In an election three candidates participated the loosing candidates get `10%` votes. What could we the minimum abosolute margins of what by which the winning candidates led by the nearest the rival. If the each candidates got an integral `%` of votes.

Options

A. `1`
B. `5`
C. `2`
D. `8`

Answer

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

### Step 1: Understand the given information
We have three candidates in an election. The problem states that the losing candidates received 10% of the votes. This means that the winning candidate must have received more than 10% of the votes.

### Step 2: Define the variables
Let’s assume the total number of votes is \( V \). Since the losing candidates together received 10% of the votes, the winning candidate must have received \( V - 0.1V = 0.9V \).

### Step 3: Assign percentages to candidates
Let’s denote the votes received by the candidates as follows:
- Winning candidate: \( W \% \)
- Second candidate: \( S \% \)
- Third candidate: \( T \% \)

From the information given:
- \( T + S = 10\% \) (since the losing candidates together received 10%)
- Therefore, \( W + S + T = 100\% \)

### Step 4: Express the votes in terms of percentages
Assuming the winning candidate received \( W \% \) of the total votes:
- Votes for the winning candidate: \( \frac{W}{100}V \)
- Votes for the second candidate: \( \frac{S}{100}V \)
- Votes for the third candidate: \( \frac{T}{100}V \)

### Step 5: Calculate the margin
The margin by which the winning candidate leads over the nearest rival (second candidate) can be expressed as:
- Margin = Votes for winning candidate - Votes for second candidate
- Margin = \( \frac{W}{100}V - \frac{S}{100}V \)
- Margin = \( \frac{(W - S)}{100}V \)

### Step 6: Find the minimum absolute margin
To find the minimum margin, we need to maximize \( S \) (the votes of the second candidate) while keeping \( S + T = 10\% \). The maximum \( S \) can be is 9% (which means \( T \) would be 1%).

Thus, if:
- \( W = 91\% \) (the winning candidate)
- \( S = 9\% \) (the second candidate)
- \( T = 1\% \) (the third candidate)

The margin becomes:
- Margin = \( \frac{(91 - 9)}{100}V = \frac{82}{100}V = 0.82V \)

### Step 7: Conclusion
The minimum absolute margin in terms of votes can be calculated as:
- If \( V = 100 \) (for simplicity), then the margin is \( 82 \) votes.

### Final Answer
The minimum absolute margin by which the winning candidate led the nearest rival is **82 votes**.

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