In an election between two candidates. 10% of voters did not cast their votes. 60 votes were found invalid. A candidate got 47% of votes and won by a majority of 308 votes. find the total number of votes.

Options

A. 5580
B. 3080
C. 6200
D. 6000

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the total number of votes
Let the total number of votes be represented as \( V \).

### Step 2: Calculate the number of valid votes
Since 10% of voters did not cast their votes, the percentage of votes that were cast is 90%. Therefore, the number of valid votes is:
\[ \text{Valid Votes} = 0.90V \]

### Step 3: Account for invalid votes
Out of the valid votes, 60 votes were found to be invalid. Thus, the number of valid votes that can be counted is:
\[ \text{Counted Votes} = 0.90V - 60 \]

### Step 4: Determine the votes received by each candidate
One candidate received 47% of the counted votes, while the other candidate received the remaining votes. Therefore:
- Votes for the winning candidate:
\[ \text{Votes for Candidate A} = 0.47 \times (\text{Counted Votes}) = 0.47 \times (0.90V - 60) \]
- Votes for the losing candidate:
\[ \text{Votes for Candidate B} = 0.53 \times (\text{Counted Votes}) = 0.53 \times (0.90V - 60) \]

### Step 5: Set up the equation based on the majority
The winning candidate won by a majority of 308 votes. Therefore, we can set up the equation:
\[ \text{Votes for Candidate A} - \text{Votes for Candidate B} = 308 \]
Substituting the expressions from Step 4:
\[ 0.47(0.90V - 60) - 0.53(0.90V - 60) = 308 \]

### Step 6: Simplify the equation
Combine the terms:
\[ (0.47 - 0.53)(0.90V - 60) = 308 \]
\[ -0.06(0.90V - 60) = 308 \]

### Step 7: Solve for \( V \)
Divide both sides by -0.06:
\[ 0.90V - 60 = \frac{308}{-0.06} \]
Calculating the right side:
\[ 0.90V - 60 = -5133.33 \]
Now, add 60 to both sides:
\[ 0.90V = -5133.33 + 60 \]
\[ 0.90V = -5073.33 \]
Now, divide by 0.90:
\[ V = \frac{-5073.33}{0.90} \]
Calculating \( V \):
\[ V = -5637.03 \]
Since the total number of votes cannot be negative, we must have made an error in the assumption or calculation. Let's re-evaluate the steps.

### Correcting the calculation:
Let's go back to the equation:
\[ 0.06(0.90V - 60) = 308 \]
Now we need to multiply by -1:
\[ 0.06(0.90V - 60) = 308 \]
Now, divide both sides by 0.06:
\[ 0.90V - 60 = \frac{308}{0.06} \]
Calculating the right side:
\[ 0.90V - 60 = 5133.33 \]
Now, add 60 to both sides:
\[ 0.90V = 5133.33 + 60 \]
\[ 0.90V = 5193.33 \]
Now, divide by 0.90:
\[ V = \frac{5193.33}{0.90} \]
Calculating \( V \):
\[ V = 5770.37 \]
Since the total number of votes must be a whole number, we round to the nearest whole number:
\[ V = 5770 \]

### Final Answer
The total number of votes is approximately **5770**.