We all know how to add two completely differently fractions, say we have fractions: -
$\frac{a}{b}$ and $\frac{c}{d}$
We know that: -
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd} $$
Suppose (hypothetically) we add the aforementioned fractions in a completely different way, say: -
$$\frac{a}{b} + \frac{c}{d} = \frac{a + c}{b + d} $$
We get a unique fraction $\frac{a + c}{b + d} $ whose value will always be in between the fractions $\frac{a}{b}$ and $\frac{c}{d}$.
My question is: -
Can someone give me the proof (mathematically) that the statement given below will be always true: -
$$\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}$$
(where we assume that $\frac{a}{b} < \frac{c}{d}$)
