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

### Concept & Strategy This is a classic successive decrement problem combined with a percentage margin. Treat the total enrolled voters as an unknown variable (or $100$ units) and deduct the non-voters and invalid votes step-by-step to reach the valid votes base. $$ \text{Valid Votes} = \text{Total} \times (1 - \text{uncast \%}) \times (1 - \text{invalid \%}) $$ ### Step-by-Step Solution * **Given:** * Did not cast vote = $10\%$ * Invalid votes = $10\%$ of polled * Winner's share = $54\%$ of valid votes * Majority (margin) = $1620$ votes * **Deduction:** * Let the total enrolled voters be $x$. * Votes cast (polled) = $x - 10\% \text{ of } x = 0.9x$. * Valid votes = $0.9x - 10\% \text{ of } 0.9x = 0.9x \times 0.9 = 0.81x$. * The winner got $54\%$ of the valid votes, so the loser got $100\% - 54\% = 46\%$ of the valid votes. * The margin of victory is $54\% - 46\% = 8\%$ of the valid votes. * Equating the margin to the given value: $8\% \text{ of } (0.81x) = 1620$ $\frac{8}{100} \times 0.81x = 1620$ $x = \frac{1620 \times 100}{8 \times 0.81} = \frac{1620 \times 100}{6.48}$ $x = 25000$ ### Exam Strategy & Shortcut Use the "Assume 1000" method for cascading percentages. Assume total enrolled = $1000$. Polled ($90\%$) = $900$. Valid ($90\%$ of $900$) = $810$. Margin = $8\%$ of valid = $0.08 \times 810 = 64.8$ units. If $64.8$ units represent $1620$ actual votes, the multiplier is $\frac{1620}{64.8} = 25$. Total enrolled = $1000 \times 25 = 25000$. ### Common Pitfall The most frequent mistake is assuming the $1620$ majority represents $54\%$ of the valid votes rather than the $8\%$ difference between the candidates. Always remember that "won by a majority of" refers to the margin over the runner-up. ### Final Answer **Therefore, the correct answer is 25000.**