Question

C/m Nếu  \(\frac{m}{n}< \frac{p}{q}\)thì \(\frac{m}{n}< \frac{m+p}{n+q}< \frac{p}{q}\)

Answer 1

\(\frac{m}{n}< \frac{p}{q}\Leftrightarrow mq< pn\)

\(\Rightarrow mq+mn< pn+mn\)

\(\Rightarrow m\left(n+q\right)< n\left(m+p\right)\)

\(\Rightarrow\frac{m}{n}< \frac{m+p}{n+q}\) (1)

\(mq< pn\)

\(\Rightarrow mq+pq=pn+pq\)

\(\Rightarrow q\left(m+p\right)< p\left(n+q\right)\) 

\(\Rightarrow\frac{m+p}{n+q}< \frac{p}{q}\) (2)

Từ (1) và (2) suy ra: \(\frac{m}{n}< \frac{m+p}{n+q}< \frac{p}{q}\)