Question

If $r$ varies jointly as $p$ and $q$ and inversely as $t$, then how do you find an equation for r if $r=6$ when $p=8,q=-3$ and $t=3$ ?

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 four variables where r varies jointly as p and q and inversely as t. The inversely proportional number is actually directly proportional to the inverse of the given number. It’s given r varies directly as p and q which gives $r\propto pq$.Also, $r$ varies inversely as $t$ which can be expressed as $r\propto \dfrac{1}{t}$.

So, the combined relation is $r\propto \dfrac{pq}{t}$. To get rid of the proportionality we use the proportionality constant which gives $r=k\dfrac{pq}{t}$.Here, the number k is the proportionality constant. It’s given $r=6$ when $p=8,q=-3$ and $t=3$. We put the values in the equation $r=k\dfrac{pq}{t}$to find the value of k. So,
$6=k\dfrac{8\times \left( -3 \right)}{3}$
Simplifying we get
\[6=k\dfrac{8\times \left( -3 \right)}{3} \\
\Rightarrow k=\dfrac{3\times 6}{8\times \left( -3 \right)}\\
\Rightarrow k =\dfrac{18}{-24}\\
\Rightarrow k =-\dfrac{3}{4} \\ \]
Therefore, the equation becomes with the value of k as $r=\left( -\dfrac{3}{4} \right)\dfrac{pq}{t}$.
Now we simplify the equation to get,
$r=\left( -\dfrac{3}{4} \right)\dfrac{pq}{t} \\
\Rightarrow 4tr=-3pq \\
\therefore 4tr+3pq=0 \\ $
Then the equation for the variables is $4tr+3pq=0$.

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.