Percentage question on voting

Question

In an election, 2 candidates participated.10% did not vote. 300 votes were declared invalid and the winner gets 60% of voting list and wins by 900 votes. Find no of valid votes.

Ans. 1500

What I tried:

Winner=60%;Not voted=10%;=>Loser=30%

ATQ:

(60-30)%=30%=900

=> Valid votes=90%-300$=\frac{90}{30}*900-300=2700-300=2400$

Where am I wrong?

Answer 1

Winner get 60% of the voting list and not total.

So it total votes are T, valid votes are T-300, and winner got 60% of that ie 0.6*(T-300)

Hope it clears your doubt.

Answer 2

Let $T$ be the total number eligible.

Votes cast: $(9/10)T.$

Valid votes: $(9/10)T-300$.

Winner gets : $(6/10)[(9/10)T-300]$.

Wins by $900$ votes:

$(6/10)[(9/10)T-300] =$

$450 +(1/2)[(9/10)T-300]$;

$(1/10)[9/10T-300] =450$;

$9/10T - 300 = 4500.$

Valid votes: $(9/10)T- 300= 4500.$

Answer 3

Problem:

In an election, 2 candidates participated.10% did not vote. 300 votes were declared invalid and the winner gets 60% of voting list and wins by 900 votes. Find no of valid votes. (answer: 1500)

What exactly do they mean by "wins by 900 votes"? 60% of 1500 votes is 900 votes. 40% of 1500 votes is 600 votes. The phrase "wins by 900 votes" in English is typically taken to mean that somebody gets 900 votes more than somebody else. Is 900 votes the winner in your problem had received more than 600 votes the loser had received by 900 votes? It definitely doesn't look that way. The difference is 300 votes while it should be 900 votes according to the problem statement. Do you see the problem? 1500 can't be the answer.

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Here's my solution:

Let $N$ be the total number of people eligible to vote. We know that $10\%$ of them did not vote. This means that $90\%$ $\left(\frac{90}{100}=0.9\right)$ of them did vote and 300 of their votes were declared invalid. Thus, the total number of valid votes is equal to $0.9N-300$.

We also know that the number of people that comprise the $60\%$ of the total number of valid votes is $900$ greater than the number of people that comprise the $40\%$ of the number of valid votes. This statement can be expressed like this: $0.6(0.9N-300)=0.4(0.9N-300)+900$.

All we have to do now is to solve this equation for $N$ (the total number of people eligible to vote) and with the resulting $N$ we will be able to find the number of valid votes: $0.9N-300$.

$$ 0.6(0.9N-300)=0.4(0.9N-300)+900\implies\\ 0.18N=960\implies\\ N=\frac{960}{0.18} $$

The number of valid votes:

$$ 0.9N-300=0.9\cdot\frac{960}{0.18}-300=4500 $$

Answer: $4500$.