There are two candidates P and Q in an election. During the campaign, 40% of the voters promised to vote for P, and rest for Q. However, on the day of election 15% of the voters went back on their promise to vote for P and instead voted for Q. 25% of the voters went back on their promise to vote for Q and instead voted for P. Suppose, P lost by 2 votes, then what was the total number of voters?

Question

Prepp Team · Accepted Answer

Election Voter Distribution

Let $T$ represent the total number of voters.

Promises Analysis

Voters initially promised to vote for P: $40\%$ of $T$, which is $0.40T$.
Voters initially promised to vote for Q: The remaining $100\% - 40\% = 60\%$ of $T$, which is $0.60T$.

Vote Swing Calculation

On election day, $15\%$ of those who promised P switched to Q. Number switching from P to Q $= 0.15 \times 0.40T = 0.06T$.
Also, $25\%$ of those who promised Q switched to P. Number switching from Q to P $= 0.25 \times 0.60T = 0.15T$.

Final Vote Calculation

Final Votes for P: The initial $0.40T$ votes minus those who switched away ($0.06T$) plus those who switched in ($0.15T$). $ \text{Final P Votes} = 0.40T - 0.06T + 0.15T = 0.49T $
Final Votes for Q: The initial $0.60T$ votes minus those who switched away ($0.15T$) plus those who switched in ($0.06T$). $ \text{Final Q Votes} = 0.60T - 0.15T + 0.06T = 0.51T $

Total Voters Solution

Candidate P lost by 2 votes. The difference between Q's final votes and P's final votes is 2.

Difference in votes: $\text{Final Q Votes} - \text{Final P Votes} = 0.51T - 0.49T = 0.02T$.
Set the difference equal to the given margin: $ 0.02T = 2 $
Solve for the total number of voters, $T$: $ T = \frac{2}{0.02} = \frac{200}{2} = 100 $

The total number of voters was 100.