In an election, 2 candidates participated.10% did not vote. 300 votes were declared invalid and the winner gets 60% of voting list and wins by 900 votes. Find no of valid votes.
Ans. 1500
What I tried:
Winner=60%;Not voted=10%;=>Loser=30%
ATQ:
(60-30)%=30%=900
=> Valid votes=90%-300$=\frac{90}{30}*900-300=2700-300=2400$
Where am I wrong?
Winner get 60% of the voting list and not total.
So it total votes are T, valid votes are T-300, and winner got 60% of that ie 0.6*(T-300)
Hope it clears your doubt.
Let $T$ be the total number eligible.
Votes cast: $(9/10)T.$
Valid votes: $(9/10)T-300$.
Winner gets : $(6/10)[(9/10)T-300]$.
Wins by $900$ votes:
$(6/10)[(9/10)T-300] =$
$450 +(1/2)[(9/10)T-300]$;
$(1/10)[9/10T-300] =450$;
$9/10T - 300 = 4500.$
Valid votes: $(9/10)T- 300= 4500.$
Problem:
In an election, 2 candidates participated.10% did not vote. 300 votes were declared invalid and the winner gets 60% of voting list and wins by 900 votes. Find no of valid votes. (answer: 1500)
What exactly do they mean by "wins by 900 votes"? 60% of 1500 votes is 900 votes. 40% of 1500 votes is 600 votes. The phrase "wins by 900 votes" in English is typically taken to mean that somebody gets 900 votes more than somebody else. Is 900 votes the winner in your problem had received more than 600 votes the loser had received by 900 votes? It definitely doesn't look that way. The difference is 300 votes while it should be 900 votes according to the problem statement. Do you see the problem? 1500 can't be the answer.
Here's my solution:
Let $N$ be the total number of people eligible to vote. We know that $10\%$ of them did not vote. This means that $90\%$ $\left(\frac{90}{100}=0.9\right)$ of them did vote and 300 of their votes were declared invalid. Thus, the total number of valid votes is equal to $0.9N-300$.
We also know that the number of people that comprise the $60\%$ of the total number of valid votes is $900$ greater than the number of people that comprise the $40\%$ of the number of valid votes. This statement can be expressed like this: $0.6(0.9N-300)=0.4(0.9N-300)+900$.
All we have to do now is to solve this equation for $N$ (the total number of people eligible to vote) and with the resulting $N$ we will be able to find the number of valid votes: $0.9N-300$.
$$ 0.6(0.9N-300)=0.4(0.9N-300)+900\implies\\ 0.18N=960\implies\\ N=\frac{960}{0.18} $$
The number of valid votes:
$$ 0.9N-300=0.9\cdot\frac{960}{0.18}-300=4500 $$
Answer: $4500$.
The answer is right. It's 1500. The total percentage of voters in voting list is 100%. And the winner gets 60%. But 10% voters didn't vote. That's means 90% of total votes are casted. So the loser will get 30%. But it's also said that 300 votes are invalid which means the actual number of votes of the loser will be 30% - 300. Now, 900= 60% - (30% - 300) 900= 60% - 30% + 300 900 - 300= 30% 600= 30% Then, 100% = 2000 Hence, total number of valid votes = 90% of 2000 - 300 = 1800 - 300 = 1500 Hope that's help you.
Let the number of votes won by the loser be $l.$ Also, let the total number of votes be $v.$
Then we have that $60\% v=900+l.$ Also, we have that $v=300+60\% v+l.$
This is a linear $2×2$ system which may be solved for $v.$ Then you want $v-300.$
Let the number of possible voters (i.e. in voting list) be $N$. Then since $10\%$ didn't vote, $\frac{9}{10}N$ did vote, and among them, 300 votes were invalid, so there are $\frac{9}{10}N-300$ valid votes.
Let the number of votes of the winner be $N_1$, and the number of votes of the loser be $N_2$.
We also know that the winner had $60\%$ of voting list, i.e. $N_1=\frac{6}{10}N$.
Then:
$$\begin{eqnarray}N_1-N_2 &=&900\\ N_1+N_2&=&\frac{9}{10}N-300\\ N_1&=&\frac{6}{10}N \end{eqnarray}$$
Then eliminate: $N_2=N_1-900$ so, from the second equation,
$$2N_1-900=\frac{9}{10}N-300$$
And $N_1=\frac{6}{10}N$, so
$$2\frac{6}{10}N-900=\frac{9}{10}N-300$$
That is,
$$\frac{3}{10}N=600$$
Hence $N=2000$, the number of persons in voting list. The winner had $60\%$ of them or $1200$, and the loser had $1200-900=300$, so the total number of valid votes is $1200+300=1500$, as expected.
Since the question goes discussing 2 candidates who participated in an election. In order to make it simple, we keep the total equal to 100. If 10% didn't vote that means 90% of people voted. The winner gets 60% of voting list so the looser must have got 40% of votes. 100%-60%= 40% The difference of percentages of the winner and the looser = 20% Now, divide the number of votes got by the winner by the difference of their percentages to get total votes. 900/20*100= 4500 which is equal to total votes cast in the election. Once you've found the total votes look at the question again. It says that 10% did not vote, so find 90% of 4500 which is 4050-300 because 300 votes were invalid and you'll get the answer= 3750 Answer=3750
