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Teacher Guide: Half-Life Problems

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.

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For Teachers

Learning Objectives
  • Use the half-life formula N(t)=N0(12)t/TN(t) = N_0 \left(\frac{1}{2}\right)^{t/T} to find the amount remaining
  • Find the fraction and the percent of a substance left after a number of half-lives
  • Find the elapsed time when the fraction left is a power of 12\frac{1}{2}
  • Find the half-life of a substance from two measurements
  • Apply half-life to carbon-14 dating, medicine and iodine-131
Prerequisites
  • • Introduction to Exponential Functions: the form f(x)=a⋅bxf(x) = a \cdot b^x
  • • Exponential Growth: multiplying by a constant factor in equal time periods
  • • Powers of 2 and powers of fractions, such as (12)3=18\left(\frac{1}{2}\right)^{3} = \frac{1}{8}
  • • Converting fractions to decimals and percents
Discussion Starters
  • 1. If a medicine has a half-life of 6 hours, why do doctors not tell you to take it every 12 hours at double the dose?
  • 2. Will a radioactive sample ever reach exactly zero? Why or why not?
  • 3. Why can carbon-14 date a 10000-year-old bone but not a 10-million-year-old dinosaur fossil?
  • 4. How is halving every half-life similar to doubling every doubling time?
Common Misconceptions

After two half-lives, the whole sample is gone

Remediation: Build a halving table together: 100→50→25→12.5100 \to 50 \to 25 \to 12.5. Ask what is left after the second half-life. Use the coin toss simulation to show that some coins always remain.

The exponent in the formula is the time tt

Remediation: Ask: how many times did the amount get halved in 12 hours if the half-life is 4 hours? Students count 3 halvings, which shows the exponent is tT\frac{t}{T}.

A fraction left of 116\frac{1}{16} means 16 half-lives

Remediation: List 12,14,18,116\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} and number them 1 to 4. The position in the list, not the denominator, is the number of half-lives.

Differentiation Ideas

For Struggling Students:

  • • Use halving tables (time, number of half-lives, amount) before the formula
  • • Keep to whole numbers of half-lives and starting amounts that are powers of 2
  • • Use the sequence visualizer to see each halving as a bar

For On-Level Students:

  • • Solve for the elapsed time from the fraction left
  • • Find the half-life from two measurements
  • • Convert between fraction left, percent left and percent decayed

For Advanced Students:

  • • Work with non-whole numbers of half-lives using a calculator
  • • Write the decay factor per time unit, such as (12)1/8\left(\frac{1}{2}\right)^{1/8} per day for iodine-131
  • • Model repeated medicine doses and find the amount just before the third dose
Standards Alignment
  • HSF-LE.A.1c (CCSS.MATH.CONTENT.HSF.LE.A.1.C)

    Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another

  • HSF-LE.A.2 (CCSS.MATH.CONTENT.HSF.LE.A.2)

    Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs

  • HSF-LE.B.5 (CCSS.MATH.CONTENT.HSF.LE.B.5)

    Interpret the parameters in a linear or exponential function in terms of a context

  • HSF-IF.C.8b (CCSS.MATH.CONTENT.HSF.IF.C.8.B)

    Use the properties of exponents to interpret expressions for exponential functions

Lesson Resources
  • visualHalving Sequence

    Geometric sequence starting at 64 with ratio 0.5, showing how the bars shrink by half each half-life

  • activityCoin Decay Simulation

    Students toss 100 coins, remove the heads each round and record how many remain, then compare with 100 times one half per round

  • worksheetHalf-Life Detective

    Carbon-14 dating, medicine doses and iodine-131 problems with whole numbers of half-lives

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The half-life TT 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 12\frac{1}{2}.
If N0N_0 is the starting amount, the amount left after time tt is:
N(t)=N0(12)t/TN(t) = N_0 \left(\frac{1}{2}\right)^{t/T}
where:
  • N0N_0 = initial amount (at t=0t = 0)
  • TT = half-life (same time unit as tt)
  • tT\frac{t}{T} = the number of half-lives that have passed
After nn whole half-lives, the amount left is N02n\frac{N_0}{2^n}:
Half-lives012345
Fraction left1112\frac{1}{2}14\frac{1}{4}18\frac{1}{8}116\frac{1}{16}132\frac{1}{32}
Percent left100%100\%50%50\%25%25\%12.5%12.5\%6.25%6.25\%3.125%3.125\%
This is exponential decay with decay factor 12\frac{1}{2} per half-life: the amount gets smaller and smaller, but it never reaches exactly zero.

