Understanding the Election Scenario

This problem involves calculating the total number of voters in an election based on initial promises made by voters to candidates P and Q, and subsequent changes in their voting intentions. We are given that candidate P lost the election by 2 votes.

Step-by-Step Calculation of Total Voters

Let's denote the total number of voters as $N$.

Initial Voter Promises:

• Percentage of voters promised to vote for P: 40%
• Percentage of voters promised to vote for Q: The rest, which is $100\% - 40\% = 60\%$

Changes in Voting Intentions:

• Voters who promised P but switched to Q: 15% of those who promised P.
• Voters who promised Q but switched to P: 25% of those who promised Q.

Calculating Actual Votes for P and Q:

Let's calculate the number of voters switching:

• Number of voters switching from P to Q = $15\%$ of $40\%$ of $N$ $= 0.15 \times (0.40 \times N)$ $= 0.06 \times N$
• Number of voters switching from Q to P = $25\%$ of $60\%$ of $N$ $= 0.25 \times (0.60 \times N)$ $= 0.15 \times N$

Now, let's find the final actual votes for each candidate:

• Actual Votes for P = (Voters initially promised to P) - (Voters who switched from P to Q) + (Voters who switched from Q to P) Actual Votes for P = $(0.40 \times N) - (0.06 \times N) + (0.15 \times N)$ Actual Votes for P = $(0.40 - 0.06 + 0.15) \times N$ Actual Votes for P = $0.49 \times N$
• Actual Votes for Q = (Voters initially promised to Q) - (Voters who switched from Q to P) + (Voters who switched from P to Q) Actual Votes for Q = $(0.60 \times N) - (0.15 \times N) + (0.06 \times N)$ Actual Votes for Q = $(0.60 - 0.15 + 0.06) \times N$ Actual Votes for Q = $0.51 \times N$

Using the Election Outcome to Find Total Voters:

We are given that P lost the election by 2 votes. This means the number of votes for P is 2 less than the number of votes for Q.

Mathematically, this can be written as:

Actual Votes for P = Actual Votes for Q - 2

Substituting the expressions we found:

$0.49 \times N = (0.51 \times N) - 2$

Now, we solve for $N$:

$2 = (0.51 \times N) - (0.49 \times N)$

$2 = (0.51 - 0.49) \times N$

$2 = 0.02 \times N$

$N = \frac{2}{0.02}$

$N = \frac{200}{2}$

$N = 100$

Summary of Votes:

With $N=100$:

• Actual Votes for P = $0.49 \times 100 = 49$
• Actual Votes for Q = $0.51 \times 100 = 51$

The difference is $51 - 49 = 2$ votes, which matches the condition that P lost by 2 votes.

Conclusion

The total number of voters was 100.