In an election, only two candidates contested. 20% of the voters on the voter list did not vote, and 120 votes were declared invalid. The winner got 200 more votes than the loser and thereby secured 41% of the total voters on the voter list. What percentage of the total votes cast did the defeated candidate receive?

Introduction / Context:
This is a comprehensive election problem involving percentages, valid and invalid votes, as well as the relationship between total voters, votes cast and the distribution of votes between two candidates. The winner's vote share is given as a percentage of total voters (not of total votes cast), some votes are invalid and some people do not vote at all. We must determine the percentage of total votes cast that were received by the defeated candidate. This requires setting up equations and carefully interpreting each condition.

Given Data / Assumptions:
- Only two candidates contested the election.- 20% of the voters on the voter list did not vote.- 120 of the votes cast were invalid.- The winner got 200 more votes than the loser.- The winner secured 41% of the total voters on the voter list as votes.- We must find the percentage of total votes cast that went to the defeated candidate.

Concept / Approach:
Let the total number of voters be V. The winner received 41% of V votes. The loser received 200 fewer votes than the winner. A certain percentage of voters did not vote and some of the votes cast were invalid, so we must distinguish between total voters, total votes cast, valid votes and the distribution of valid votes. Using the given relationships, we can derive equations to find V, compute how many valid votes each candidate received and then compute the loser's vote share as a percentage of total votes cast.

Step-by-Step Solution:
Step 1: Let total number of voters on the list be V.Step 2: Winner secured 41% of total voters, so winner's votes W = 0.41 * V.Step 3: Let loser's votes be L. It is given that W = L + 200.Step 4: Also, 20% of voters did not vote, so votes cast = 80% of V = 0.80 * V.Step 5: Among votes cast, 120 were invalid, so valid votes = votes cast - invalid votes = 0.80 * V - 120.Step 6: Valid votes are exactly the sum of votes of winner and loser, so:W + L = 0.80 * V - 120.Step 7: Substitute W = 0.41 * V and L = W - 200 into this equation.W + L = 0.41V + (0.41V - 200) = 0.82V - 200.Step 8: Set this equal to valid votes: 0.82V - 200 = 0.80V - 120.Step 9: Rearrange: 0.82V - 0.80V = -120 + 200.Step 10: 0.02V = 80, so V = 80 / 0.02 = 4,000.Step 11: Winner's votes W = 0.41 * 4,000 = 1,640.Step 12: Loser's votes L = W - 200 = 1,640 - 200 = 1,440.Step 13: Votes cast (valid + invalid) = 0.80 * V = 0.80 * 4,000 = 3,200.Step 14: Valid votes = W + L = 1,640 + 1,440 = 3,080. Invalid votes = 3,200 - 3,080 = 120, consistent with given data.Step 15: Required percentage = loser's votes as a percentage of total votes cast.Step 16: Percentage for defeated candidate = (L / votes cast) * 100% = (1,440 / 3,200) * 100%.Step 17: 1,440 / 3,200 = 0.45, so the percentage is 45%.

Verification / Alternative check:
Check that all conditions are satisfied with V = 4,000.Non-voters = 20% of 4,000 = 800.Votes cast = 4,000 - 800 = 3,200.Invalid votes = 120, so valid votes = 3,200 - 120 = 3,080.Winner's votes = 1,640, loser's votes = 1,440, difference = 200.Winner's votes as percentage of total voters = 1,640 / 4,000 = 0.41 or 41%.All conditions check out, confirming V = 4,000 and the calculated percentages.

Why Other Options Are Wrong:
- 47.5%: This would correspond to 0.475 * 3,200 = 1,520 votes, which would change the difference and the winner's percentage of total voters.- 41%: This is the winner's share of total voters, not the defeated candidate's share of total votes cast.- 38%: Would give 1,216 votes to the loser, which is inconsistent with the 200-vote difference and other constraints.

Common Pitfalls:
- Confusing percentages of total voters with percentages of votes cast or valid votes.- Forgetting to subtract invalid votes when forming the equation for valid votes.- Assuming the 41% is of total votes cast instead of total voters, leading to incorrect equations.- Arithmetic errors when solving 0.82V - 200 = 0.80V - 120.

Final Answer:
The defeated candidate received 45% of the total votes cast.