In a local municipal election involving two contestants, 1,200 ballots were declared invalid. The winning candidate secured 75% of all valid votes, beating the opponent by a difference of 500 votes. Find the total number of votes polled in the election.

The correct answer is 2,200.

Step 1: Understand the given information
Let us break down the information provided in the question:
• Number of invalid votes = 1,200
• Percentage of valid votes won by the winning candidate = 75%
• Winning margin (difference in votes between winner and opponent) = 500 votes

Step 2: Find the opponent's percentage of valid votes
The total valid votes represent 100% of all valid votes. Since the winning candidate secured 75% of these valid votes, the percentage of valid votes secured by the opponent is:

$100\%-75\%=25\%$

Step 3: Calculate the percentage difference between the two candidates
The difference in percentage of valid votes between the winner and the opponent is:

$75\%-25\%=50\%$

This means the winner secured 50% more of the total valid votes than the opponent.

Step 4: Calculate the total number of valid votes
We are given that this 50% difference corresponds to 500 votes.
Let $V$ represent the total number of valid votes.

$50\%\text{of}V=500$

$\frac{50}{100}\times V=500$

$0.5\times V=500$

$V=\frac{500}{0.5}=\mathrm{1,000}$

Therefore, the total number of valid votes cast in the election is 1,000.

Step 5: Calculate the total number of votes polled
The total number of votes polled in an election includes both valid votes and invalid votes:

$\text{Total Votes Polled}=\text{Valid Votes}+\text{Invalid Votes}$

$\text{Total Votes Polled}=\mathrm{1,000}+\mathrm{1,200}=\mathrm{2,200}$

Hence, the total number of votes polled in the election is 2,200.