All solvers

    Half-life calculations: worked examples

    By Engr. Mehmood Kakar · Updated

    Quick answer

    Count the half-lives, then halve that many times. Number of half-lives = time ÷ half-life, with both times in the same unit. After n half-lives the activity is the starting activity × (½)n. To find a half-life, count how many times the activity halved and divide the time by that number.

    The one idea you need

    Every half-life, the activity of a sample halves. Not "drops by 100", not "drops by a quarter": it halves, whatever it started at. The half-life calculator shows this as A = A0 × (½)n, but at GCSE you can almost always just halve step by step and write each step down.

    The numbers in these examples are made up to show the method.

    Finding the activity left

    A sample has an activity of 640 Bq. Its half-life is 15 minutes. Calculate its activity after 1 hour. (3 marks)

    1. Same units: 1 hour = 60 minutes
    2. Number of half-lives = 60 ÷ 15 = 4
    3. 640 → 320 → 160 → 80 → 40
    4. Activity = 40 Bq

    Writing the chain of halvings earns the method marks even if you slip on the last step.

    Net decline as a fraction

    Some Higher questions ask what fraction of the original is left, or the ratio of what is left to what you started with. Multiply ½ by itself once for each half-life:

    Half-lives passedFraction leftPercentage left
    11/250%
    21/425%
    31/812.5%
    41/166.25%
    51/323.125%

    After 5 half-lives, the ratio of remaining to original nuclei is 1 : 32.

    Finding the half-life

    The activity of a sample falls from 4800 Bq to 600 Bq in 24 days. Calculate its half-life. (3 marks)

    1. 4800 → 2400 → 1200 → 600
    2. That is 3 half-lives
    3. Half-life = 24 ÷ 3
    4. Half-life = 8 days

    A sample has a half-life of 6 hours. How long does it take for its activity to fall from 3200 Bq to 200 Bq? (3 marks)

    1. 3200 → 1600 → 800 → 400 → 200: 4 half-lives
    2. Time = 4 × 6
    3. Time = 24 hours

    Half-life from a graph

    On a graph of activity (or count rate) against time:

    1. Read the starting activity where the curve meets the y-axis.
    2. Halve it, and find that value on the y-axis.
    3. Go across to the curve, then straight down to the time axis.
    4. The time you read off is one half-life.

    Draw your construction lines on the graph. Examiners often give a mark for them, and repeating the method from half to a quarter lets you check your answer.

    Background count

    Detectors also pick up background radiation, so a question may give you a background count to subtract first.

    A detector records 520 counts per minute from a source. The background count is 20 counts per minute. What will the detector record after 2 half-lives? (3 marks)

    1. Corrected count = 520 − 20 = 500 counts per minute
    2. After 2 half-lives: 500 → 250 → 125
    3. Add the background back on: 125 + 20
    4. Recorded count = 145 counts per minute

    Halving 520 directly gives 130, which is wrong: the background doesn't decay.

    Carbon dating

    Carbon-14 has a half-life of 5700 years. A piece of wood has 1/16 of the carbon-14 activity of living wood. Estimate its age. (3 marks)

    1. 1 → 1/2 → 1/4 → 1/8 → 1/16: 4 half-lives
    2. Age = 4 × 5700
    3. Age = 22,800 years

    Practice

    1. A sample has an activity of 2000 Bq and a half-life of 10 minutes. Calculate its activity after 30 minutes.

    30 ÷ 10 = 3 half-lives → 2000 → 1000 → 500 → 250 → Activity = 250 Bq

    2. The count rate of a source falls from 900 to 225 counts per minute in 16 hours. Calculate its half-life.

    900 → 450 → 225: 2 half-lives → 16 ÷ 2 → Half-life = 8 hours

    3. What percentage of a radioactive isotope is left after 4 half-lives?

    (½)⁴ = 1/16 → 100 ÷ 16 = 6.25%

    Frequently asked questions

    How do you calculate half-life?

    Count how many times the activity halves over a known time, then divide the time by that number. For example, 4800 Bq falling to 600 Bq in 24 days is 3 halvings, so the half-life is 24 ÷ 3 = 8 days.

    What is net decline in half-life questions?

    The fraction of the original activity left after a number of half-lives: 1/2 after one, 1/4 after two, 1/8 after three, and so on. In AQA GCSE Physics, calculating it is Higher tier.

    Do I need logarithms for GCSE half-life questions?

    No. GCSE questions are set so the activity halves a whole number of times, so you can halve step by step.

    Should I subtract background radiation?

    Yes, if the question gives a background count. Subtract it from each reading first, do the half-life calculation on the corrected values, then add it back if the question asks for the recorded count.

    Engr. Mehmood Kakar

    Written by Engr. Mehmood Kakar

    Physics educator and engineer (MSc Physics, B.Ed, M.Ed) with 8 years of hands-on teaching, from college students to teacher training. About the author · LinkedIn (opens in a new tab)

    Checked against the AQA GCSE Physics (8463) specification on 14 November 2026.