3.12 Election / Voting Problems

Election problems combine percentage-of-a-whole (valid votes as % of total votes, since some votes are invalid) with percentage-of-valid-votes (each candidate’s share). Always work in two steps: first strip out invalid votes to find the valid vote pool, then split that pool by the given percentages.

Valid Votes = Total Votes × (1 - (Invalid%)/(100))

Worked Example 1: In an election between two candidates, the winner secures 60% of the total valid votes and wins by 480 votes. Find the total number of valid votes. Winner’s share − Loser’s share = 60% − 40% = 20% of valid votes = 480. Valid votes = 480 × (100/20) = 2,400. (Winner got 1,440; loser got 960.)

Worked Example 2: In an election, 10% of the total votes cast were declared invalid. The total number of voters is 50,000, and the winning candidate secured 55% of the valid votes. Find the number of votes the winner received. Valid votes = 50,000 × 0.9 = 45,000. Winner’s votes = 45,000 × 0.55 = 24,750.

Worked Example 3: In an election with two candidates, 15% of the total 12,000 voters cast invalid votes. The winning candidate secured 54% of the valid votes. Find the winning margin. Valid votes = 12,000 × 0.85 = 10,200. Winner = 10,200 × 0.54 = 5,508. Loser = 10,200 × 0.46 = 4,692. Margin = 5,508 − 4,692 = 816 votes.

Worked Example 4: In a three-cornered election, candidate A got 50% of the total valid votes, candidate B got 30%, and the rest went to candidate C. If the total valid votes were 40,000, find by how many votes A defeated the combined total of B and C. A’s votes = 50% of 40,000 = 20,000. B + C’s votes = 50% of 40,000 = 20,000 (since percentages must sum to 100%). Margin over combined B + C = 20,000 − 20,000 = 0 — A exactly ties the combined opposition, illustrating that when a candidate crosses exactly 50% of valid votes, the margin over “all other candidates combined” is always zero; only above 50% does a positive margin over the rest exist.

Worked Example 5 (three-candidate election, no invalid votes): In an election among three candidates, A got 6,000 votes, B got 3,000 votes, and C got 1,000 votes, all votes being valid. Find A’s percentage share of the total votes, and by what percentage A’s votes exceed the combined votes of B and C. Total votes = 6,000 + 3,000 + 1,000 = 10,000. A’s share = (6,000/10,000) × 100 = 60%. B + C combined = 4,000. A exceeds this by (6,000 − 4,000)/4,000 × 100 = 2,000/4,000 × 100 = 50%.

Worked Example 6 (reverse: find total votes from margin and invalid %): In an election between two candidates, 20% of the total votes polled were invalid. The winning candidate secured 60% of the valid votes and won by 3,600 votes. Find the total number of votes polled. Winner − Loser (as % of valid votes) = 60% − 40% = 20% of valid votes = 3,600. Valid votes = 3,600 × (100/20) = 18,000. Since valid votes = 80% of total votes polled (20% being invalid): Total votes = 18,000/0.80 = 22,500.