In a village, two contestants (A and B) are contesting in an election. 70% of the registered voters cast their votes in the election and A wins the election by 400 votes. If A had received 12.5% less votes, A's votes would have been equal to B's votes. How many registered voters are there in the village?

Let the total number of registered voters in the village be \( x \). 

Since 70% of the registered voters cast their votes, the total number of votes cast in the election is:
\[
0.7x
\]

Let \( A \) and \( B \) be the number of votes received by contestant A and B, respectively. From the problem, we know two important facts:
1. A wins by 400 votes, so:
\[
A = B + 400
\]
2. If A had received 12.5% fewer votes, A's votes would be equal to B's votes. This implies:
\[
A - 0.125A = B
\]
Simplifying this equation:
\[
0.875A = B
\]

Now we have the system of two equations:
1. \( A = B + 400 \)
2. \( B = 0.875A \)

### Step 1: Substitute \( B = 0.875A \) into \( A = B + 400 \):
\[
A = 0.875A + 400
\]
Subtract \( 0.875A \) from both sides:
\[
A - 0.875A = 400
\]
\[
0.125A = 400
\]
Solve for \( A \):
\[
A = \frac{400}{0.125} = 3200
\]

### Step 2: Find \( B \):
Using \( B = 0.875A \):
\[
B = 0.875 \times 3200 = 2800
\]

### Step 3: Find the total number of votes cast:
The total number of votes cast is \( A + B \):
\[
A + B = 3200 + 2800 = 6000
\]

### Step 4: Find the total number of registered voters:
Since 70% of the registered voters cast their votes, we have:
\[
0.7x = 6000
\]
Solve for \( x \):
\[
x = \frac{6000}{0.7} = 8571.43
\]
Rounding to the nearest whole number, the total number of registered voters in the village is:
\[
x = 8571
\]

Thus, the total number of registered voters in the village is 8571.

---

Would you like more details or further explanation on any step? Here are some related questions you might find helpful:

1. How do you solve a system of linear equations?
2. What is the importance of percentages in real-life situations like elections?
3. Can you explain how rounding works when dealing with real-world data?
4. What are other ways to approach this type of election problem?
5. How does changing percentages affect outcomes in problems like this?

**Tip:** When solving word problems, carefully translate the text

Solve an election problem where contestant A wins by 400 votes and 70% of registered voters cast their votes. Learn to calculate the total number of registered voters by solving systems of linear equations and applying percentage calculations. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation