Percentage question. People voting in an election.

Question

In an election, $10\%$ voters did not participate and $1200$ votes are found invalid. The winner gets $68\%$ of total voting list and he won by $56400$ votes. Find the votes polled in favor of losing candidate.

I can't understand what should I do with the number of invalid votes.

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Edit:

Can anybody tell why 1200 has been subtracted from 56400 in

$$46units=56400-1200$$

Shouldn't $$46 units =56400$$

Because 1200 votes are not counted in voting, how can it be a part of the votes by which the candidate won?

Answer 1

Let total number of voters be $x$. Then, we have:

Total votes = $\frac{9x}{10}$ and total number of valid votes = $\frac{9x}{10}-1200$

Because, the winner got $68\%$ votes of total list, the number of votes he got = $\frac{34x}{50}$ and thus, number of votes the loser got = $\frac{34x}{50}-56400$

Thus, $$\text{ Total valid votes } = \text{ Votes secured by winner } + \text{ Votes secured by loser } $$ $$\implies \frac{9x}{10}-1200=\frac{34x}{50}+\frac{34x}{50}-56400$$ $$\implies x= \, ?$$

Answer 2

Let $x$ be the number of voters on the list. The total number of votes is equal to the number of votes received by the winner plus the number of votes received by the loser plus the number of voters who did participate in the election plus the number of voters whose votes were invalidated.

The winner receives $0.68x$.

Since the loser receives $56~400$ fewer votes, the loser receives $0.68x - 56~400$.

The number of voters who did participate in the election is $0.1x$.

There were $1200$ invalid votes.

Hence, \begin{align*} 0.68x + 0.68x - 56~400 + 0.1x + 1200 & = x\\ 1.46x - 55~200 & = x\\ 0.46x & = 55~200\\ 46x & = 5~520~000\\ x & = 120~000 \end{align*} Therefore, the loser receives $$0.68(120~000) - 56~400 = 81~600 - 56~400 = 25~200$$ votes.