UPSC Prelims 2001 Q113: Science & Technology – Energy & Environment
A radioactive substance has a half - life of four months. Three fourth of the substance would
decay in
- (a) 3 months
- (b) 4 months
- (c) 8 months
- (d) 12 months
Answer: (c) 8 months
Explanation
Time for three-fourths of a radioactive substance to decay
| Time | Amount left |
|---|---|
| Start | 1 (whole) |
| After 1 half-life (4 months) | 1/2 left, 1/2 decayed |
| After 2 half-lives (8 months) | 1/4 left, 3/4 decayed |
| After 3 half-lives (12 months) | 1/8 left, 7/8 decayed |
Given: a radioactive substance with a half-life of four months. Three-fourths decayed means one-fourth is left, and 1/4 = (1/2)². That needs two half-lives, so 2 × 4 = 8 months.
Answer: (c) 8 months.
Why each option is right or wrong
- (a) Incorrect. In 3 months less than half has decayed.
- (b) Incorrect. In 4 months only half has decayed.
- (c) CORRECT. Two half-lives (8 months) leave one-fourth.
- (d) Incorrect. In 12 months seven-eighths has decayed.
Key point
Amount left after n half-lives = (1/2)ⁿ. Carbon-14 half-life is about 5,730 years.
Remember
3/4 decayed = 1/4 left = 2 half-lives.
Common mistakes
- Thinking 3/4 decay takes 1.5 half-lives (6 months).
- Thinking decay is linear in time, not halving.
- Mixing up the fraction decayed with the fraction left.
Exam tip
Convert 'decayed' to 'left', then count halvings.
Sources
- NCERT Class XII — Physics Part II, chapter 'Nuclei' (radioactive decay, half-life)
- H. C. Verma — Concepts of Physics Part 2, chapter 'The Nucleus'