In an election, 3 candidate participated, the loosing candidate got `30%` votes. What would be the minimum absolute margin votes by which the winning candidate led by the nearest rival if each candidate got an integral percent of votes.

Options

A. `2`
B. `3`
C. `1`
D. `5`

Answer

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

### Step 1: Understand the Problem
We have three candidates in an election, and the losing candidate received 30% of the total votes. We need to find out the minimum absolute margin of votes by which the winning candidate led the nearest rival, given that each candidate received an integral percentage of the votes.

### Step 2: Assume Total Votes
Let's assume the total number of votes is 100. This simplifies our calculations since percentages can be directly converted to whole numbers.

### Step 3: Calculate Votes for the Losing Candidate
The losing candidate received 30% of the votes:
\[ \text{Votes for losing candidate} = 30\% \text{ of } 100 = 30 \text{ votes} \]

### Step 4: Calculate Remaining Votes
The remaining votes, which are shared between the winning candidate and the nearest rival, can be calculated as follows:
\[ \text{Remaining votes} = 100 - 30 = 70 \text{ votes} \]

### Step 5: Distribute Remaining Votes
We need to distribute these 70 votes between the winning candidate and the nearest rival in such a way that the absolute margin of votes is minimized. Let's denote the votes for the winning candidate as \( W \) and for the nearest rival as \( R \).

### Step 6: Set Up the Equation
We know:
\[ W + R = 70 \]
To minimize the margin \( |W - R| \), we can express the margin as:
\[ \text{Margin} = W - R \]

### Step 7: Find Values for W and R
To minimize the margin, we can try to make \( W \) and \( R \) as close as possible. If we let:
- \( W = 36 \)
- \( R = 34 \)

Then:
\[ W + R = 36 + 34 = 70 \quad \text{(This is valid)} \]
And the margin is:
\[ \text{Margin} = 36 - 34 = 2 \]

### Step 8: Conclusion
Thus, the minimum absolute margin by which the winning candidate led the nearest rival is:
\[ \text{Minimum Absolute Margin} = 2 \]

### Final Answer
The minimum absolute margin of votes is **2**.

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