Analyzing the Election Vote Calculation

This problem involves calculating the total number of votes based on percentages of cast votes, invalid votes, and valid votes received by a candidate.

Step-by-Step Solution Breakdown

1. Calculate Total Valid Votes

We are given that a candidate received 9,261 votes, which represents 75% of the total valid votes cast in the election. Let '$T_{valid}$' represent the total number of valid votes.

The relationship can be written as:

$$9,261 = 75\% \text{ of } T_{valid}$$

In decimal form, this is:

$$9,261 = 0.75 \times T_{valid}$$

To find the total valid votes, we rearrange the equation:

$$T_{valid} = \frac{9,261}{0.75}$$

$$T_{valid} = \frac{9,261}{\frac{3}{4}}$$

$$T_{valid} = 9,261 \times \frac{4}{3}$$

$$T_{valid} = 3,087 \times 4$$

$$T_{valid} = 12,348$$

So, there were 12,348 valid votes in total.

2. Calculate Total Votes Cast

Next, we know that 2% of the total votes cast were declared invalid. This means the valid votes constitute 98% of the total votes cast (100% - 2% = 98%). Let '$T_{cast}$' represent the total number of votes cast.

The relationship is:

$$T_{valid} = 98\% \text{ of } T_{cast}$$

In decimal form:

$$12,348 = 0.98 \times T_{cast}$$

To find the total votes cast, we rearrange the equation:

$$T_{cast} = \frac{12,348}{0.98}$$

$$T_{cast} = \frac{12,348}{\frac{98}{100}}$$

$$T_{cast} = 12,348 \times \frac{100}{98}$$

$$T_{cast} = \frac{1,234,800}{98}$$

Performing the division:

$$T_{cast} = 12,600$$

Therefore, a total of 12,600 votes were cast.

3. Calculate Total Number of Voters

Finally, we are told that 75% of the total voters cast their votes. Let '$T_{voters}$' represent the total number of registered voters.

The relationship is:

$$T_{cast} = 75\% \text{ of } T_{voters}$$

In decimal form:

$$12,600 = 0.75 \times T_{voters}$$

To find the total number of voters, we rearrange the equation:

$$T_{voters} = \frac{12,600}{0.75}$$

$$T_{voters} = \frac{12,600}{\frac{3}{4}}$$

$$T_{voters} = 12,600 \times \frac{4}{3}$$

$$T_{voters} = 4,200 \times 4$$

$$T_{voters} = 16,800$$

Thus, the total number of votes (total registered voters) is 16,800.