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?

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.