Logical Reasoning

Optimization

Hard

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 $20 \times(2+2) \%$ , 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 both of them run staid campaigns attacking the other, then what percentage of students will vote in the election?

$60\%$

$64\%$

$40\%$

$36\%$

✅ Correct Option: 4

Slide 1/4

Turnout Table

Amiya ↓ Ramya →	Staid	Vigorous
Staid	Turnout = 40% A, R = {½, ½}	Turnout = 60% A, R = {⅓, ⅔}
Vigorous	Turnout = 60% A, R = {⅔, ⅓}	Turnout = 80% A, R = {½, ½}

From the first two statements, we can make the following table :-

In this table first column shows type of campaign amiya chose and first row shows type of campaign ramya chose.

according to clue 1, percentage of students voting in the election will be 20 times the sum of the levels of campaigning of the two students

where level 1 = staid campaign

and, level 2 = vigorous campaign

therefore in the table - voter turnout will be - 20* (level of campaign of ramya + level of campaign of amiya)

therefore in the table for example if amiya had chose a staid campaign (level 1)

and Ramya also chose a staid campaign (level 1)

then their voter turnout will be = ( 20 \times (1 + 1) = 40% )

Turnout Table

Amiya ↓ Ramya →	Staid	Vigorous
Staid	Turnout = 40% A, R = {½, ½}	Turnout = 60% A, R = {⅓, ⅔}
Vigorous	Turnout = 60% A, R = {⅔, ⅓}	Turnout = 80% A, R = {½, ½}

A and R is the percentage of votes for each candidate which is proportional to the levels of their campaigns.

i.e., if amiya chose vigorous campaign (level 2) while ramya chose staid campaign (level 1)

Then, percentage of votes for each amiya will be 2/3 and for ramya will be 1/3.

Strategy Table

Amiya ↓ Ramya →	Attack	Issues
Attack	10% - A → No Vote 10% - R → No Vote	10% - A → R 10% - A → No Vote
Issues	20% - R → A 5% - R → No Vote	No change

This is the table we can form ,from the further information given in the question directly.

Forming a table like this will help us to rationalize and comprehend the data more easily.

Turnout Table

Amiya ↓ Ramya →	Staid	Vigorous
Staid	Turnout = 40% A, R = {½, ½}	Turnout = 60% A, R = {⅓, ⅔}
Vigorous	Turnout = 60% A, R = {⅔, ⅓}	Turnout = 80% A, R = {½, ½}

Strategy Table

Amiya ↓ Ramya →	Attack	Issues
Attack	10% - A → No Vote 10% - R → No Vote	10% - A → R 10% - A → No Vote
Issues	20% - R → A 5% - R → No Vote	No change

from the table above it is clear that if both Amiya and Ramya had run staid campaign, total votes turnout = 20 * (1 + 1)% = 40%.

Amiya receives ½ (40%) = 20% whereas Ramya also receives ½ (40%) = 20%.

Since both are attacking, 10% of the Amiya and Ramya votes don't turn up. Therefore, each will receive only 20% - (10% of 20%) = 18%.

Percentage of students who will vote in the election = 18% + 18% = 36%.

Question 19

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

Question 20

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

can get?

Question 21

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

Question 22

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

If both of them run staid campaigns attacking the other, then what percentage of students will vote in the election?

Solution

Turnout Table

Turnout Table

Strategy Table

Turnout Table

Strategy Table

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