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A, B and C are three parties which contests in local election (people vote for one of the three parties alone). A and B forms an alliance thinking that they will get 50% more votes than C due to which 20% of voters from party A vote for party C and 15000 votes from party B vote for party C. Now C gets 56 2/3% of the total vote casted whereas only 16 2/3% of the voters only voted for party B? What is the initial amount of vote bank that party A had before alliance?

A, B and C are three parties which contests in local election (people vote for one of the three parties alone). A and B forms an alliance thinking that they will get 50% more votes than C due to which 20% of voters from party A vote for party C and 15000 votes from party B vote for party C. Now C gets 56 2/3% of the total vote casted whereas only 16 2/3% of the voters only voted for party B? What is the initial amount of vote bank that party A had before alliance? Correct Answer 50000

Let the amount of original vote bank for party A, B and C be a, b and c respectively.

Due to alliance 20% of voters from A votes to party C and 15000 from party B to party C

So, Remaining vote for A = 80% of a = 0.8a

Remaining vote for B = b – 15000

Vote for C = c + 0.2a + 15000

% of vote of party C = ((c + 0.2a + 15000) / (a + b + c)) × 100

⇒ (170/3) = ((c + 0.2a + 15000) / (a + b + c)) × 100

⇒ 17/3 = ((c + 0.2a + 15000) / (a + b + c)) × 10

⇒ 17a + 17b + 17c = 30c + 6a + 450000

⇒ 11a + 17b = 13c + 450000     ---- 1

% vote by party B = ((b – 15000) / (a + b + c)) × 100

⇒ 16 2/3 = ((b – 15000) / (a + b + c)) × 100

⇒ 50/3 = ((b – 15000) / (a + b + c)) × 100

⇒ a + b + c = 6b – 90000

⇒ a + c = 5b – 90000      ---- 2

Given that (a + c) = 50% more than B

a + b = 1.5c

⇒ a = 1.5c – b

Substituting in equation 1 and 2,

16.5c – 11b + 17b = 13c + 450000

⇒ 3.5c + 6b = 450000     ---- 3

⇒ 1.5c – b + c = 5b – 90000

⇒ 6b – 2.5c = 90000      ---- 4

Subtracting the equations

6c = 360000

⇒ c = 60000

Substituting in equation 3

210000 + 6b = 450000

⇒ b = 40000

∴ a = 1.5c – b = 50000

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