Can we say that $\frac{a+b}{c+d} \leq \frac{a}{c} + \frac{b}{d} $ If $a,b,c,d$ are positive real numbers, can we say that $\frac{a+b}{c+d} \leq \frac{a}{c} + \frac{b}{d} $ is always true? If no, can you please give insignts on under which conditions this might be true. 
Any references to a similar type of inequalities are also welcome
Thank you, 

After the latest edit:
Wlog. $\frac ac\le\frac bd$. Then $$\frac ac\le\frac{a+b}{c+d}\le\frac bd<\frac ac+\frac bd$$
Indeed, $\frac ac\le \frac bd$ implies $\frac{bc-ad}{cd}\ge 0$, i.e., $bc-ad\ge 0$.
Then 
$$ \frac bd-\frac{a+b}{c+d}=\frac{b(c+d)-d(a+b)}{d(c+d)}=\frac{bc-ad}{d(c+d)}\ge0$$
and
$$ \frac{a+b}{c+d}-\frac ac=\frac{c(a+b)-a(c+d)}{(c+d)a}=\frac{bc-ad}{(c+d)a}
\ge 0$$
