In series LR circuit, $X_L = 3 R$. Now a capacitor with $X_C=R$ is added in series. The ratio of new to old power factor is

$\sqrt{2}$

$\text {Power factor Initially } cos\phi = \frac{R}{Z_i} = \frac{R}{\sqrt {R^2 + X_L^2}} = \frac{1}{\sqrt 10}$

$ \text{New Power Factor }cos\phi'= \frac{R}{Z_f} = \frac{R}{\sqrt {R^2 + (X_L-X_C)^2}} = \frac{1}{\sqrt 5}$

$\text{Ratio }= \sqrt 2$