TITLE: Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$?
QUESTION [9 upvotes]: We were trying to come up with an easy way to generate a rational number in between two existing rational numbers with a fairly low numerator and denominator (the way we were doing this earlier was to find the average of the two rationals, but that results in a denominator of up to $c * d$. Does this inequality hold for all values of $a, b, c, d$?

REPLY [2 votes]: Weighted average of two numbers with positive weight lies between the two numbers.
$$
x<y\phantom{x}\leftrightarrow\phantom{x}x<\frac{b}{b+d}\cdot x+\frac{d}{b+d}\cdot y<y
$$
Now substitute $x=\frac{a}{b}$ and $y=\frac{c}{d}$