Question

If $\frac{a}{b}=\frac{2}{3}$ then find the values of the following expressions.

(i) $\frac{4a+3b}{3b}$ (ii) $\frac{5a^2+2b^2}{5a^2-2b^2}$ (iii) $\frac{a^3+b^3}{b^3}$ (iv) $\frac{7b-4a}{7b+4a}$

Answer 1

(i)
$$\begin{gathered} \frac{a}{b}=\frac{2}{3} \\ \Rightarrow \frac{4a}{3b}=\frac{4\times 2}{3\times 3}=\frac{8}{9} \end{gathered}$$
Applying componendo, we get

$$\begin{gathered} \frac{4a+3b}{3b}=\frac{8+9}{9} \\ \Rightarrow \frac{{\displaystyle 4a+3b}}{{\displaystyle 3b}}=\frac{{\displaystyle 17}}{{\displaystyle 9}} \end{gathered}$$
(ii)
$$\begin{gathered} \frac{a}{b}=\frac{2}{3} \\ \Rightarrow \frac{a^2}{b^2}=\frac{4}{9}\left(\mathrm{Squaring}\mathrm{both}\mathrm{sides}\right) \\ \Rightarrow \frac{5a^2}{2b^2}=\frac{5\times 4}{2\times 9} \\ \Rightarrow \frac{5a^2}{2b^2}=\frac{20}{18} \end{gathered}$$
Applying componendo and dividendo, we get

$$\begin{gathered} \frac{5a^2+2b^2}{5a^2-2b^2}=\frac{20+18}{20-18} \\ \Rightarrow \frac{5a^2+2b^2}{5a^2-2b^2}=\frac{38}{2} \\ \Rightarrow \frac{5a^2+2b^2}{5a^2-2b^2}=19 \end{gathered}$$
(iii)
$$\begin{gathered} \frac{a}{b}=\frac{2}{3} \\ \Rightarrow \frac{a^3}{b^3}=\frac{8}{27}\left(\mathrm{Cubing}\mathrm{both}\mathrm{sides}\right) \end{gathered}$$
Applying componendo, we get

$$\begin{gathered} \frac{a^3+b^3}{b^3}=\frac{8+27}{27} \\ \Rightarrow \frac{a^3+b^3}{b^3}=\frac{35}{27} \end{gathered}$$
(iv)
$\frac{a}{b}=\frac{2}{3}$

Applying invertendo, we get

$$\begin{gathered} \frac{b}{a}=\frac{3}{2} \\ \Rightarrow \frac{7b}{4a}=\frac{7\times 3}{4\times 2}=\frac{21}{8} \end{gathered}$$
Applying componendo and dividendo, we get

$$\begin{gathered} \frac{7b+4a}{7b-4a}=\frac{21+8}{21-8} \\ \Rightarrow \frac{7b+4a}{7b-4a}=\frac{29}{13} \end{gathered}$$
Applying invertendo, we get

$\frac{7b-4a}{7b+4a}=\frac{13}{29}$