In an election, a candidate received 35% of the votes polled and lost to his rival by 2250 votes. Assuming there were only two candidates, how many total votes were cast in the election?

Introduction / Context:
This election problem combines percentage vote shares with an absolute margin of defeat to determine the total number of votes polled. Such questions are common in quantitative aptitude to test proportional reasoning and understanding of complementary percentages that sum to 100 when there are only two candidates.

Given Data / Assumptions:

There are two candidates in the election.

The losing candidate receives 35% of the total votes.

The winning candidate receives the remaining 65% of the votes.

The margin of defeat is 2250 votes.

We must find the total number of votes cast.

Concept / Approach:
If total votes are V, the losing candidate has 0.35V votes, and the winner has 0.65V votes. The difference between their votes is 0.65V − 0.35V = 0.30V. This difference is given as 2250 votes. Thus we set 0.30V equal to 2250 and solve for V. This is a direct algebraic application of the definition of percentage and difference.

Step-by-Step Solution:
Step 1: Let total votes be V.Step 2: Votes received by the losing candidate = 35% of V = 0.35V.Step 3: Votes received by the winning candidate = 65% of V = 0.65V.Step 4: Margin of victory = votes of winner − votes of loser = 0.65V − 0.35V = 0.30V.Step 5: We are told this margin equals 2250 votes, so 0.30V = 2250.Step 6: Solve for V: V = 2250 / 0.30.Step 7: 0.30 is 3 / 10, so V = 2250 * 10 / 3 = 22500 / 3 = 7500.

Verification / Alternative check:
Check with the found total: 35% of 7500 = 0.35 * 7500 = 2625 votes. 65% of 7500 = 0.65 * 7500 = 4875 votes. The difference 4875 − 2625 = 2250 votes, exactly as stated. Thus, the result is consistent and correct.

Why Other Options Are Wrong:
If total votes were 6200, then 30% would be 1860, not 2250.
For 5600, 30% is 1680, again not equal to the given margin.
For 4700, 30% is 1410 votes, which is too small.
9000 votes would give a margin of 2700 votes, which overshoots the given value.

Common Pitfalls:
Some students incorrectly compute the margin as 35% of votes instead of the difference between 65% and 35%. Others mis-handle the proportion, taking 2250 as 35% instead of 30%. Always remember that when only two candidates are present, their vote percentages add up to 100%, and the margin corresponds to the difference between these percentages applied to the total votes.

Final Answer:
The total number of votes cast in the election was 7500.