in an election contested by two candidates a and b only 80% of the registered voters cast their votes.among the votes polled only 90% of them were valid. a got 35% of the valid votes and lost by a margin of 5454 votes.find the number of registered voters.

Let's solve the problem step by step.

### Given:
- 80% of the registered voters cast their votes.
- 90% of the votes cast were valid.
- Candidate A received 35% of the valid votes.
- Candidate A lost by a margin of 5454 votes.
- We need to find the total number of registered voters.

### Step 1: Define Variables
Let \( N \) be the total number of registered voters.

### Step 2: Calculate the Number of Votes Cast
The number of votes cast is 80% of the registered voters:
\[
\text{Votes Cast} = 0.8N
\]

### Step 3: Calculate the Number of Valid Votes
The number of valid votes is 90% of the votes cast:
\[
\text{Valid Votes} = 0.9 \times 0.8N = 0.72N
\]

### Step 4: Calculate the Votes Received by Candidates A and B
Candidate A received 35% of the valid votes:
\[
\text{Votes for A} = 0.35 \times 0.72N = 0.252N
\]

Let \( \text{Votes for B} \) be \( \text{Votes for A} + 5454 \) (since B won by 5454 votes):
\[
\text{Votes for B} = 0.252N + 5454
\]

### Step 5: Set Up the Equation for Total Valid Votes
The total number of valid votes is the sum of the votes for A and B:
\[
\text{Valid Votes} = \text{Votes for A} + \text{Votes for B}
\]
Substitute the values:
\[
0.72N = 0.252N + (0.252N + 5454)
\]
Simplifying:
\[
0.72N = 0.504N + 5454
\]

### Step 6: Solve for \( N \)
Subtract \( 0.504N \) from both sides:
\[
0.72N - 0.504N = 5454
\]
\[
0.216N = 5454
\]
\[
N = \frac{5454}{0.216} = 25250
\]

### Final Answer
The total number of registered voters is **25,250**.

Learn how to calculate the total number of registered voters in an election scenario using percentage calculations and equations. This problem involves determining the total voters based on votes cast and valid votes received by candidates. Suitable for high school level mathematics.

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