Question

If $p$ varies as $q$ and if $p=5$ when $q=10$, find $q$ when $p=20$

Answer 1

The statement " $x$ varies directly as $y$ ," means that when $y$ increases, $x$ increases by the same factor. In other words, $y$ and $x$ always have the same ratio that is:

$x=ky$

where $k$ is the constant of variation.

Here, it is given that $p$ varies as $q$ if $p=5$ and $q=10$, therefore,

$p=kq\Rightarrow 5=k\times 10\Rightarrow 5=10k\Rightarrow k=\frac{5}{10}=\frac{1}{2}$

Thus, the general equation is $p=\frac{1}{2}q$.

Since $p=20$, therefore,

$p=\frac{1}{2}q\Rightarrow 20=\frac{1}{2}q\Rightarrow q=20\times 2=40$

Hence, $q=40$ when $p=20$.