How does this numerators and denominators relate with the fraction?

Question

Suppose if we have two fractions $\frac{a}{b}$ and $\frac{c}{d}$ then how are their values related with the fraction $\frac{a+c}{b+d}$ ?

I have observed this inequality: $\frac{a}{b}\le\frac{a+c}{b+d}\le\frac{c}{d}$.

Does this hold true $\forall a,b,c,d \in N$ ? Is there any proof?

I am sorry if this question is too basic, but I want help regarding this.

Answer 1

HINT:

$$\dfrac{a+c}{b+d}-\dfrac ab=\dfrac{bc-ad}{b(b+d)}=\dfrac{\dfrac cd-\dfrac ab}{b^2d(b+d)}$$

This will be $>0$ if $d(b+d)>0$ and $\dfrac cd-\dfrac ab>0$

Answer 2

we have $$\frac{a+c}{b+d}\le \frac{c}{d}$$ if $$\frac{c}{d}-\frac{a+c}{b+d}=\frac{bc-ad}{d(b+d)}>0$$ and this is true since $$\frac{a}{b}<\frac{c}{d}$$ if $$ad<bc$$ further we have $$\frac{a+c}{b+d}-\frac{a}{b}=\frac{bc-ad}{b(b+d)}$$