The half life of a radioactive substance is 5 years. After x years a given sample of the radioactive substance gets reduced to 6.25% of its initial value. The value of x is ____________.

Solution

Physics sir · 3 steps

Turn 6.25% into a fraction: it is 1/16, which is four halvings, so four half-lives.

1. Percent to fraction. 6.25% = 6.25/100 = 1/16 of the starting amount is left.

   N/N₀ = (1/2)ⁿ = 1/16 = (1/2)⁴

2. Count half-lives. So n = 4 half-lives have passed: 100% → 50% → 25% → 12.5% → 6.25%.
3. So. x = n × T_(1/2) = 4 × 5 = 20 years.

The common slip: 4. Stopping at the number of half-lives gives 4, not 20. Multiply by the 5-year half-life to get the time.

The idea to keep. When the remaining fraction is a power of 1/2, just count the halvings. Then time = number of half-lives × one half-life, no logarithms needed.

Answer from the NTA final answer key. Solution by PrepWiser.