Question

If $p$ is directly proportional to $q$ and $p=9$ when $q=7.5$, how do you find q when $p=24$?

Answer 1

Hint:We first try to form the proportionality equation for the variables. We take an arbitrary constant. We use the given values of the variables to find the value of the constant. Finally, we put the constant’s value to find the equation.

Complete step by step answer:
We have been given the relation between two variables where p is directly proportional to $q$.The inversely proportional number is actually directly proportional to the inverse of the given number.It’s given $p$ is directly proportional to $q$ which gives $p\propto q$.To get rid of the proportionality we use the proportionality constant which gives $p=kq$.Here, the number $k$ is the proportionality constant.

It’s given $p=9$ when $q=7.5$. We put the values in the equation $p=kq$ to find the value of k. So, $9=k\left( 7.5 \right)$. Simplifying we get
\[9=k\left( 7.5 \right) \\
\Rightarrow k=\dfrac{9}{7.5}\\
\Rightarrow k =\dfrac{90}{75}\\
\Rightarrow k =\dfrac{6}{5} \\ \]
Therefore, the equation becomes with the value of k as $p=\dfrac{6}{5}q$.
Now we simplify the equation to get
\[p=\dfrac{6}{5}q \\
\Rightarrow 5p=6q \]
Now we find q when $p=24$.
So, \[5\times 24=6q\] which gives \[q=\dfrac{5\times 24}{6}=5\times 4=20\].

The value of q is 20.

Note: In a direct proportion, the ratio between matching quantities stays the same if they are divided. They form equivalent fractions. In an indirect (or inverse) proportion, as one quantity increases, the other decreases. In an inverse proportion, the product of the matching quantities stays the same.