If the fraction of the atoms of a radioactive material left after 120years is $\frac{1}{64}$, what is the half-life of the material?  $\frac{1}{64}$

Analyze the question After 120 years, only 1/64 of the original radioactive atoms remain. We need the half-life, which is the time for half of the atoms to decay.

Concept After n half-lives, the fraction remaining is

and the number of half-lives is n = total time ÷ half-life, so n = t / T½.

Solution

1. Set the fraction remaining equal to (1/2)ⁿ:

2. Write 64 as a power of 2: 64 = 2⁶, so

3. So 6 half-lives have passed in 120 years:

Check: 120 → after 20 years, 1/2; after 40, 1/4; after 60, 1/8; after 80, 1/16; after 100, 1/32; after 120, 1/64. ✓

Tip: For half-life questions, convert the fraction to a power of ½ to find the number of half-lives, then divide the total time by it. Useful values: 1/2 = 1 half-life, 1/4 = 2, 1/8 = 3, 1/16 = 4, 1/32 = 5, 1/64 = 6.