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 $\dfrac{a+c}{b+d}$?

One nice thing to notice is that
$$
\frac{a}{b}=\frac{c}{d} \Leftrightarrow \frac{a}{b}=\frac{a+c}{b+d}
$$
no matter the values of $a$, $b$, $c$ and $d$. The $(\Rightarrow)$ is because $c=xa, d=xb$ for some $x$, so $\frac{a+c}{b+d}=\frac{a+xa}{b+xb}=\frac{a(1+x)}{b(1+x)}=\frac{a}{b}$. The other direction is similar.
The above is pretty easy to remember, and with that intuition in mind it is not hard to imagine that
$$
\frac{a}{b}<\frac{c}{d} \Leftrightarrow \frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}
$$
and similar results.
