The rate of data transfer, $r$, over a particular network is directly proportional to the bandwidth, $b$, and inversely proportional to the square of the number of networked computers, $n$.
Quantity A = The resulting rate of data transfer if the bandwidth is quadrupled and the number of networked is more than tripled.
Quantity B = $\dfrac{4r}{9}$
Which is greater?
From propotionality I have got that $r=\alpha b$ and $r=\dfrac{\beta}{n^2}$ where $\alpha$ and $\beta$ are constants. Here I am stuck. But in official solution they have written that $r=k\dfrac{b}{n^2}$ where $k$ is also constant. How they got the last relation?
And can anyone explain how they derived it?
You cannot say $r=\alpha b$ if you know $r$ depends on another variable as well (in this case, $n$). For example, consider that the volume of a cylinder is directly proportional to both the height and the square of the radius. Then it does not follow that $V=\alpha h$ for some constant $\alpha$. Rather, $\alpha$ must itself depend on $r^2$.
For this reason, your $\alpha$ must depend on $n^{-2}$. If we hold $b$ constant, we know $r$ must be directly proportional to $n^{-2}$, so we conclude $\alpha=k\cdot n^{-2}$.