Adding constants to the numerator and denominator of a fraction

Question

I have a basic question. If $\frac{a}{b} \leq \frac{c}{d}$ and we also have $\frac{e}{f}=\frac{g}{h}$, can we say:

$$ \frac{a+e}{b+f} \leq \frac{c+g}{d+h}$$

Basically, can we add constants to the numerator and enumerator and conclude the above inequality? Does it hold? If so, does anybody know how to prove this? Regards, Ali

Answer 1

No: $\frac{2}{1}\leq\frac{1}{\frac 12}$ (in fact, this is an equality) but $\frac{2+2}{1+1}\not\leq\frac{1+1}{\frac 12+\frac 12}$.

Answer 2

This is not correct. For example: $$\frac{1}{2}=0.5 < 0.66\approx \frac{1}{1.5}.$$ If you add $\frac{5}{5}=\frac{1}{1}$ you get $$\frac{6}{7} \approx 0.85 > 0.8=\frac{2}{2.5}$$

Answer 3

No, $\frac{1}{2} \leq \frac{3}{4}$ and $\frac{4}{2} = \frac{2}{1}$, but $\frac{1+4}{2+2} = \frac{5}{4} > \frac{3+2}{4+1}=\frac{5}{5}$