In an election between two candidates, $10 %$ of the voters did not cast their vote. $10 %$ of the votes polled were found invalid. The successful candidates got $54 %$ of the valid votes and won by a majority of $1620$ votes. Find the number of votes enrolled on the voter's list.

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Solution

Let the total voters enrolled be $x$ .
Given: Percentage of vote not casted $= 10 %$
Percentage of invalid votes from casted ones $= 10 %$
Successful candidates got $54 %$ from valid casted votes and win by a majority of $1620$ votes.
No. of casted votes $= (100 - 10) % \times x = 0.9 x$
No. of votes valid $= (100 - 10) % \times 0.9 x = 0.81 x$
The successful candidate got $= 54 %$ of number of valid votes
So, the unsuccessful candidate will get $= 46 %$ of the number of valid votes
Number of votes successful candidate got - Number of votes Unsuccessful candidate got $= 1620$
$∴ 54 %$ of $0.81 x - 46 %$ of $0.81 x = 1620$
$⟹ 0.54 \times 0.81 x - .46 \times 0.81 x = 1620$
$⟹ (0.54 - 0.46) \times 0.81 x = 1620$
$⟹ x = \frac{1620}{0.81 \times 0.08}$
$⟹ x = 25000$
Hence, total no.of voters enrolled $= 25000$

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