Half-Life Calculations in Radioactive Decay

1. Best grade 10 summary: half-life calculations usually use:

Adding a constant each time

Multiplying by 1/2 each half-life (exponential decay)

Random guessing

Temperature changes

Concept: exponential model. It’s repeated halving. Multiplying by the same factor each step is what makes the process exponential.

Explanation

Concept: exponential model. It’s repeated halving. Multiplying by the same factor each step is what makes the process exponential.

2. Each half-life multiplies the remaining amount by 1/2.

True

False

Concept: repeated-halving rule. That’s the repeated-halving rule. In each half-life interval, the remaining amount is multiplied by 0.5.

Explanation

Concept: repeated-halving rule. That’s the repeated-halving rule. In each half-life interval, the remaining amount is multiplied by 0.5.

3. Starting with 120 g, after 3 half-lives you have:

60 g

40 g

15 g

10 g

Concept: stepwise halving. 120 → 60 → 30 → 15. Three halvings correspond to three half-lives, leaving one eighth of the original.

Explanation

Concept: stepwise halving. 120 → 60 → 30 → 15. Three halvings correspond to three half-lives, leaving one eighth of the original.

4. If half-life is constant, the time from 100% to 50% equals the time from 50% to 25%.

True

False

Concept: equal halving intervals. Each halving takes one half-life. Because half-life is constant, each 50% reduction takes the same amount of time.

Explanation

Concept: equal halving intervals. Each halving takes one half-life. Because half-life is constant, each 50% reduction takes the same amount of time.

5. A sample has half-life 3 days. Starting at 200 g, how much after 9 days?

100 g

50 g

25 g

12.5 g

Concept: multiple half-lives. 9 days = 3 half-lives → 200 → 100 → 50 → 25. Three halvings reduce the amount to one eighth.

Explanation

Concept: multiple half-lives. 9 days = 3 half-lives → 200 → 100 → 50 → 25. Three halvings reduce the amount to one eighth.

6. Starting with 96 units, after 2 half-lives you have:

72

24

48

12

Concept: two halvings. 96 → 48 → 24. Two half-lives means the amount is reduced to one quarter.

Explanation

Concept: two halvings. 96 → 48 → 24. Two half-lives means the amount is reduced to one quarter.

7. If you know the half-life and starting amount, you can estimate the remaining amount after a whole number of half-lives.

True

False

Concept: discrete half-life steps. Repeated halving works. For whole numbers of half-lives, you can halve step-by-step or use (1/2)^n.

Explanation

Concept: discrete half-life steps. Repeated halving works. For whole numbers of half-lives, you can halve step-by-step or use (1/2)^n.

8. After 1 half-life, activity becomes:

Double

About half

About one quarter

Zero

Concept: activity vs number of nuclei. Activity is proportional to undecayed nuclei (roughly). If the number of undecayed nuclei halves, the decay rate (activity) also halves.

Explanation

Concept: activity vs number of nuclei. Activity is proportional to undecayed nuclei (roughly). If the number of undecayed nuclei halves, the decay rate (activity) also halves.

9. To reach 1/32 of the original, you need 5 half-lives.

True

False

Concept: powers of 1/2. (1/2)^5 = 1/32. Each additional half-life adds one more factor of 1/2.

Explanation

Concept: powers of 1/2. (1/2)^5 = 1/32. Each additional half-life adds one more factor of 1/2.

10. Half-life = 2 hours. After 6 hours, the remaining fraction is:

1/2

1/4

1/8

1/16

Concept: time to half-life conversion. 6 hours = 3 half-lives → (1/2)^3. Three half-lives means three halvings, giving 1/8 remaining.

Explanation

Concept: time to half-life conversion. 6 hours = 3 half-lives → (1/2)^3. Three half-lives means three halvings, giving 1/8 remaining.

11. Half-life = 4 years. How many years to reduce from 100% to 12.5%?

4

8

12

16

Concept: converting percent to fraction and steps. 12.5% = 1/8 = 3 half-lives → 3×4=12. Three halvings take you from 1 to 1/8.

Explanation

Concept: converting percent to fraction and steps. 12.5% = 1/8 = 3 half-lives → 3×4=12. Three halvings take you from 1 to 1/8.

12. If an isotope has a half-life of 1 day, it always fully disappears after 1 day.

True

False

Concept: half-life does not mean 'gone.' Only half decays; decay continues over time. After 1 day, about half remains, and after more days it keeps halving again.

Explanation

Concept: half-life does not mean 'gone.' Only half decays; decay continues over time. After 1 day, about half remains, and after more days it keeps halving again.

13. Half-life = 5 hours. How long for a sample to drop to 1/16?

5 h

10 h

15 h

20 h

Concept: matching fraction to halvings. 1/16 = 4 half-lives → 4×5=20 hours. Each half-life halves the amount, and four halvings produce 1/16.

Explanation

Concept: matching fraction to halvings. 1/16 = 4 half-lives → 4×5=20 hours. Each half-life halves the amount, and four halvings produce 1/16.

14. After 6 half-lives, the fraction remaining is ______.

Concept: (1/2)^n calculation. (1/2)^6 = 1/64. Six halving steps multiply the original by 1/2 six times.

Explanation

Concept: (1/2)^n calculation. (1/2)^6 = 1/64. Six halving steps multiply the original by 1/2 six times.

Submit

15. Half-life = 12 minutes. Starting with 640 counts, after 36 minutes you expect about:

320

160

80

40

Concept: converting time to half-lives. 36 minutes = 3 half-lives → 640 → 320 → 160 → 80. Three halvings give one eighth of the starting count.

Explanation

Concept: converting time to half-lives. 36 minutes = 3 half-lives → 640 → 320 → 160 → 80. Three halvings give one eighth of the starting count.

16. If 25% of the original remains, that corresponds to:

1 half-life

2 half-lives

3 half-lives

4 half-lives

Concept: fractions and exponents. 25% = 1/4 = (1/2)^2. Two halvings take you from 1 to 1/4.

Explanation

Concept: fractions and exponents. 25% = 1/4 = (1/2)^2. Two halvings take you from 1 to 1/4.

17. If the half-life is 8 days, after 16 days the fraction remaining is:

1/2

1/4

1/8

1/16

Concept: counting half-lives. 16 days = 2 half-lives → (1/2)^2. Two half-lives means the amount is halved twice, leaving one quarter.

Explanation

Concept: counting half-lives. 16 days = 2 half-lives → (1/2)^2. Two half-lives means the amount is halved twice, leaving one quarter.

18. A half-life of 1000 years means the isotope is not radioactive.

True

False

Concept: long half-life still means radioactive. It’s radioactive, just decays slowly. A long half-life means it remains active for a long time, not that it stops being radioactive.

Explanation

Concept: long half-life still means radioactive. It’s radioactive, just decays slowly. A long half-life means it remains active for a long time, not that it stops being radioactive.