CAT 2024 Slot 1 — DILR Question 21

Mixed PracticeEasy

Passage / Data

Answer the following questions based on the information given below.

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, then

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?

Answer & solution

A
12%
B
18%
15%
D
30%

Solution

Hard

Ramya attacks Amiya. Her guaranteed minimum comes from the worst case for her: Amiya stays on issues (triggering the 20%+5% leakage from Ramya) and Ramya is at the lowest level.

Rule. When Ramya attacks Amiya and Amiya focuses on issues, 20% of Ramya’s would-be voters switch to Amiya and another 5% stop voting — Ramya keeps 75%. Her issues baseline is $20\,\ell_R$ .

Ramya’s baseline. In the issues split she would have $20\,\ell_R\%$ .

\begin{aligned} &\text{Ramya baseline}=20\,\ell_R \end{aligned}

Worst case: Amiya on issues. Then Ramya loses 20%+5% = 25% of her base, keeping 75%. (If Amiya also attacked, Ramya would keep 90% — more — so the issues case is the worst for her.)

\begin{aligned} &\text{Ramya keeps}=0.75\times20\,\ell_R=15\,\ell_R \end{aligned}

Minimise over Ramya’s level. Smallest at $\ell_R=1$ .

\begin{aligned} &\text{Guaranteed minimum}=15\times1=15\% \end{aligned}

15\%