ACT Science: Calculate Radioactive Decay Using Half-Life Without Confusion

Half-life (t₁/₂) is the time required for half of a radioactive sample to decay. After one half-life, half the original sample remains (and has decayed). After two half-lives, one-quarter remains (three-quarters have decayed). Formula: Amount remaining=Initial amount×(1/2)^n, where n=number of half-lives. Example: Carbon-14 has a half-life of 5,730 years. If a sample starts with 100 grams, after 5,730 years (one half-life), 50 grams remain. After 11,460 years (two half-lives), 25 grams remain. The key is recognizing that half-life is exponential decay; the amount is repeatedly halved, not reduced linearly by a fixed quantity.

Why it matters: Half-life is used in carbon dating, medical applications (radioactive tracers), and nuclear waste disposal. ACT Science questions test whether you understand the exponential nature of decay and can predict remaining amounts or calculate elapsed time given decay data.

Mistake 1: Linear thinking instead of exponential. If half-life is 10 years and you have 100 grams, after 20 years, the amount is NOT 0 (100 minus 50 minus 50). It is 25 grams (half of half). Half-life is exponential: 100→50→25→12.5..., not 100→50→0. Mistake 2: Calculating time incorrectly. If half-life is 5,730 years and elapsed time is 11,460 years, that is 11,460/5,730=2 half-lives, not 4. Mistakes with division lead to wrong exponents and wrong answers. Always divide elapsed time by half-life to find n. Then apply the formula with the correct n.

Before calculating, write down: Initial amount, half-life, elapsed time, and n (number of half-lives). Verify n by dividing elapsed time by half-life. This setup prevents careless errors.

Problem 1: Iodine-131 has a half-life of 8 days. Starting with 80 grams, how much remains after 16 days? n=16/8=2 half-lives. Amount=80×(1/2)²=80×(1/4)=20 grams. Problem 2: A sample of Uranium-238 decays from 100 grams to 12.5 grams. If the half-life is 4.5 billion years, how much time has elapsed? Work backward: 100→50→25→12.5 is three half-lives. Time=3×4.5=13.5 billion years. Problem 3: Tritium has a half-life of 12.3 years. After 49.2 years, what percent of the original sample remains? n=49.2/12.3=4 half-lives. Amount=Original×(1/2)⁴=Original×(1/16)=6.25% of the original. Problem 4: Polonium-210 decays such that after 138 days, 1/8 of the original remains. What is the half-life? If 1/8 remains, three half-lives have passed (1→1/2→1/4→1/8). Half-life=138/3=46 days. For each problem, identify the given information, calculate n, apply the formula, and verify by working backward.

After calculating, verify by working backward. If 20 grams is the answer and the original was 80, check: 20×2=40, 40×2=80. Correct. This backward check catches errors in your formula application.

Half-life questions appear in chemistry (radioactive decay), physics (nuclear reactions), and earth science (carbon dating). Understanding exponential decay helps you answer questions about nuclear stability, predict decay products, and interpret age estimates. Once you master the half-life formula and recognize that decay is exponential (not linear), you solve these questions confidently across all science contexts.

Spend 20 minutes this week solving 10 half-life problems (vary the number of half-lives and include both forward and backward calculations). Time yourself; each should take 1-2 minutes. By test day, half-life calculations will be automatic, and you will answer radioactive decay questions with the confidence that comes from understanding exponential processes.