In an election between two candidates, 86% of the registered voters cast their votes and 6% of the votes polled were found invalid. The winning candidate got 58% of the valid votes and won the election by a margin of 202100 votes. How many voters were registered?

The correct option is 1562500.

Let us solve the problem step-by-step by defining the variables and relations given in the question.

Step 1: Define the total registered voters
Let the total number of registered voters be represented by:

$V$

Step 2: Calculate the number of votes polled
According to the question, 86% of the registered voters cast their votes. Therefore, the number of polled votes is:

$\text{Polled Votes}=86\%\text{of}V=0.86V$

Step 3: Calculate the number of valid votes
Since 6% of the polled votes were found to be invalid, the percentage of valid votes is:

$100\%-6\%=94\%$

Now, we calculate the number of valid votes as 94% of the polled votes:

$\text{Valid Votes}=94\%\text{of}0.86V=0.94\times 0.86V$

Step 4: Find the margin of victory in terms of valid votes
The election is between two candidates. The winning candidate received 58% of the valid votes. Therefore, the losing candidate must have received the remaining percentage of the valid votes:

$100\%-58\%=42\%$

The margin by which the winning candidate won the election is the difference between their vote percentages:

$\text{Winning Margin Percentage}=58\%-42\%=16\%$

Thus, the winning candidate won by 16% of the valid votes.

Step 5: Set up the equation and solve for V
We are given that the winning margin is 202,100 votes. We can set up the equation as follows:

$16\%\text{of Valid Votes}=202100$

Substituting the expression for valid votes into the equation:

$0.16\times 0.94\times 0.86V=202100$

First, calculate the product of the decimals:

$0.16\times 0.94\times 0.86=0.129344$

Now, substitute this back to solve for V:

$0.129344V=202100$

Divide both sides by 0.129344 to find the total registered voters:

$V=\frac{202100}{0.129344}$

Simplifying the division gives:

$V=1562500$

Therefore, the total number of registered voters is 1,562,500.