Question

If $bc:ac:ab=1:3:5,$ then find $\frac{a}{bc}:\frac{b}{ca}$.

Answer 1

Given:

$bc:ac:ab=1:3:5$

Let say $\to bc=x$ ; $ac=3x$ ; $ab=5x$

If we multiply all then -

$bc\times ac\times ab=\left(x\right)\left(3x\right)\left(5x\right)$

$a^2{b}^2{c}^2=15x^3$

$abc=x\sqrt{15x}$ ___ (i)

$\because bc=x$; so, $a\left(x\right)=x\sqrt{15x}\Rightarrow a=\sqrt{15x}$

similarly $\because ac=3x$;

$abc=x\sqrt{15x}$

$b\left(ac\right)=x\sqrt{15x}$

$b\left(3x\right)=x\sqrt{15x}\Rightarrow b=\frac{1}{3}\sqrt{15x}$

Ratio :

$\Rightarrow \frac{a}{bc}:\frac{b}{ca}$

$\Rightarrow \frac{\sqrt{15x}}{x}:\frac{\frac{1}{3}\sqrt{15x}}{3x}\Rightarrow \frac{1}{1}:\frac{1}{3\times 3}\Rightarrow 1:\frac{1}{9}$

$\Rightarrow 9:1$ Answer