In a village `70%` registered voters cast their votes in the election. Only two candidates (A and B) contrasted the election. A won the election by 400 votes and A received `12.5%` less votes had A received `12.5%` less votes the result would have been tie how many registered voters are their in village.

Options

A. `2800`
B. `3000`
C. `4000`
D. `2500`

Answer

To solve the problem step by step, we will break down the information given in the question and derive the required values systematically.

### Step 1: Define Variables
Let the number of votes received by candidate A be \( x \). Since A won by 400 votes, the number of votes received by candidate B will be \( x - 400 \).

**Hint:** Start by defining variables to represent the unknown quantities in the problem.

### Step 2: Understand the Impact of Reduced Votes
According to the problem, if candidate A had received 12.5% less votes, the result would have been a tie. Therefore, if A received 12.5% less, the votes would be:
\[ x - 0.125x = 0.875x \]
In this case, the votes for A would equal the votes for B:
\[ 0.875x = (x - 400) \]

**Hint:** Set up an equation based on the conditions given in the problem.

### Step 3: Solve the Equation
Now, we can solve the equation:
\[ 0.875x = x - 400 \]
Rearranging gives:
\[ x - 0.875x = 400 \]
\[ 0.125x = 400 \]
Dividing both sides by 0.125:
\[ x = \frac{400}{0.125} = 3200 \]

**Hint:** Isolate the variable to find its value.

### Step 4: Calculate the Total Votes Cast
Now that we know \( x = 3200 \) (the votes received by A), we can find the votes received by B:
\[ \text{Votes for B} = x - 400 = 3200 - 400 = 2800 \]

**Hint:** Use the value of \( x \) to find related quantities.

### Step 5: Determine the Total Registered Voters
Since 70% of registered voters cast their votes, we can let \( R \) be the total number of registered voters. The equation for the total votes cast is:
\[ 0.7R = \text{Votes for A} + \text{Votes for B} = 3200 + 2800 = 6000 \]
Thus, we can find \( R \):
\[ R = \frac{6000}{0.7} = 8571.43 \]

Since the number of registered voters must be a whole number, we round it to:
\[ R = 8571 \]

**Hint:** Use the percentage of registered voters to relate the total votes cast to the total number of registered voters.

### Final Answer
The total number of registered voters in the village is approximately **8571**.