Question

Suppose $y$ is inversely proportional to $x$. If $y=6$ when $x=4$, how do you find the constant of proportionality and write the formula for $y$ as a function of $x$ and use your formula to find $x$ when $y=8$.

Answer 1

Hint: In the problem we have the relation between the two variables $x$ and $y$ as they are inversely proportional. We will represent it mathematically. In the mathematical form we will remove the proportionality symbol by replacing it with a proportionality constant and equal to symbol. In the problem we have the values of both the variables, so we will substitute those values in the above obtained equation and calculate the proportionality constant. After calculating the proportionality constant, we will substitute this constant in the proportionality equation and that’s our required equation. After getting the equation we will substitute $y=8$ in the equation to calculate the value of $x$.

Complete step-by-step solution:
Given that, $y$ is inversely proportional to $x$. We can represent it mathematically as
$y\propto \dfrac{1}{x}$
We can replace the proportionality symbol with equality symbol and introducing the proportionality constant $k$, then we will get
$y=k\times \dfrac{1}{x}....\left( \text{i} \right)$
We have the values of $x$ and $y$ as $x=4$, $y=6$. Substituting these values in the above equation, then we will get
$\begin{align}
  & y=k\times \dfrac{1}{x} \\
 & \Rightarrow 6=k\times \dfrac{1}{4} \\
 & \Rightarrow k=24 \\
\end{align}$
Substituting the value of $k$ in equation $\left( \text{i} \right)$, then we will get
$y=\dfrac{24}{x}$
Hence the relation between the $x$ and $y$ is given by $y=\dfrac{24}{x}$ or $xy=24$.
Now we have $y=8$, substituting this value in the above equation, then we will get
$\begin{align}
  & x\left( 8 \right)=24 \\
 & \Rightarrow x=3 \\
\end{align}$

Note: In the problem they have mentioned that the variables $x$ and $y$ are inversely proportional, so we have followed the above method to solve the problem. If they have mentioned that the variables $x$ and $y$ are directly proportional they we need to use the representation $y\propto x$ and follow the same method to get the result.