Question

If $y$ varies inversely as $x$ and $y=3$ when $x=5$ how do you find $y$ when $x=2.5?$

Answer 1

Hint: The statement $y$ varies inversely as $x$ implies that $y$ is inversely proportional to $x.$ This relation says that when $x$ increases, $y$ decreases and when $x$ decreases, $y$ increases. We will convert this proportionality to equation using the proportionality constant.

Complete step by step solution:
Consider the given statement that says the relationship between the two variables $y$ and $x.$
Suppose that $y$ varies inversely as $x.$
This implies that $y$ is inversely proportional to $x.$
It means that $y$ decreases as $x$ increases and $y$ increases as $x$ decreases.
So, the proportionality can be written as, $y\propto \dfrac{1}{x}.$ This is our initial statement.
We need to convert this proportionality into an equation so as to find the value of $y$ when $x=2.5.$
To convert the proportionality into an equation, we need to multiply the left-hand side of the initial statement with a constant which is called the proportionality constant.
So, we will get the equation $y=k\dfrac{1}{x}=\dfrac{k}{x},$ where $k$ the proportionality constant.
Now, suppose we are transposing $x$ from the right-hand side of the equation to the left-hand side of the equation.
So, we will get $k=xy.$
Using the above equation, we can find the value of $k.$ For finding the value of $k,$ we need to apply the given values of $x$ and $y.$
We are going to put the values $x=5$ and $y=3$ in the equation $k=xy.$
So, we will get $k=5\times 3=15.$
Now, we will use the value of $k$ which we have obtained to find the value of $y$ when $x=2.5.$
Substituting the value of $k$ in the equation $y=\dfrac{k}{x}$ will give us $y=\dfrac{15}{x}.$
Now, in this equation we need to substitute the value of $x$ for which the value of $y$ to be determined.
So, $y=\dfrac{15}{2.5}=6.$

Hence when $x=2.5,$ $y=6.$

Note: If two quantities are directly proportional to each other, then that means they both increase or decrease together. We can write mathematically, if two quantities $x$ and $y$ are directly proportional to each other, then $x\propto y.$ This proportionality can be converted into equality by introducing the proportionality constant.