Question

5 $m-n$$=3m+4n$ then find the values of the following expressions.

(i) $\frac{m^2+n^2}{m^2-n^2}$

(ii) $\frac{3m+4n}{3m-4n}$

Answer 1

$$\begin{gathered} 5m-n=3m+4n \\ \Rightarrow 5m-3m=4n+n \\ \Rightarrow 2m=5n \\ \Rightarrow \frac{m}{n}=\frac{5}{2} \end{gathered}$$
(i)
​$$\begin{gathered} \frac{m^2+n^2}{m^2-n^2} \\ =\frac{{\displaystyle \frac{m^2}{n^2}+\frac{n^2}{n^2}}}{{\displaystyle \frac{m^2}{n^2}-\frac{n^2}{n^2}}} \\ =\frac{{\displaystyle \frac{25}{4}+1}}{{\displaystyle \frac{25}{4}-1}}\left(\frac{m}{n}=\frac{5}{2}\right) \\ =\frac{29}{21} \end{gathered}$$
(ii)
$$\begin{gathered} \frac{3m+4n}{3m-4n} \\ =\frac{{\displaystyle \frac{3m}{n}+\frac{4n}{n}}}{{\displaystyle \frac{3m}{n}-\frac{4n}{n}}} \\ =\frac{3\times {\displaystyle \frac{5}{2}}+4}{3\times {\displaystyle \frac{5}{2}}-4} \\ =\frac{15+8}{15-8} \\ =\frac{23}{7} \end{gathered}$$