In 15 days, the earth picks up

Let $x$ be the number of days taken by the earth to pick up $4.8 \times 10^{8} \mathrm{~kg}$ of dust.

Since the amount of dust picked up by the earth and the number of days are in direct variation, we get:

$\frac{15}{x}=\frac{1.2 \times 10^{8}}{4.8 \times 10^{8}}$

$\Rightarrow x=15 \times \frac{4.8}{1.2}$

$\Rightarrow x=60$

Thus, the required number of days will be 60.