The distance between object and its $$3$$ times magnified virtual image as produced by a convex lens is $$20$$ cm. The focal length of the lens used is ________ cm.

$$m = \frac{v}{u} \implies 3 = \frac{v}{u} \implies v = 3u$$

For a virtual image in a convex lens: $$\text{Distance} = |v| - |u| = |3u| - |u| = 2|u|$$

$$2|u| = 20 \implies |u| = 10\text{ cm}$$

Using sign convention: $$u = -10\text{ cm} \implies v = -30\text{ cm}$$

Lens formula: $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$

$$\frac{1}{f} = \frac{1}{-30} - \frac{1}{-10} = -\frac{1}{30} + \frac{1}{10} = \frac{2}{30} = \frac{1}{15} \implies f = 15\text{ cm}$$