`10%` of the voters did not cast their votes in an election between two candidates. The successful candidate got `60%` of the used votes and won by 360 votes. Find the number of votes polled.

Options

A. `3000`
B. `2000`
C. `4500`
D. `2500`

Answer

To solve the problem step by step, we will denote the total number of voters as \( V \).

### Step 1: Calculate the number of voters who did not cast their votes.
Since 10% of the voters did not cast their votes, the number of voters who did not vote is:
\[ \text{Voters who did not vote} = 0.10 \times V = 0.1V \]

### Step 2: Calculate the number of voters who cast their votes.
The number of voters who actually cast their votes is:
\[ \text{Voters who voted} = V - \text{Voters who did not vote} = V - 0.1V = 0.9V \]

### Step 3: Determine the votes received by the successful candidate.
The successful candidate received 60% of the votes that were cast. Therefore, the number of votes received by the successful candidate is:
\[ \text{Votes for successful candidate} = 0.60 \times (0.9V) = 0.54V \]

### Step 4: Determine the votes received by the unsuccessful candidate.
Since the successful candidate won by 360 votes, the number of votes received by the unsuccessful candidate can be expressed as:
\[ \text{Votes for unsuccessful candidate} = \text{Votes for successful candidate} - 360 = 0.54V - 360 \]

### Step 5: Set up the equation based on the total votes.
The total votes cast can also be expressed as the sum of the votes received by both candidates:
\[ \text{Votes for successful candidate} + \text{Votes for unsuccessful candidate} = 0.9V \]
Substituting the expressions we have:
\[ 0.54V + (0.54V - 360) = 0.9V \]

### Step 6: Simplify the equation.
Combine like terms:
\[ 0.54V + 0.54V - 360 = 0.9V \]
\[ 1.08V - 360 = 0.9V \]

### Step 7: Solve for \( V \).
Rearranging gives:
\[ 1.08V - 0.9V = 360 \]
\[ 0.18V = 360 \]
\[ V = \frac{360}{0.18} = 2000 \]

### Step 8: Calculate the number of votes polled.
The number of votes polled (cast) is:
\[ \text{Votes polled} = 0.9V = 0.9 \times 2000 = 1800 \]

### Final Answer:
The number of votes polled is **1800**.

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