GeographyISC Class 11 · Origin and Age of the Earth

Radioactive Dating and Half-life Calculations (ISC Class 11 Geography)

Updated 17 October 2026 · Fact-checked

Radioactive atoms decay at a fixed rate, so the half-life is the time for half of them to decay. After n half-lives the fraction left is (1/2)^n. Comparing daughter and parent atoms in a rock gives its age; applied to meteorites and the oldest rocks this gives about 4.6 billion years for the earth.

Each half-life halves the parent atoms and adds daughter atoms; their ratio tells a rock's age, and meteorites give about 4.6 billion years.

Illustration drawn for QuestionBuddy.

Understand Radioactive Dating and Half-life Calculations

Principle. A radioactive parent isotope breaks down into a daughter isotope. The rate is constant and unaffected by temperature, pressure or chemistry. The half-life is the time taken for half the parent atoms in a sample to decay. When a rock crystallises it traps parent atoms, and daughter atoms accumulate afterwards, so the ratio of daughter to parent atoms and the known half-life give the time since the rock formed.

Half-lives (rounded). Uranium-238 to lead-206: 4.47 billion years. Uranium-235 to lead-207: 704 million years. Potassium-40 to argon-40: 1.25 billion years. Rubidium-87 to strontium-87: about 49 billion years. Carbon-14 to nitrogen-14: 5,730 years, usable only for remains of once-living material up to about 50,000 years old.

Age of the earth. The oldest earth rocks are about 4 billion years old and some zircon grains about 4.4 billion years, but earth rocks have been recycled, so meteorites that formed with the solar system are also dated. Together they give about 4.5 to 4.6 billion years.

Calculation rules.
1. Fraction of parent left after n half-lives = (1/2)^n, where n = t divided by T.
2. Time elapsed = n x T.
3. If all daughter atoms come from decay and the daughter-to-parent ratio is D:P, the parent fraction = P/(P + D).
4. First write the fraction as a power of 1/2 (1/2, 1/4, 1/8, 1/16, ...), read off n, then multiply by T.

Radioactive Dating and Half-life Calculations: key facts

Fraction remaining: $N / N 0 = (1/2)^{n} ∣ n = t / T$; T is the half-life.
Time elapsed: $t = n x T ∣ n = n u mb er o f ha l f - l i v es$; Find n first.
Parent fraction from ratio: $P a r e n t f r a c t i o n = P / (P + D) ∣ D : P i s d a ug h t er t o p a r e n t$; D:P is daughter to parent.
Half-lives: U-238 4.47 billion yr
U-235 704 million yr
K-40 1.25 billion yr
C-14 5,730 yr

U-238	4.47 billion yr
U-235	704 million yr
K-40	1.25 billion yr
C-14	5,730 yr

How to answer Radioactive Dating and Half-life Calculations questions

Numericals are 3 to 4 marks. Show formula, substitution, answer with units.

1Find the parent fraction, using P/(P + D) if a ratio is given.
2Write it as (1/2)^n and read n.
3Multiply n by the half-life.
4Write the answer with units and check it against the age of the earth.

Radioactive Dating and Half-life Calculations in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Sample left after decay

Suppose a lab sample of an invented isotope halves in mass every 8 days, and 200 g is stored for 24 days.

How much of the isotope remains?

Try it first, then show the solutionSolution

n = 24/8 = 3 half-lives.
Remaining = 200 x (1/2)^3 = 200/8 = 25 g.

Answer: 25 g.

Memory hook: Three halvings: 200, 100, 50, 25.

Which clock?

Suppose a team wants the age of an ancient wooden beam about 9,000 years old and of a mineral about 3 billion years old.

Which clocks suit each?

Try it first, then show the solutionSolution

The beam once lived, so carbon-14 with its 5,730-year half-life suits it.
The mineral needs a long-lived pair such as uranium-lead or potassium-argon.

Answer: Carbon-14 for the beam; uranium-lead or potassium-argon for the mineral.

Memory hook: Short clock for young things.

Tricks to remember Radioactive Dating and Half-life Calculations

Short memory aids. Say each one aloud once, then use it on a question the same day.

Halve and count
1, 1/2, 1/4, 1/8, 1/16, 1/32: count the halvings.
Use it when: Fraction remaining.
Parent over total
P over P + D.
Use it when: Parent fraction from ratio.

Radioactive Dating and Half-life Calculations: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Half-life meaning · 1 mark · MCQBoard level

The half-life of a radioactive element is the time taken for:
(a) all atoms to decay
(b) half the parent atoms to decay
(c) the rock to melt
(d) the daughter atoms to double

Try it first, then show the solutionSolution

Definition.

Answer: (b) Half the parent atoms to decay.

Example 2: Sample mass · 2 marks · Short answerBoard level

A 192 g sample of an invented isotope undergoes 6 half-lives. Find the mass of the parent isotope left.

Try it first, then show the solutionSolution

Fraction remaining = (1/2)^6 = 1/64.
Mass left = 192/64 = 3 g.

Answer: 3 g.

