At an election between two candidates, 68 votes were declared invalid. The winning candidate secures `52%` of the valid votes and wins by 354 votes. Find the total number of votes polled.

Answer

To find the total number of votes polled in the election, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The number of invalid votes = 68
- The winning candidate secured 52% of the valid votes.
- The winning candidate won by a margin of 354 votes.

### Step 2: Define valid votes
Let the total number of votes be \( V \). The valid votes would then be:
\[ \text{Valid Votes} = V - \text{Invalid Votes} = V - 68 \]

### Step 3: Calculate the percentage of votes for both candidates
Since the winning candidate secured 52% of the valid votes, the losing candidate secured:
\[ \text{Losing Candidate's Votes} = 100\% - 52\% = 48\% \]

### Step 4: Express the votes in terms of valid votes
Let’s denote the valid votes as \( V_v \):
- Votes for the winning candidate = \( 0.52 \times V_v \)
- Votes for the losing candidate = \( 0.48 \times V_v \)

### Step 5: Set up the equation based on the winning margin
The difference in votes between the winning and losing candidate is given as 354 votes:
\[ 0.52 \times V_v - 0.48 \times V_v = 354 \]

### Step 6: Simplify the equation
This simplifies to:
\[ (0.52 - 0.48) \times V_v = 354 \]
\[ 0.04 \times V_v = 354 \]

### Step 7: Solve for valid votes
Now, we can solve for \( V_v \):
\[ V_v = \frac{354}{0.04} = 8850 \]

### Step 8: Calculate the total votes
Now that we have the valid votes, we can find the total votes \( V \):
\[ V = V_v + \text{Invalid Votes} = 8850 + 68 = 8918 \]

### Final Answer
The total number of votes polled is **8918**.

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