In a two-candidate election, 10% of the voters did not cast their ballots. 10% of the votes cast were found invalid. The winning candidate received 54% of the valid votes and a 1620-vote majority. Find the number of people on the voter list who have registered to vote.

Solution (By Examveda Team)

Given
10% of the voters did not cast their vote and 10% of the polled vote were found invalid
The winner candidate got 54% of the valid vote and beat the opposition by 1620 votes
Concept used:
The percentage is calculated as based on 100 i.e. 100 is the base
40% means 40 out of 100
Calculation:
Let, total enrolled voter be $$x$$
10% did not cast vote means casted or polled vote = $$\frac{{9x}}{{10}}$$
10% vote is invalid
That means valid vote = $$\frac{{90}}{{100}} \times \frac{{9x}}{{10}}$$
$$ \Rightarrow \frac{{81x}}{{100}}$$
The winner candidate got 54% of the polled vote means loosed got (100 - 54) = 46% vote
Winner candidate got total $$\left( {\frac{{54}}{{100}} \times \frac{{81x}}{{100}}} \right)$$   vote
And the looser candidate got $$\left( {\frac{{46}}{{100}} \times \frac{{81x}}{{100}}} \right)$$   vote
Accordingly,
$$\eqalign{ & \left( {\frac{{54}}{{100}} \times \frac{{81x}}{{100}}} \right) - \left( {\frac{{46}}{{100}} \times \frac{{81x}}{{100}}} \right) = 1620 \cr & \Rightarrow \left( {\frac{{81x}}{{100}}} \right) \times \left( {\frac{{54 - 46}}{{100}}} \right) = 1620 \cr & \Rightarrow \frac{{\left( {81 \times 8} \right)x}}{{10000}} = 1620 \cr & \Rightarrow x = \frac{{1620 \times 10000}}{{81 \times 8}} \cr & \Rightarrow x = 25000 \cr} $$
∴ Total 25000 voters registered in the voter list.