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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?

1. $100$
2. $110$
3. $90$
4. $95$

edited | 353 views

Let $x$ be the number of votes

• Number of votes $P$ was supposed to get $=0.4x$
• Number of votes $Q$ was supposed to get $=0.6x$

Now according to the given conditions,

• Number of votes $P$ got $=\frac{85}{100}(0.4x)+\frac{25}{100}(0.6 x)$
• Number of votes $Q$ has got $= \frac{75}{100}(0.6x)+\frac{15}{100}(0.4x)$

$P-Q=2$

$\implies \frac{75}{100}(0.6x)+\frac{15}{100}(0.4x)-\frac{85}{100}(0.4x)-\frac{25}{100}(0.6 x)=2$

$\implies 45x + 6x - 34x - 15x = 200$

$\implies x = 100$

Correct Option: A.

by Boss (31.4k points)
selected by

Ans a. 100

Exp:-

Suppose no. Of voters =100

So Promised to vote P =40

Promised to vote Q=60

Then 15% of the voter promised to vote p went back and voted for Q = 15% of 40= 6

Rest out of 40 voted for P=34

similarly, 25% of 60 voted for P= 15

Rest out of 60 voted for P=45

So voted for P= 49

So voted for Q=51 voters

So, no of voters=100

by Loyal (7.1k points)
used the technique from paging and tlb acess. let total were x

for p = 0.40x

q=0.60x

15% of q = 0.60 * 0.15 x ; =0.09x

they have not voted for q. so

(0.60x-0.09x) - (0.40x+0.09x) = 2

0.02x=2

x=100
by Boss (16.1k points)
0
you could have answered in step 3 as 15% of q that is .09x cant be a fraction as they are voters so x is 100.

Let $V_P$, $V_Q$ be the total votes received by the candidate $P$ and $Q$ respectively and the total number of voters be $x$.

Now

$V_P=(40\%-40\%\times 15\%+60\%\times 25\%)x=\left(\frac{40}{100}-\frac{40}{100}\times\frac{15}{100}+\frac{60}{100}\times\frac{25}{100}\right)x=\frac{49x}{100}\\V_Q=(60\%+40\%\times 15\%-60\%\times 25\%)x=\left(\frac{60}{100}+\frac{40}{100}\times\frac{15}{100}-\frac{60}{100}\times\frac{25}{100}\right)x=\frac{51x}{100}$

According to the question finally,

\begin{align} V_P &=V_Q-2\\ \Rightarrow V_Q-V_P &=2\\ \Rightarrow \frac{51x}{100}-\frac{49x}{100}&=2\\\ \Rightarrow \frac{2x}{100}&=2\\ \therefore x&=100\end{align}

Therefore, the number of voters is $100$.

So the correct answer is A.

by Active (3.6k points)
+1 vote
Ans -> A) 100
by Boss (41.9k points)

Let the number of voters be 100.

Then, the number of candidates which promised to vote for P = 40

and the no. of candidates promised to vote for Q = 100 – 40 = 60

According to question,

15% of 40 = 6 voters went back on their promise to vote for
P and instead voted for Q.

and 25% of 60 = 15 voters went back on their promise to vote
for Q and instead voted for P.

Number of Votes for P = 40 – 6 + 15 = 49

and Number of Votes for Q = 60 + 6 – 15 = 51

Difference of Votes = 51 – 49 = 2.

Also P lost by 2 Votes.

Number of Voters were 100.

by Active (4.7k points)

Let Total number of voters be $100$

 Actions Votes of P Votes of Q During the campaign,40% of the voters promised to vote for P, and rest for Q. 40 60 15% of the voters went back on their promise to vote for P and instead voted for Q 40 - (15% of 40) = 40 -6 = 34 60 + (15% of 40) = 60 + 6 = 66. 25% of the voters went back on their promise to vote for Q and instead voted for P 34 + (25% of 60) = 34 + 15 = 49 66 - (25% of 60) = 66 -15 = 51

Given: P lost by 2 votes i.e Q-P = 2.

In our assumption also Q-P = 51 -49 =2.

$\therefore$ Option A. $100$ is the correct answer.

by Boss (24.2k points)