At an election between two candidates 53 votes were declared invalid. The winning candidate secures `58%` of the valid votes and wins by 588 votes. Find the total number of votes polled.

Answer

To solve the problem step by step, let's break it down:

### Step 1: Identify the given information
- Invalid votes = 53
- Winning candidate's percentage of valid votes = 58%
- Winning margin = 588 votes

### Step 2: Calculate the losing candidate's percentage of valid votes
The losing candidate's percentage can be calculated as:
\[ \text{Losing percentage} = 100\% - \text{Winning percentage} = 100\% - 58\% = 42\% \]

### Step 3: Set up the equation for valid votes
Let the total valid votes be \( V \). According to the problem, the difference in votes between the winning and losing candidates is given by:
\[ \text{Winning votes} - \text{Losing votes} = 588 \]
The winning votes can be expressed as:
\[ \text{Winning votes} = 58\% \text{ of } V = 0.58V \]
The losing votes can be expressed as:
\[ \text{Losing votes} = 42\% \text{ of } V = 0.42V \]
Substituting these into the equation gives:
\[ 0.58V - 0.42V = 588 \]

### Step 4: Simplify the equation
\[ 0.16V = 588 \]

### Step 5: Solve for valid votes \( V \)
To find \( V \), divide both sides by 0.16:
\[ V = \frac{588}{0.16} \]
Calculating this gives:
\[ V = 3675 \]

### Step 6: Calculate the total votes polled
The total votes polled (T) is the sum of valid votes and invalid votes:
\[ T = V + \text{Invalid votes} = 3675 + 53 \]
Calculating this gives:
\[ T = 3728 \]

### Final Answer
The total number of votes polled is **3728**.

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