Adding a different constant to numerator and denominator Suppose that $a$ is less than $b$ , $c$ is less than $d$. 
What is the relation between $\dfrac{a}{b}$ and  $\dfrac{a+c}{b+d}$? Is  $\dfrac{a}{b}$ less than, greater than or equal to $\dfrac{a+c}{b+d}$?

Note that if $b$ and $d$ have the same sign, then
$$
\frac{a}{b}-\frac{a+c}{b+d}=\frac{ad-bc}{b(b+d)}
$$
and
$$
\frac{a+c}{b+d}-\frac{c}{d}=\frac{ad-bc}{d(b+d)}
$$
also have the same sign.
Therefore, if $b$ and $d$ have the same sign, then $\dfrac{a+c}{b+d}$ is between $\dfrac{a}{b}$ and $\dfrac{c}{d}$.
Comment: As Srivatsan points out, if $b$ and $d$ are both positive,
$$
\frac{a}{b}\lesseqqgtr\frac{a+c}{b+d}\text{ if }\frac{a}{b}\lesseqqgtr\frac{c}{d}
$$
