Understanding the Election Problem
This problem requires us to determine the total number of voters enrolled for an election. We are given details about the participation rate (percentage of voters who cast votes), the rate of invalid votes, and the performance of one candidate relative to the valid votes cast.
Step-by-Step Calculation Breakdown
Let's denote the total number of voters enrolled as \(V\). We can work backward from the information given:
- Votes Cast: 75% of the enrolled voters cast their votes. Number of votes cast = \(75\%\) of \(V = 0.75 \times V\).
- Invalid Votes: Out of the votes cast, 2% were invalid. Number of invalid votes = \(2\%\) of (Votes Cast) = \(0.02 \times (0.75 \times V)\).
- Valid Votes: The percentage of valid votes is \(100\% - 2\% = 98\%\) of the votes cast. Number of valid votes = \(98\%\) of (Votes Cast) = \(0.98 \times (0.75 \times V)\).
- Candidate's Share: One candidate secured 9261 votes, which accounted for 75% of the total valid votes. Votes for the candidate = \(75\%\) of (Valid Votes) So, \(9261 = 0.75 \times (\text{Number of valid votes})\).
Calculating Total Enrolled Voters (\(V\))
Now, let's substitute the expression for the number of valid votes into the equation:
We know \(9261 = 0.75 \times (0.98 \times 0.75 \times V)\).
Let's first calculate the combined product of the percentages:
\(0.75 \times 0.98 \times 0.75 = \frac{3}{4} \times \frac{98}{100} \times \frac{3}{4}\) \(= \frac{3}{4} \times \frac{49}{50} \times \frac{3}{4}\) \(= \frac{3 \times 49 \times 3}{4 \times 50 \times 4}\) \(= \frac{441}{800}\) \(= 0.55125\)Our equation simplifies to:
\(9261 = 0.55125 \times V\)To find the total number of voters enrolled (\(V\)), we rearrange the equation:
\(V = \frac{9261}{0.55125}\) \(V = 16800\)Final Answer
The calculation shows that the total number of voters enrolled in the election was 16,800.
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