rational numbers of the form $\frac{a^3+b^3}{c^3+d^3}$ Show that all positive rational numbers can be written in the form
$$\frac{a^3+b^3}{c^3+d^3}$$
where $a,b,c,d$ are positive integers.

Xam commented on 2017 Feb 13 that an answer was given here by zabelman on 2005 Dec 17. Also on 2017 Feb 13, timon92 mentioned the following answer found in a post by mathlove on 2015 Mar 11:

For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that 
  $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$
For $r=p/q$ where $p,q$ are positive integers, we can take 
  $$(a,b,c,d)=(3ps^3t+9qt^4,\ 3ps^3t-9qt^4,\ 9qst^3+ps^4,\ 9qst^3-ps^4)$$
  where $s,t$ are positive integers such that $3\lt r\cdot(s/t)^3\lt 9$.
For $r=2014/89$, for example, since we have $(2014/89)\cdot(2/3)^3\approx 6.7$, taking $(p,q,s,t)=(2014,89,2,3)$ gives us $$\frac{2014}{89}=\frac{209889^3+80127^3}{75478^3+11030^3}.$$

