\times v_R$
The maximum difference in baseline votes is 20% (when one is 40% and the other is 20%).
Voting Margin: $0.9\times (40\%-20\%)=18\%$

Conclusion:
Comparing the outcomes, the maximum possible voting margin is 29%, which occurs when Amiya runs a vigorous campaign focusing on issues while Ramya runs a staid campaign attacking Amiya.