In the Presidency College two candidates contested a presidential election. `15%` of the voters did not vote amd 41 votes were invalid. The elected contestant got 314 votes more than the other candidate. If the elected candidate got `45%` of the total eligible voters, which is equal to the no of all the students of the college. The individual votes of each candidate are :

Options

A. 2250 and 1936
B. 3568 and 3254
C. 2442 and 2128
D. 2457 and 2143

Answer

To solve the problem step by step, we will break down the information given and use it to find the votes for each candidate.

### Step 1: Determine the total number of eligible voters
Let the total number of eligible voters (students in the college) be \( V \).

### Step 2: Calculate the number of voters who actually voted
Since 15% of the voters did not vote, the percentage of voters who did vote is:
\[ 100\% - 15\% = 85\% \]
Thus, the number of voters who actually voted is:
\[ 0.85V \]

### Step 3: Account for invalid votes
Out of the voters who voted, 41 votes were invalid. Therefore, the number of valid votes is:
\[ \text{Valid votes} = 0.85V - 41 \]

### Step 4: Determine the votes received by the elected candidate
We know that the elected candidate received 45% of the total eligible voters:
\[ \text{Votes for elected candidate} = 0.45V \]

### Step 5: Determine the votes received by the other candidate
Let the votes received by the other candidate be \( x \). According to the problem, the elected candidate received 314 votes more than the other candidate:
\[ 0.45V = x + 314 \]

### Step 6: Set up the equation for valid votes
The total number of valid votes can also be expressed as the sum of the votes received by both candidates:
\[ 0.85V - 41 = 0.45V + x \]

### Step 7: Substitute for \( x \) in the valid votes equation
From the equation \( 0.45V = x + 314 \), we can express \( x \) as:
\[ x = 0.45V - 314 \]
Now substitute this into the valid votes equation:
\[ 0.85V - 41 = 0.45V + (0.45V - 314) \]

### Step 8: Simplify the equation
Combine like terms:
\[ 0.85V - 41 = 0.9V - 314 \]
Rearranging gives:
\[ 0.85V - 0.9V = -314 + 41 \]
\[ -0.05V = -273 \]
Thus, solving for \( V \):
\[ V = \frac{273}{0.05} = 5460 \]

### Step 9: Calculate the votes for each candidate
Now that we have \( V \), we can find the votes for each candidate:
1. Votes for the elected candidate:
\[ \text{Votes for elected candidate} = 0.45 \times 5460 = 2457 \]
2. Votes for the other candidate using \( x \):
\[ x = 0.45V - 314 = 2457 - 314 = 2143 \]

### Final Votes
- Votes for the elected candidate: **2457**
- Votes for the other candidate: **2143**