Rn-222 has a half life of 3.82 days. How long before only ⅛ of the original remains?

Three half-lives reduce the quantity remaining to ⅛ of the original sample size; 1→¹/₂ → ¹/₄ → ¹/₈.

3 half-lives x 3.82 days/half-life = 11.5 days

Every 3.82 days, you will lose 1/2 of however much you had 3.82 days ago.

Since 1/8th if half of a quarter (a cycle), and a quarter is half of a half (another cycle), and half is half the original (another cycle), you can expect it to take 3 cycles (or 11.46 days) to have 1/8th of your original amount

in 3.82 day 1/2 of the sample remains, in 7.64 days 1/2 of 1/2 or 1/4 remains, in 11.46 days 1/2 of 1/4 or 1/8 remains