Worked Examples

A lab sample contains 8080 mg of a substance with a half-life of 44 hours. How much is left after 1212 hours?
1

Find the number of half-lives

tT=124=3\frac{t}{T} = \frac{12}{4} = 3 → 3 half-lives
2

Write the formula

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

Compute the power

(12)3=18\left(\frac{1}{2}\right)^{3} = \frac{1}{8} → 18\frac{1}{8} of the sample is left
4

Multiply

80×18=1080 \times \frac{1}{8} = 10 (or halve three times: 80→40→20→1080 \to 40 \to 20 \to 10) → 1010 mg

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(12)1280 \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 tT=124=3\frac{t}{T} = \frac{12}{4} = 3, so the answer is 80(12)3=1080 \left(\frac{1}{2}\right)^{3} = 10.

Subtracting half of the ORIGINAL amount each half-life

Why it's wrong: Thinking 100→50→0100 \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→50→25→12.5100 \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 18\frac{1}{8} is left, some students say 8 half-lives passed. But 8 half-lives would leave 1256\frac{1}{256}.

Correct: Write the denominator as a power of 2: 8=238 = 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, 18\frac{1}{8} is left, so 78\frac{7}{8} has decayed. Questions can ask for either one.

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

Why It Matters

Half-life is how scientists, doctors and engineers measure how fast something disappears:
  • Archaeology: Carbon-14 has a half-life of 57305730 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 88 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 14\frac{1}{4} of its original carbon-14 is 2×5730=114602 \times 5730 = 11460 years old.

1Try It Yourself

Charcoal from an ancient fireplace contains 116\frac{1}{16} of its original carbon-14. The half-life of carbon-14 is 57305730 years.

How old is the charcoal?

Step 1: Write the mathematical expression

Find the number of half-lives, then multiply by 57305730:

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 400400 mg with a half-life of 66 hours leaves 400(12)12/6=100400 \left(\frac{1}{2}\right)^{12/6} = 100 mg after 1212 hours.

2Try It Yourself

A patient takes 600600 mg of a medicine with a half-life of 44 hours.

How many mg are left after 1616 hours?

Step 1: Write the mathematical expression

Calculate: 600(12)16/4600 \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:

100100 mg of iodine-131 becomes 100(12)16/8=25100 \left(\frac{1}{2}\right)^{16/8} = 25 mg after 1616 days.

3Try It Yourself

A hospital stores 160160 mg of iodine-131 (half-life 88 days).

How many mg are left after 2424 days?

Step 1: Write the mathematical expression

Calculate: 160(12)3160 \left(\frac{1}{2}\right)^{3}

Key Takeaways

  • 1The half-life TT is the time it takes for half of a substance to disappear
  • 2The amount left is N(t)=N0(12)t/TN(t) = N_0 \left(\frac{1}{2}\right)^{t/T}, where tT\frac{t}{T} is the number of half-lives
  • 3After nn half-lives, the fraction left is 12n\frac{1}{2^n}: 12,14,18,116,…\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots
  • 4To find the elapsed time, write the fraction left as 12n\frac{1}{2^n} and multiply nn 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

Does a substance ever completely disappear?

Mathematically, no: (12)n\left(\frac{1}{2}\right)^{n} is never zero. In practice, after about 10 half-lives less than 0.1%0.1\% is left (11024\frac{1}{1024}), which is usually treated as gone.

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

The factor 12\frac{1}{2} applies once per half-life, not once per time unit. Dividing the time tt by the half-life TT 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(12)1.5400 \left(\frac{1}{2}\right)^{1.5}. Use a calculator (type 400 × 0.5^1.5) or a given value like (12)1.5≈0.3536\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−rb = 1 - r every time unit. Half-life uses the same idea but chooses the time unit so that the factor is exactly 12\frac{1}{2}, which makes mental calculations much easier.

Glossary

Half-life
The time TT it takes for half of a substance to decay or disappear
Initial amount
The amount N0N_0 at the start, when t=0t = 0
Number of half-lives
The ratio tT\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 57305730 years)

Formula Card

Half-Life Formula

N(t)=N0(12)t/TN(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=N02n,n=tTN = \frac{N_0}{2^n}, \quad n = \frac{t}{T}

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

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