Radioactive decay arithmetic: after a duration equal to four half-lives, what percentage of the original amount of a radioisotope remains undecayed?

Introduction / Context:
Half-life problems are common in nuclear chemistry and radiation protection. They test exponential decay understanding and the ability to translate half-life intervals into remaining fractions or percentages of a radionuclide.

Given Data / Assumptions:

• Ideal first-order decay law applies: N = N0 * (1/2)^(t/t_half).
• No production or replenishment of the isotope during the interval.
• We measure remaining, not decayed, fraction.

Concept / Approach:

Each half-life reduces the quantity by half. After n half-lives, remaining fraction is (1/2)^n. Converting to percentage involves multiplying by 100. For four half-lives, n = 4, so remaining fraction is (1/2)^4 = 1/16 = 0.0625 = 6.25% of the original amount.

Step-by-Step Solution:

Verification / Alternative check:

Successive halving: 100% → 50% (1st) → 25% (2nd) → 12.5% (3rd) → 6.25% (4th). Matches the calculation.

Why Other Options Are Wrong:

12.50%: This is after three half-lives, not four. 3.125%: This is after five half-lives. 25%: After two half-lives. 50%: After one half-life only.

Common Pitfalls:

Confusing the amount decayed with amount remaining; stopping one half-life too early; arithmetic errors when converting fractions to percentages.

Final Answer:

6.25%