Election arithmetic with valid/invalid votes: In a constituency, 75% of voters cast votes; 2% of those were rejected as invalid. The winner got 75% of the valid votes and polled 9,261 votes. Find the total number of voters.

Introduction / Context:
This is a layered percentage problem: turnout rate, rejection rate of cast ballots, and the winner’s share of valid votes. We track each stage multiplicatively from total registered voters to the winner’s votes, then invert to find the total electorate.

Given Data / Assumptions:

• Total voters = T.
• Cast votes = 75% of T = 0.75T.
• Rejected = 2% of cast ⇒ valid votes = 98% of cast = 0.98 * 0.75T = 0.735T.
• Winner’s share = 75% of valid = 0.75 * 0.735T = 0.55125T.
• Winner’s votes = 9,261.

Concept / Approach:
Set 0.55125T = 9261 and solve for T. Working with decimals or their fractional equivalents both lead to the same result.

Step-by-Step Solution:

Verification / Alternative check:
Forward check with T = 16800: Cast = 12600; Valid = 12348; Winner’s 75% = 9261. All figures align exactly.

Why Other Options Are Wrong:

• 14500, 18900, 24000: Do not satisfy the composite proportion 0.55125T = 9261.

Common Pitfalls:
Applying 2% rejection to total voters instead of to cast votes, or compounding percentages in the wrong order. Always multiply stage by stage.

Final Answer: