In an election, the winner was supported by 46% of all the voters in the list, and he got 410 votes more than his only rival. 10% of the voters on the voters’ list did not cast their votes, and 60 voters cast their ballot papers blank. How many voters were on the list?

The correct answer is 17500.

Step-by-Step Explanation:

Step 1: Define the variables and understand the given conditions.

Let the total number of voters on the voters' list be $x$.

From the problem statement, we have the following data:

1. Total percentage of voters who did not cast their vote = $10\%$ of $x$.

2. Number of blank votes cast = $60$.

3. Percentage of total voters who supported the winner = $46\%$ of $x$.

Therefore, the number of votes received by the winner is:

$\text{Winner's votes}=0.46x$

Step 2: Calculate the votes received by the rival (runner-up).

The winner received 410 votes more than his only rival. Therefore, the number of votes received by the rival is:

$\text{Rival's votes}=0.46x-410$

Step 3: Set up the total equation for all voters on the list.

The total number of voters on the list ($x$) is composed of four parts:

• Voters who did not cast their vote ($10\%$ of $x$ = $0.10x$)

• Voters who cast blank ballot papers ($60$)

• Votes received by the winner ($0.46x$)

• Votes received by the rival ($0.46x-410$)

Summing these components gives the total number of voters on the list:

$x=0.10x+60+0.46x+(0.46x-410)$

Step 4: Solve the linear equation for $x$.

Combine the terms containing $x$ and the numerical constants on the right side of the equation:

$x=(0.10+0.46+0.46)x+(60-410)$

$x=1.02x-350$

Now, rearrange the terms to isolate $x$:

$350=1.02x-x$

$350=0.02x$

Solve for $x$:

$x=\frac{350}{0.02}$

$x=\frac{35000}{2}=17500$

Conclusion:

There were 17500 voters on the list.