Why is the exponent $\frac{t}{T}$ and not just $t$?

The factor $\frac{1}{2}$ applies once per half-life, not once per time unit. Dividing the time $t$ by the half-life $T$ counts how many half-lives have passed.

What if the time is not a whole number of half-lives?

The formula still works, for example $400 \left(\frac{1}{2}\right)^{1.5}$. Use a calculator (type 400 × 0.5^1.5) or a given value like $\left(\frac{1}{2}\right)^{1.5} \approx 0.3536$. Finding an unknown time in that case needs logarithms, which come in a later lesson.

How is half-life related to the decay rate in exponential decay?

In the Exponential Decay lesson, an amount is multiplied by $b = 1 - r$ every time unit. Half-life uses the same idea but chooses the time unit so that the factor is exactly $\frac{1}{2}$, which makes mental calculations much easier.

Teacher Resources

Half-Life Problems

Q: Does a substance ever completely disappear?

Mathematically, no: $\left(\frac{1}{2}\right)^{n}$ is never zero. In practice, after about 10 half-lives less than $0.1\%$ is left ($\frac{1}{1024}$), which is usually treated as gone.

Learn how to use half-life to find how much of a substance remains, how much time has passed, and how long a half-life is.

Advanced25 minLesson

Printable worksheet

Definition

The half-life

T

of a substance is the time it takes for half of it to disappear (decay, break down or leave the body). After every half-life, the amount is multiplied by

\frac{1}{2}

N_0

is the starting amount, the amount left after time

t

is:

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}

where:

$N_0$ = initial amount (at $t = 0$ )
$T$ = half-life (same time unit as $t$ )
$\frac{t}{T}$ = the number of half-lives that have passed

After

n

whole half-lives, the amount left is

\frac{N_0}{2^n}

Half-lives	0	1	2	3	4	5
Fraction left	$1$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{1}{16}$	$\frac{1}{32}$
Percent left	$100\%$	$50\%$	$25\%$	$12.5\%$	$6.25\%$	$3.125\%$

This is exponential decay with decay factor

\frac{1}{2}

per half-life: the amount gets smaller and smaller, but it never reaches exactly zero.

Try it now

A substance has a half-life of

10

years. What fraction of it is left after

2

half-lives?

Worked Examples

A lab sample contains $80$ mg of a substance with a half-life of $4$ hours. How much is left after $12$ hours?

Find the number of half-lives

$\frac{t}{T} = \frac{12}{4} = 3$ → 3 half-lives

Write the formula

$N(12) = 80 \left(\frac{1}{2}\right)^{3}$ → Formula set up

Compute the power

$\left(\frac{1}{2}\right)^{3} = \frac{1}{8}$ → $\frac{1}{8}$ of the sample is left

Multiply

$80 \times \frac{1}{8} = 10$ (or halve three times: $80 \to 40 \to 20 \to 10$ ) → $10$ mg

Answer: After 12 hours, 10 mg of the substance is left.

Common Mistakes

Halving once for every hour (or day, or year) instead of once per half-life

Why it's wrong: With a half-life of 4 hours, after 12 hours some students compute $80 \left(\frac{1}{2}\right)^{12}$ . But the amount only halves every 4 hours, not every hour.

Correct: The exponent is the number of half-lives $\frac{t}{T} = \frac{12}{4} = 3$ , so the answer is $80 \left(\frac{1}{2}\right)^{3} = 10$ .

Subtracting half of the ORIGINAL amount each half-life

Why it's wrong: Thinking $100 \to 50 \to 0$ treats the decay as linear. Each half-life removes half of what is left NOW, not half of the starting amount.

Correct: Halve the current amount each time: $100 \to 50 \to 25 \to 12.5$ . A sample never reaches exactly zero.

Using the denominator as the number of half-lives

Why it's wrong: If $\frac{1}{8}$ is left, some students say 8 half-lives passed. But 8 half-lives would leave $\frac{1}{256}$ .

Correct: Write the denominator as a power of 2: $8 = 2^3$ , so 3 half-lives have passed.

Mixing up the fraction left and the fraction that decayed

Why it's wrong: After 3 half-lives, $\frac{1}{8}$ is left, so $\frac{7}{8}$ has decayed. Questions can ask for either one.

Correct: Read the question carefully: fraction decayed $= 1 -$ fraction left. After 3 half-lives: $12.5\%$ left, $87.5\%$ decayed.

Interactive Visual

Sequence Explorer

First Term (a₁ = 64)

Common Ratio (r = 0.5)

64,

32,

16,

, ...

Formula

a_n = 64 × 0.5^(n-1)

a_1

a_5

a_10

0.125

a_20

0.000122

Function graph

N = 80T = 4

Tap or hover over the graph to read off values.

