In an election between two candidates. `10%` of the voters did not cast their votes while `10%` of the vote polled were declared invalid. If the winning candidate of got `54%` of the valid votes and won by `1620` votes then find the total number of voters in the voter list.

Options

A. `22700`
B. `25000`
C. `22500`
D. `25500`

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define Variables
Let the total number of voters in the voter list be \( x \).

### Step 2: Calculate Votes Not Cast
10% of the voters did not cast their votes, so the number of voters who did not vote is:
\[ 0.10x \]
Thus, the number of voters who cast their votes is:
\[ x - 0.10x = 0.90x \]

### Step 3: Calculate Invalid Votes
10% of the votes polled were declared invalid. Therefore, the number of invalid votes is:
\[ 0.10 \times 0.90x = 0.09x \]
The number of valid votes is:
\[ 0.90x - 0.09x = 0.81x \]

### Step 4: Votes Received by Each Candidate
The winning candidate received 54% of the valid votes:
\[ \text{Votes for winning candidate} = 0.54 \times 0.81x = 0.4374x \]
The losing candidate received the remaining votes:
\[ \text{Votes for losing candidate} = 0.46 \times 0.81x = 0.3726x \]

### Step 5: Calculate the Vote Difference
The difference in votes between the winning and losing candidate is given as 1620 votes:
\[ 0.4374x - 0.3726x = 1620 \]
This simplifies to:
\[ 0.0648x = 1620 \]

### Step 6: Solve for Total Voters
To find \( x \), divide both sides by 0.0648:
\[ x = \frac{1620}{0.0648} \approx 25000 \]

### Final Answer
The total number of voters in the voter list is:
\[ \boxed{25000} \]

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