At an election involving two candidates, $68$ votes were declared invalid. The winning candidate secures $52\%$ and wins by $98$ votes. The total number of votes polled is

### Concept & Logic In an election with two candidates, the total number of valid votes represents $100\%$. The margin of victory is the difference in the percentage of valid votes secured by the winner and the loser. The total votes polled includes both valid and invalid votes. $$ \text{Margin} = \text{Winner's \%} - \text{Loser's \%} $$ $$ \text{Total Polled} = \text{Valid Votes} + \text{Invalid Votes} $$ ### Step-by-Step Solution * **Given:** * Winner's share of valid votes = $52\%$ * Winning margin = $98$ votes * Invalid votes = $68$ * **Deduction:** * Since the winner secured $52\%$ of the valid votes, the loser secured the remaining: $100\% - 52\% = 48\%$ of the valid votes. * The margin of victory in percentage terms is: $52\% - 48\% = 4\%$ * This $4\%$ difference represents the $98$ vote margin. * Let the total valid votes be $V$. * $4\%$ of $V = 98$ * $V = \frac{98 \times 100}{4} = 98 \times 25 = 2450$ valid votes. * The total number of votes polled is the sum of valid and invalid votes: $\text{Total Polled} = 2450 + 68 = 2518$ ### Exam Strategy & Shortcut Use proportional reasoning for the valid votes. If $4\% = 98$, you can find $100\%$ by multiplying by $25$. $98 \times 25$ is easily calculated as $98 \times \frac{100}{4} = 2450$. Then, simply add the $68$ invalid votes: $2450 + 68 = 2518$. ### Common Pitfall The most common mistake is stopping after calculating the valid votes ($2450$) and selecting it as the final answer (Option b). Always re-read the final question line: it asks for the "total number of votes polled," which must include the invalid votes. ### Final Answer **Therefore, the correct answer is 2518.**