N(t) = N₀·(1/2)^(t/T)Asymptote

Table of values

t	0	4	8	12	16
y	80	40	20	10	5

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems

Problem 1 of 18

Easy

A substance has a half-life of $10$ years. What fraction of it is left after $2$ half-lives?

Why It Matters

Half-life is how scientists, doctors and engineers measure how fast something disappears:

Archaeology: Carbon-14 has a half-life of $5730$ years, so the carbon left in bones and wood tells us how old they are
Medicine: The half-life of a medicine tells doctors how often a patient should take the next dose
Hospitals: Iodine-131, used to treat thyroid illness, has a half-life of $8$ days, which tells staff when a patient is safe to go home
Energy and safety: Nuclear waste must be stored until enough half-lives have passed

The good news: when the time is a whole number of half-lives, you can solve these problems by halving, without any advanced tools.

Real World Applications

Carbon-14 Dating

Living things take in carbon-14. After they die, it decays with a half-life of 5730 years, so the fraction left shows the age of bones, wood and charcoal.

Example:

A piece of wood with $\frac{1}{4}$ of its original carbon-14 is $2 \times 5730 = 11460$ years old.

1Try It Yourself

Charcoal from an ancient fireplace contains $\frac{1}{16}$ of its original carbon-14. The half-life of carbon-14 is $5730$ years.

How old is the charcoal?

Step 1: Write the mathematical expression

Find the number of half-lives, then multiply by $5730$ :

Medicine in the Bloodstream

Doctors use the half-life of a medicine to decide how often to give the next dose, so the amount in the body stays safe and effective.

Example:

A dose of $400$ mg with a half-life of $6$ hours leaves $400 \left(\frac{1}{2}\right)^{12/6} = 100$ mg after $12$ hours.

2Try It Yourself

A patient takes $600$ mg of a medicine with a half-life of $4$ hours.

How many mg are left after $16$ hours?

Step 1: Write the mathematical expression

Calculate: $600 \left(\frac{1}{2}\right)^{16/4}$

Iodine-131 Treatment

Iodine-131 is used to treat thyroid illness. Its half-life of 8 days tells the hospital how fast the radiation drops and when it is safe to store or dispose of leftovers.

Example:

$100$ mg of iodine-131 becomes $100 \left(\frac{1}{2}\right)^{16/8} = 25$ mg after $16$ days.

3Try It Yourself

A hospital stores $160$ mg of iodine-131 (half-life $8$ days).

How many mg are left after $24$ days?

Step 1: Write the mathematical expression

Calculate: $160 \left(\frac{1}{2}\right)^{3}$

Key Takeaways

1The half-life $T$ is the time it takes for half of a substance to disappear
2The amount left is $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}$ , where $\frac{t}{T}$ is the number of half-lives
3After $n$ half-lives, the fraction left is $\frac{1}{2^n}$ : $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$
4To find the elapsed time, write the fraction left as $\frac{1}{2^n}$ and multiply $n$ by the half-life
5To find the half-life, count how many halvings fit in the data and divide the time by that number
6Each half-life removes half of what is left now, so the amount never reaches exactly zero

Frequently Asked Questions

Mathematically, no:

\left(\frac{1}{2}\right)^{n}

is never zero. In practice, after about 10 half-lives less than

0.1\%

is left (

\frac{1}{1024}

), which is usually treated as gone.

Glossary

Half-life: The time $T$ it takes for half of a substance to decay or disappear
Initial amount: The amount $N_0$ at the start, when $t = 0$
Number of half-lives: The ratio $\frac{t}{T}$ : how many times the amount has been halved
Radioactive decay: The process in which unstable atoms change into other atoms, losing a fixed fraction in every equal time period
Carbon-14 dating: A method that finds the age of once-living material from the fraction of carbon-14 left (half-life $5730$ years)

Formula Card

Half-Life Formula

N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}

$N_0$ = initial amount, $T$ = half-life, $t$ = time passed (same unit as $T$).

After n Half-Lives

N = \frac{N_0}{2^n}, \quad n = \frac{t}{T}

Divide by 2 once for every half-life. The elapsed time is $t = n \cdot T$.

Half-Life Problems

Definition

Worked Examples

Common Mistakes

Interactive Visual

Sequence Explorer

Formula

Function graph

Table of values

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Carbon-14 Dating

Medicine in the Bloodstream

Iodine-131 Treatment

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Half-Life Formula

After n Half-Lives

More in This Topic

Introduction to Exponential Functions

Exponential Growth

Graphing Exponential Functions

Exponential Decay