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**Hint:**In this problem we have the two variables say $x$ and $y$, at the same time we have the relation between the two variables as inversely proportional. We will first represent this in mathematical form. Now we will replace the proportionality symbol with a constant and an equal to symbol. We will consider this as an equation and numbered as equation one. Now we have that if the variable $x$ increases by $20\%$, so we will calculate the changed value and denote it as ${{x}^{'}}$ and we will assume the variable that is proportional to ${{x}^{'}}$ as ${{y}^{'}}$. Now we will form a mathematical equation by using all the values we have up to now in the problem. After simplifying the obtained equation, we will get the required result.

**Complete step by step solution:**

Given that, $x$ and $y$ are inversely proportional.

We can represent the above statement mathematically as

$x\propto \dfrac{1}{y}$

Removing the proportionality symbol by introducing the proportionality constant say $k$ in the above equation, then we will get

$\begin{align}

& \Rightarrow x=k\times \dfrac{1}{y} \\

& \Rightarrow xy=k....\left( \text{i} \right) \\

\end{align}$

Now we have that $x$ increases by $20\%$. So, the new changed value is

$\begin{align}

& {{x}^{'}}=x+20\%x \\

& \Rightarrow {{x}^{'}}=x+\dfrac{20}{100}x \\

& \Rightarrow {{x}^{'}}=x+0.2x \\

& \Rightarrow {{x}^{'}}=1.2x \\

\end{align}$

Now the proportional value for ${{x}^{'}}$ from the equation $\left( \text{i} \right)$ is given by

$\Rightarrow {{x}^{'}}\times {{y}^{'}}=k$

From equation $\left( \text{i} \right)$ substituting the value $k=xy$ in the above equation, then we will get

$\begin{align}

& \Rightarrow 1.2x\times {{y}^{'}}=xy \\

& \Rightarrow {{y}^{'}}=\dfrac{y}{1.2} \\

\end{align}$

Now the decrease in the variable $y$ will be

$\Rightarrow d=y-{{y}^{'}}$

Substituting ${{y}^{'}}=\dfrac{y}{1.2}$ in the above equation, then we will have

$\begin{align}

& \Rightarrow d=y-\dfrac{y}{1.2} \\

& \Rightarrow d=y\left( 1-\dfrac{1}{1.2} \right) \\

& \Rightarrow d=y\left( \dfrac{1.2-1}{1.2} \right) \\

& \Rightarrow d=\dfrac{y}{6} \\

\end{align}$

Now the percentage decrease will be given by

$\begin{align}

& \Rightarrow d\%=\dfrac{d}{y}\times 100\% \\

& \Rightarrow d\%=\dfrac{\dfrac{y}{6}}{y}\times 100\% \\

& \Rightarrow d\%=\dfrac{100}{6}\% \\

& \Rightarrow d\%=16\dfrac{2}{3}\% \\

\end{align}$

**Hence Option – B is the correct answer.**

**Note:**In this problem they have mentioned that the two variables are inversely proportional so we have represented it as $x\propto \dfrac{1}{y}$ and solved the problem. If they have mentioned that the two variables are directly proportional, then we must use the representation $x\propto y$ and solve the problem.

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