Let the total number of voters be $100x.$
Now, the number of voters for $P$ and $Q:$
- The number of voters for $P : 100x \times \frac{2}{5} = 40x$
- The number of voters for $Q : 100x \times \frac{3}{5} = 60x$
On the last day, $15\%$ of the voters went to vote for $Q$ (they didn’t vote for $P$, they broke their promise) and $25\%$ of the voters went to vote for $P$ (they also didn’t vote for $Q$, they broke their promise).
Now,
- $15\% \; \text{of} \; 40x = 40x \times \frac{15}{100} = 6x$
- $25\% \; \text{of} \; 60x = 60x \times \frac{25}{100} = 15x$
So,
- The number of voters for $P : 40x – 6x + 15x = 49x$
- The number of voters for $Q : 60x – 15x + 6x = 51x$
$P$ lost by $2$ votes, that means, $51x – 49x = 2$
$ \Rightarrow 2x = 2$
$ \Rightarrow \boxed{x = 1}$
$\therefore$ The total number of voters $100x = 100 \times 1 = 100.$
Correct Answer$: 100$
Short Method$:$
Let the total number of voters be $100.$
- For $P : 100 \underline{\frac{2}{5}}$
- For $Q : 100$
We have vote for $Q – $ vote for $P = 2$
$ \Rightarrow 51 – 49 = 2$
$ \Rightarrow \boxed{2 = 2 \; (\text{Always True})}$
$\therefore$ The total number of voters is $100.$
Correct Answer$: 100$