Question

If a : b = 2 : 5, find the value of $\frac{3a+2b}{4a+b}$.

Answer 1

It is given that,
$\frac{a}{b}=\frac{2}{5}$ ...(1)

Now,
$$\begin{gathered} \frac{3a+2b}{4a+b}=\frac{\left(3a+2b\right)÷b}{\left(4a+b\right)÷b} \\ =\frac{3\left(a÷b\right)+2}{4\left(a÷b\right)+1} \\ =\frac{3\left({\displaystyle \frac{a}{b}}\right)+2}{4\left({\displaystyle \frac{a}{b}}\right)+1} \\ =\frac{3\left({\displaystyle \frac{2}{5}}\right)+2}{4\left({\displaystyle \frac{2}{5}}\right)+1}\left[\mathrm{From}\left(1\right)\right] \\ =\frac{{\displaystyle \frac{6+2\times 5}{5}}}{{\displaystyle \frac{8+1\times 5}{5}}} \\ =\frac{6+10}{8+5} \\ =\frac{16}{13} \end{gathered}$$

Hence, $\frac{3a+2b}{4a+b}=\frac{16}{13}$.