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.

If Amiya runs a campaign focusing on issues, then what is the maximum percentage of votes that she can get?

The correct option is 48%.

To find the maximum percentage of votes that Amiya can get, let us analyze the campaign rules and the options available for both candidates. Since Amiya runs a campaign focusing on issues, her campaign type is fixed. Let us denote Amiya's campaign level as $L_A$ and Ramya's campaign level as $L_R$. The campaign levels can be either 1 (staid) or 2 (vigorous).

Ramya has two options for her campaign type: she can either focus on issues or attack Amiya. We will analyze the votes Amiya gets (as a percentage of the total class strength) under both scenarios:

Scenario 1: Both focus on issues
If both Amiya and Ramya focus on issues, the percentage of students voting in the election is:
$20\times (L_A+L_R)\%$
Among the voting students, the votes are proportional to the levels of their campaigns. Therefore, Amiya's share of the votes is:
$\frac{L_A}{L_A+L_R}$
Thus, the percentage of total class votes that Amiya receives is:
$20\times (L_A+L_R)\times \frac{L_A}{L_A+L_R}\%=20\times L_A\%$
For Amiya to maximize her votes in this scenario, she must run a vigorous campaign ($L_A=2$), which gives her:
$20\times 2\%=40\%$

Scenario 2: Ramya runs a campaign attacking Amiya
If Ramya runs an attacking campaign while Amiya focuses on issues, the voting percentages shift based on the "otherwise" scenario (the baseline where both would have focused on issues).
In the baseline case:
- Amiya would get $20\times L_A\%$ of the total votes.
- Ramya would get $20\times L_R\%$ of the total votes.
According to the rules, since Ramya attacks Amiya, 20% of the students who would have otherwise voted for Ramya will now vote for Amiya, and 5% of Ramya's baseline voters will not vote at all.
Therefore, Amiya's actual percentage of votes becomes:
$\text{Amiya's Votes}=\left(\text{Baseline for Amiya}\right)+20\%\text{of}\left(\text{Baseline for Ramya}\right)$
$\text{Amiya's Votes}=20L_A\%+0.20\times \left(20L_R\right)\%$
$\text{Amiya's Votes}=(20L_A+4L_R)\%$

To maximize this expression, we choose the maximum campaign levels for both candidates: $L_A=2$ and $L_R=2$.
Substituting these values into the equation:
$\text{Maximum Votes}=20\left(2\right)\%+4\left(2\right)\%=40\%+8\%=48\%$

Comparing both scenarios, the maximum percentage of votes Amiya can get is 48%.