The half-life of a radio active element is 5 years. The fraction of the radioactive substance that remains after 20 years is

Half-life is defined as the time required for half of a radioactive sample to decay, so after each half‑life the remaining quantity is halved [1]. The quantitative expression for decay after time t with half‑life T1/2 is N(t)=N0*(1/2)^(t/T1/2); thus every complete half‑life multiplies the remainder by 1/2 [2]. Here T1/2 = 5 years and t = 20 years, so the number of half‑lives elapsed is 20/5 = 4. Therefore the fraction remaining is (1/2)^4 = 1/16. Hence the correct choice is option 4 (1/16).

Sources

1. [1] Environment, Shankar IAS Acedemy .(ed 10th) > Chapter 5: Environmental Pollution > Period of Radioactivity > p. 83
2. [2] https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/31:_Radioactivity_and_Nuclear_Physics/31.05:_Half-Life_and_Activity