Example 3: Potassium-argon age · 3 marks · Short answerHarder board level

A mineral still holds 12.5 per cent of its original potassium-40 (half-life 1.25 billion years). Find its age.

Try it first, then show the solutionSolution

12.5 per cent = 1/8 = (1/2)^3, so n = 3.
Age = n x T = 3 x 1.25 = 3.75 billion years.

Answer: 3.75 billion years.

Example 4: Ratio method · 4 marks · Long answerHarder board level

A mineral has three atoms of lead-207 for each atom of uranium-235 (half-life 704 million years). Assume all lead came from decay. Find the age.

Try it first, then show the solutionSolution

D:P = 3:1, so parent fraction = 1/(1 + 3) = 1/4.
1/4 = (1/2)^2, so n = 2.
Age = 2 x 704 = 1,408 million years.
That is 1.408 billion years.

Answer: 1,408 million years (1.408 billion years).

Example 5: Reasoning on method · 5 marks · Case-basedHOTS / competency

Case (invented): A wooden artefact retains 12.5 per cent of its original carbon-14. (a) Find its age. (b) Calculate roughly how many carbon-14 half-lives fit into 4.6 billion years. (c) Explain why this shows carbon-14 cannot date the earth.

Try it first, then show the solutionSolution

(a) 12.5 per cent = 1/8, n = 3; age = 3 x 5,730 = 17,190 years.
(b) 4,600,000,000 / 5,730 is about 802,800 half-lives.
(c) After so many halvings essentially no carbon-14 would remain, so it cannot give an age for the earth; it is used only for once-living remains up to about 50,000 years.

Answer: (a) 17,190 years (b) about 802,800 half-lives (c) no carbon-14 would be left to measure.

Where students lose marks in Radioactive Dating and Half-life Calculations

The slips examiners see most often, and what to write instead.

Mistake: Using the number of half-lives as the age
n is written as the age.
Wrong working
Age = 3 (half-lives)
Right working
Age = 3 x 1.25 = 3.75 billion years
Do this: Multiply n by the half-life T.
Usually costs: 1 mark.
Mistake: Taking D/P as the remaining fraction
Ratio is used directly.
Wrong working
3:1 gives 1/3 remaining
Right working
3:1 gives 1/4 remaining
Do this: Use P/(P + D).
Usually costs: 2 marks.
Mistake: Units dropped
Years or million years omitted.
Do this: Always write units and keep one unit throughout.
Usually costs: 1 mark.

Exam tips

Convert every fraction into a power of one half.
Quote the half-life you use, with units.
Say carbon-14 is only for once-living remains.
Check that your answer is smaller than 4.6 billion years if it is an earth rock.

Practice questions on Radioactive Dating and Half-life Calculations

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
What fraction of the parent remains after five half-lives?
(a) 1/10
(b) 1/16
(c) 1/32
(d) 1/64
Hint
Halve five times.
Answer
(c) 1/32.
Q2 · 3 marks · Short answerHarder board level
A 640 g sample of an invented isotope has a half-life of 20 days. How much remains after 100 days?
Hint
n = 100/20.
Answer
n = 5; fraction = 1/32; mass = 640/32 = 20 g.
Q3 · 4 marks · Long answerHarder board level
A meteorite fragment has 15 atoms of lead-207 for each atom of uranium-235 (half-life 704 million years). Taking all lead as radiogenic, find the age.
Hint
Parent fraction 1/16.
Answer
P/(P + D) = 1/16 = (1/2)^4, so n = 4; age = 4 x 704 = 2,816 million years, about 2.816 billion years.
Q4 · 3 marks · Short answerHarder board level
Explain why heating a rock does not change the decay rate, and state one way strong heating can still upset a radiometric age.
Hint
Decay rate is constant.
Answer
The decay rate is fixed and not affected by temperature, pressure or chemical conditions, so heating does not change the rate at which parent atoms decay. However, strong heating can let daughter atoms such as argon escape from the rock and reset the clock, so dating assumes a closed system.
Q5 · 4 marks · Case-basedHOTS / competency
Case: One laboratory uses a uranium-lead pair and another a potassium-argon pair on pieces of the same meteorite and both get about 4.5 billion years. (a) Say why agreement raises confidence. (b) Say why scientists date meteorites. (c) Name the age accepted for the earth.
Hint
Independent clocks; recycled earth rocks.
Answer
(a) Two independent decay systems with different half-lives agree. (b) Earth rocks have been recycled, whereas meteorites formed with the solar system. (c) About 4.6 billion years.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Half-life of C-14?Show answer
5,730 years.
2.Half-life of K-40?Show answer
1.25 billion years.
3.Fraction after three half-lives?Show answer
1/8.
4.Earth age by dating?Show answer
About 4.6 billion years.
5.Parent fraction formula?Show answer
P/(P + D).

Radioactive Dating and Half-life Calculations: frequently asked questions

Does decay speed up in a hot rock?

No, the decay rate is constant.

Why meteorites?

They formed with the solar system and have not been recycled like earth rocks.

Can carbon-14 date a granite?

No, it is used for once-living material up to about 50,000 years old.