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 Ramya runs a campaign attacking Amiya, then what is the minimum percentage of votes that she is guaranteed to get?

The correct option is 15%.

To find the minimum percentage of votes that Ramya is guaranteed to get if she runs a campaign attacking Amiya, let us analyze the voting outcomes under different scenarios.

Let:
_A be the campaign level of Amiya (which can be 1 for a staid campaign or 2 for a vigorous campaign).
_R be the campaign level of Ramya (which can be 1 for a staid campaign or 2 for a vigorous campaign).

First, let us calculate the percentage of votes each candidate would get if they both focused on issues (the baseline scenario):
The total percentage of students voting is given by:
$$\text{Total votes cast}=20\times (L_A+L_R)\%$$ Among these voting students, the votes are divided proportionally to their campaign levels. Therefore, the baseline vote shares (as a percentage of the total student body) are:
$$\text{Baseline votes for Amiya}=\frac{L_A}{L_A+L_R}\times 20(L_A+L_R)\%=20L_A\%$$ $$\text{Baseline votes for Ramya}=\frac{L_R}{L_A+L_R}\times 20(L_A+L_R)\%=20L_R\%$$

Now, we analyze the cases when Ramya runs a campaign attacking Amiya. Amiya has two choices for her campaign type:

Case 1: Amiya runs a campaign focusing on issues
According to the rules, if Ramya attacks and Amiya focuses on issues, then:
- 20% of the students who would have otherwise voted for Ramya will vote for Amiya.
- 5% of the students who would have otherwise voted for Ramya will not vote at all.
Thus, Ramya loses a total of 25% of her baseline votes, meaning she retains:
$$\text{Ramya's vote percentage}=20L_R\%\times (1-0.20-0.05)=20L_R\%\times 0.75=15L_R\%$$ - If Ramya runs a staid campaign (_R = 1), she gets:
$$15\times 1\%=15\%$$ - If Ramya runs a vigorous campaign (_R = 2), she gets:
$$15\times 2\%=30\%$$

Case 2: Amiya also runs an attacking campaign
If both candidates run campaigns attacking each other, then 10% of the students who would have otherwise voted for them under the baseline scenario will not vote at all.
Therefore, Ramya retains 90% of her baseline votes:
$$\text{Ramya's vote percentage}=20L_R\%\times 0.90=18L_R\%$$ - If Ramya runs a staid campaign (_R = 1), she gets:
- If Ramya runs a vigorous campaign (_R = 2), she gets:
$$18\times 2\%=36\%$$

Comparing all possible outcomes when Ramya attacks, the percentage of votes she can get are 15%, 18%, 30%, or 36%. Therefore, the absolute minimum percentage of votes that she is guaranteed to get is 15% (which occurs when Ramya runs a staid campaign of level 1 and Amiya runs a campaign focusing on issues).