Question

2/5 of the voters promised to vote for P and the rest promised to vote for Q. Amongst these, on the last day 15% of the voters went back on their promises to vote for P and 25% of voters went back on their promise to vote for Q and P lost by 2 votes. Then the total number of voters.
$
  (a){\text{ 100}} \\
  (b){\text{ 110}} \\
  (c){\text{ 90}} \\
  (d){\text{ 95}} \\
 $

Answer 1

Hint – In this problem let the total numbers of voters be x, use the constraints given in question to find the total number of people out of this x that promised to vote for P and Q separately. The voters that went back to an individual means that they voted the other candidate so use this to formulate an equation and finally as P lost by 2 votes this means that the votes of Q must be greater than that of P by 2.

Complete step-by-step solution-
Let us assume that the total voters are x.
Now it is given that (2/5) of the voters promise to vote for P.
Therefore voters of P = $\dfrac{2}{5} \times x = \dfrac{{2x}}{5}$.
Therefore voters of Q = x – $\dfrac{{2x}}{5}$ = $\dfrac{{3x}}{5}$.
Now on the last day 15 % of the voters went back on their promise to vote for P and 25 % of the voters went back on their promise to vote for Q.
Therefore at the time of voting 15 % of P votes goes to Q and 25 % of Q vote goes to P.
So 15 % of P vote = $\dfrac{{15}}{{100}} \times \dfrac{{2x}}{5} = \dfrac{{3x}}{{50}}$
And 25 % of Q vote = $\dfrac{{25}}{{100}} \times \dfrac{{3x}}{5} = \dfrac{{3x}}{{20}}$
So the total votes for P = $\dfrac{{2x}}{5}$ – 15 % of P vote + 25 % of Q vote
                                          = $\dfrac{{2x}}{5} - \dfrac{{3x}}{{50}} + \dfrac{{3x}}{{20}} = \dfrac{{40x - 6x + 15x}}{{100}}$
                                          = $\dfrac{{49x}}{{100}}$
And the total votes for Q = $\dfrac{{3x}}{5}$ – 25 % of Q vote + 15 % of P vote
                                              = $\dfrac{{3x}}{5} - \dfrac{{3x}}{{20}} + \dfrac{{3x}}{{50}} = \dfrac{{40x + 15x - 6x}}{{100}}$
                                              = $\dfrac{{51x}}{{100}}$
Now it is given that Q wins by 2 votes.
So votes of Q – votes of P = 2
$ \Rightarrow \dfrac{{51x}}{{100}} - \dfrac{{49x}}{{100}} = 2$
Now simplify this equation we have,
$ \Rightarrow \dfrac{{51x - 49x}}{{100}} = 2$
$ \Rightarrow \dfrac{{2x}}{{100}} = 2$
\[ \Rightarrow x = 100\]
So the total voters are 100.
Hence option (A) are correct.

Note – The tricky point here was that when some percentage of people went back from their promise than what happened. We have assumed that each and every one has voted, so when 15% people went back from their promise vote for P, this 15% must have voted for Q and the people who went off from their promise to vote for Q must have voted for P hence the total number of votes of P must be the initial percentage of people who promised to vote for P reduced with the people who went off from their promise to vote for P addition to the people who went off form their promise to vote Q as these are being gained by P now.