Hi here are my two cents for this questions
Well there can be multiple ways to solve this questions depends on which method strikes you at that moment.
However, its noteworthy to see what can we derive from the information given
We are told that ratio of \(\frac{NF}{F}= \frac{2}{1}\)
this mean\(\frac{2}{3}=\frac{NF}{NF+F}\) and \(\frac{1}{3}=\frac{F}{NF+F}\)
we are told that \(\frac{2}{9}\) of votes are counted and of which \(\frac{3}{4}\) are in favor of proposal. that means \(\frac{1}{4}\) are against the proposal.
what remains to be counted is \(\frac{7}{9}\).then let the fraction of votes against the proposal be 'a' , then number of votes for the resolution will be (1-a) and total number of votes =x
then
\(\frac{2x}{3}\) = \(\frac{2x}{9}\) \(\frac{1}{4}\) + \(\frac{7x}{9}\) a
Simplyfying the above equation we have
a= \(\frac{11}{14}\)
Alternatively if one is good at algebriac approach then
\(\frac{NF}{F}\)=\(\frac{\frac{2x}{9}\frac{1}{4} +\frac{7x}{9} a }{\frac{2x}{9}\frac{3}{4} +\frac{7x}{9}(1- a )}\)
\(\frac{2}{1}\)=\(\frac{\frac{2x}{9}\frac{1}{4} +\frac{7x}{9} a }{\frac{2x}{9}\frac{3}{4} +\frac{7x}{9}(1- a )}\)
solving we get a= \(\frac{11}{14}\)
Alternatively,
we see that if there were 36 votes (how did we come at 36 , we take the LCM of denominators of \(\frac{2}{9} and \frac{3}{4}\) )
then 8 are counted and 6 are in favor and 2 are against
remaining 28 votes x are in favor and y are against.then we have
\(\frac{2}{3}\)= y+2
solving we get y= 22
so we need \(\frac{y}{28}= \frac{22}{28}= \frac{11}{14}\)
Probus
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