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$\frac{2}{5}$ of the voters promise to vote for $\text{P}$ and the rest promised to vote for $\text{Q}$. Of these, on the last day $15\%$ of the voters went back of their promise to vote for $\text{P}$ and $25\%$ of voters went back of their promise to vote for $\text{Q}$, and $\text{P}$ lost by $2$ votes. Then the total number of voters is _________

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$: 100$

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