Question

When $\dfrac{2}{9}$ of the votes on a certain resolution have been counted, $\dfrac{3}{4}$ of those counted are in favor of the resolution. What fraction of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution?
A. $\dfrac{11}{4}$
B. $\dfrac{13}{18}$
C. $\dfrac{4}{7}$
D. $\dfrac{3}{14}$

Answer 1

Hint: To find the fraction of the remaining votes that must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution, we will assume this to be $x$ . It is given that Fraction of votes in favor of the counted votes $=\dfrac{3}{4}\times \dfrac{2}{9}=\dfrac{1}{6}$ . Hence, Fraction of votes against the counted votes $=\dfrac{2}{9}-\dfrac{1}{6}=\dfrac{1}{18}$ . Hence, we can find remaining uncounted votes $=1-\dfrac{2}{9}=\dfrac{7}{9}$ . Hence, fractions of votes that must be against the uncounted votes $=\dfrac{7}{9}x$ . Now, we can find the fraction of votes that must be against of total votes $=\dfrac{1}{18}+\dfrac{7}{9}x=\dfrac{1+14x}{18}...(i)$ and fraction of votes that must be in favor of total votes $=\dfrac{1}{6}+\dfrac{7\left( 1-x \right)}{9}=\dfrac{17-14x}{18}...(ii)$ . Now the ratio of (i) to (ii) will be equal to $\dfrac{2}{1}$ . By solving this, we will get the value of x which is the required answer.

Complete step-by-step answer:
We need to find the fraction of the remaining votes that must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution. For this, we will have to find the fraction of votes that must be against total votes and fraction of votes that must be in favor of total votes.
From the given data, we can understand that
Fraction of votes in favor of the counted votes $=\dfrac{3}{4}\text{ of }\dfrac{2}{9}$
That is,
Fraction of votes in favor of the counted votes $=\dfrac{3}{4}\times \dfrac{2}{9}=\dfrac{1}{6}$
From this, we can find the fraction of votes against the counted votes. This is given as
Fraction of votes against the counted votes $=\dfrac{2}{9}-\dfrac{1}{6}=\dfrac{1}{18}$
Now we can find the remaining uncounted votes. We know that the total of the fractions will be 1, that is, the entire votes will be denoted as 1. Thus,
Remaining uncounted votes $=1-\dfrac{2}{9}=\dfrac{7}{9}$
Let us assume $x$ to be the fraction of uncounted votes that must be against the resolution of the uncounted votes so that the total count will result in a vote of 2 to 1 against the resolution.
Hence, we can find the fraction of votes that must be against of the uncounted votes. This is shown below.
Fraction of votes that must be against of the uncounted votes $=\dfrac{7}{9}x$
Now, let us see the fraction of votes that must be in favor of the uncounted votes. This can be found by subtracting the uncounted votes from the fraction of votes that are against the uncounted votes. That is,
Fraction of votes that must be favor of the uncounted votes $=\dfrac{7}{9}-\dfrac{7}{9}x=\dfrac{7\left( 1-x \right)}{9}$
Let us now find the fraction of votes that must be against total votes. This can be computed by adding fraction of votes against the counted votes and fraction of votes that must be against the uncounted votes. This is shown below.
Fraction of votes that must be against of total votes $=\dfrac{1}{18}+\dfrac{7}{9}x=\dfrac{1+14x}{18}...(i)$
Now, we should find the fraction of votes that must be in favor of total votes. This is done by adding a fraction of votes in favor of the counted votes and fraction of votes that must be in favor of the uncounted votes. Thus,
Fraction of votes that must be in favor of total votes $=\dfrac{1}{6}+\dfrac{7\left( 1-x \right)}{9}=\dfrac{17-14x}{18}...(ii)$
Now, we should find the fraction of the remaining votes that is against the resolution so that the total count will result in a vote of 2 to 1 against the resolution. This means that
Ratio of votes that must be in favor of total votes to votes that must be against of total votes $=\dfrac{2}{1}$
From (i) and (ii), this is given as
$\dfrac{\dfrac{1+14x}{18}}{\dfrac{17-14x}{18}}=\dfrac{2}{1}$
Let us now solve this. The above equation can be written as
$\dfrac{1+14x}{17-14x}=\dfrac{2}{1}$
Let us now cross multiply. We will get
$1+14x=2\left( 17-14x \right)$
Let us now expand the RHS. We will get
$1+14x=34-28x$
By collecting the variables to one side and constants to other, we will get
$\begin{align}
  & \Rightarrow 14x+28x=34-1 \\
 & \Rightarrow 42x=33 \\
 & \Rightarrow x=\dfrac{33}{42} \\
 & \Rightarrow x=\dfrac{11}{14} \\
\end{align}$

Hence, the fraction of the remaining votes that is against the resolution so that the total count will result in a vote of 2 to 1 against the resolution $=\dfrac{11}{14}$ .

So, the correct answer is “Option A”.

Note: You may not understand the given ratio, that is, the total count will result in a vote of 2 to 1 against the resolution. Be careful when writing the ratio of votes that must be in favor of total votes to votes that must be against total votes. You may write it as $\dfrac{\dfrac{17-14x}{18}}{\dfrac{1+14x}{18}}=\dfrac{2}{1}$ .