You can’t predict when a single nucleus will decay — but with billions of them, the pattern is beautifully reliable. Activity tells you how fast a sample is decaying right now, and half-life tells you how long it takes for that activity to drop by half. The magic is that the half-life is constant: every time one passes, whatever’s left halves again. That simple “halving” rule lets you solve almost any decay problem without heavy maths.
📚 What you need to know
Activity is the number of nuclei that decay per unit time, measured in becquerels (Bq)
1 Bq = one decay per second
Half-life (t½) is the time for half the undecayed nuclei — or the activity — to halve
Half-life is constant for a given isotope
After each half-life, the amount remaining halves again
After n half-lives, the proportion remaining = (½)n
Count rate ∝ activity ∝ number of nuclei, so all three fall by the same pattern
Activity
Activity is simply how many nuclei are decaying each second. A highly radioactive source has lots of decays per second, so a high activity. It’s measured in becquerels, where 1 Bq means one nucleus decaying every second.
Activityactivity = number of nuclei decaying per second (unit: Bq)
You can’t say which nucleus will go next — that’s random. But the rate at which the activity falls over time is perfectly predictable, and that’s captured by the half-life.
Half-life
Half-life is the time it takes for half of the remaining undecayed nuclei to decay — equivalently, the time for the activity to fall to half its value. Crucially, this time is the same no matter when you start measuring.
Half-lifethe time taken for half the undecayed nuclei to decay, or for the activity to halve
Each equal step along the time axis halves the activity: A0 → A0/2 → A0/4. The curve approaches zero but never quite reaches it.
The thing students find weird at first: the half-life doesn’t get longer as the sample “runs down”. Going from 100% to 50% takes exactly as long as going from 50% to 25%, or 25% to 12.5%. Every halving takes the same time. That constant rhythm is what makes half-life such a powerful, simple tool.
The halving rule
Because each half-life halves the amount, you can jump straight to the answer using powers of a half. After n half-lives:
Proportion remaining after n half-livesproportion remaining = (½)n
Half-lives passed
Proportion remaining
Activity
0
1
A0
1
½
A0/2
2
¼
A0/4
3
⅛
A0/8
4
1/16
A0/16
WE 1
A radioactive sample has a half-life of 3 years. What is the ratio of decayed nuclei to original nuclei after 15 years?
Step 1 — how many half-lives?n = 15 ÷ 3 = 5 half-livesStep 2 — proportion remaining(½)⁵ = 1/32Step 3 — write the ratio
If 1/32 remains, then 31/32 has decayed.
decayed : original = 31 : 32The question asks for DECAYED, not remaining — a classic trap. Find the remaining fraction first, then subtract from the whole to get the decayed fraction.
WE 2
A sample contains 2 million undecayed atoms. After 1 year, only 500 000 remain undecayed. Determine the half-life of the material.
Step 1 — how many times has it halved?2 000 000 → 1 000 000 (1 half-life)1 000 000 → 500 000 (2 half-lives)
So 2 half-lives have passed in 1 year.
Step 2 — find one half-life1 year ÷ 2 = 6 monthshalf-life = 6 monthsCount the halvings by repeatedly dividing by 2 until you reach the final number, then divide the total time by that many half-lives.
Reading half-life off a decay curve
If you’re given a graph of activity against time, the method is: find the initial activityA0, go to half of it on the vertical axis, read across to the curve and down to the time axis. That time is the half-life. You can check it by confirming the next halving takes the same time again.
WE 3
A technetium sample starts at an activity of 8 × 107 Bq. From its decay graph, the activity reaches 4 × 107 Bq after 6 hours. State the half-life, and predict the activity after 18 hours.
Step 1 — read the first halving
Activity fell from 8 to 4 × 10⁷ Bq in 6 hours, so:
half-life = 6 hoursStep 2 — count half-lives in 18 hours18 ÷ 6 = 3 half-livesStep 3 — apply the halving rule8 → 4 → 2 → 1 (×10⁷ Bq)activity after 18 h = 1 × 10⁷ BqOnce you have the half-life, just halve the activity once per half-life. Three half-lives means three halvings: 8, 4, 2, 1.
⚛ Working an activity / half-life question
Find the number of half-lives:n = total time ÷ half-life.
Proportion remaining? Use (½)n.
Asked for decayed? Subtract the remaining fraction from 1.
Finding the half-life from data? Count how many times the amount halves, then divide the total time by that count.
From a graph? Drop from A₀, across at half, down to time.
💡 Top tips
Half-life is constant — every halving takes the same time.
After n half-lives, the fraction left is (½)n.
Read the question: remaining and decayed are different (they add to 1).
Count rate, activity and number of nuclei all follow the same curve.
1 Bq = one decay per second.
⚠ Common mistakes
Giving the remaining fraction when asked for the decayed fraction
Thinking the half-life gets longer as the sample runs down — it’s constant
Using the total time as n instead of time ÷ half-life
Confusing activity (Bq, per second) with total count
Misreading the graph — halve the activity, not the time
Quick recap:Activity is decays per second (in Bq), and half-life is the constant time for the activity (or number of nuclei) to halve. After n half-lives the proportion remaining is (½)n, and the decayed proportion is one minus that. To find a half-life from data, count how many times the amount halves and divide the total time by that count.
The halving rule is quick and intuitive, but it only works neatly for whole numbers of half-lives. What if you need the amount after 7 hours when the half-life is 6? For that you need a proper equation with a decay constant — the exact link between how likely a nucleus is to decay and its half-life. Next page: Decay Constant & Half-Life.
Half-life problems slowing you down?
Book a free meeting and we’ll drill the (½)n rule, decayed-vs-remaining, and reading half-life straight off a decay curve.
