Answer
$\dfrac{a}{b}\lt \dfrac{a+c}{b+d}\lt \dfrac{c}{d}$
Work Step by Step
Consider the inequality $\dfrac{a}{b}\lt \dfrac{c}{d}$
This gives: $\dfrac{ad}{b}\lt c$ ....(1)
Also, we have $\dfrac{a}{b}\lt \dfrac{c}{d}$
$a\lt \dfrac{bc}{d}$
or, $a+c\lt \dfrac{bc}{d}+c$ ...(2)
Combine the both equations (1) and (2).
This gives: $\dfrac{ad}{b}+a\lt a+c \lt \dfrac{bc}{d}+c$
or, $\dfrac{\dfrac{ad}{b}+a}{b+d}+\dfrac{a+c}{b+d}\lt \dfrac{\dfrac{bc}{d}+c}{b+d}$
Thus, $\dfrac{a}{b}\lt \dfrac{a+c}{b+d}\lt \dfrac{c}{d}$
