Prove $M,n,p,q>0$, if $\frac{m}{n}\lt\frac{p}{q},$ show $\frac{m}{n}\lt\frac{m+p}{n+q}\lt\frac{p}{q},$

Question

Prove the following inequality

Given :$M,n,p,q>0$,

$$\frac{m}{n}\lt\frac{p}{q},$$

To show:

$$\frac{m}{n}\lt\frac{m+p}{n+q}\lt\frac{p}{q},$$

I would like to solve it myself so maybe a hint would help. I am working ahead of class so I am not sure what I am doing ;) I've tried to manipulate the inequality in different ways but did not get anywhere... I am sure the solution is quite simple, and maybe with a hint, I can figure it out myself

Answer 1

The left inequality is true because $$\frac{m+p}{n+q}-\frac{m}{n}=\frac{np-mq}{n(n+q)}>0$$

and $\frac{m}{n}<\frac{p}{q}$ gives $mq<np$.

A proof of the right inequality we can get by the same way.

Answer 2

Hint:

Since all numbers are positive, you simply have to use this rule:

Let $a,b,c,d\:$ be positive numbers. Then $$\frac ab<\frac cd\iff ad<bc.$$

Indeed, your left inequality, for instance, reduces to $$m(n+q)<n(m+p)\iff mq<np, $$ which is another form of the hypothesis.