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Doubling time and half-life

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

Some things double regularly, others halve. You learn the function \(y = a \cdot b^{t/T}\) with doubling time and half-life.
📄 Open worksheet (PDF, German)

The video

In the video a medicine halves every four hours. You see the table, the graph and the function.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Half-life
  2. 0:20 The function
  3. 0:45 Doubling time
  4. 1:13 In general
  5. 1:45 Careful, mistake!
  6. 2:11 Try it yourself

The most important points

Half-life

A medicine: \(80\) mg, half as much every \(4\) hours: \(80, 40, 20, 10\). The time in which the amount halves is the half-life (die Halbwertszeit).

Table: every 4 hours times 1/2.
Table: every \(4\) hours times \(\frac{1}{2}\).

The function

\(y = 80 \cdot 0{,}5^{t/4}\). The exponent \(t/4\) counts the half-lives. For \(t = 8\) it is \(2\): \(80 \cdot 0{,}5^2 = 20\).

The graph of y = 80 · 0{,}5^{t/4}.
The graph of \(y = 80 \cdot 0{,}5^{t/4}\).

Doubling time (die Verdopplungszeit): \(100\) bacteria, times \(2\) every \(3\) hours: \(y = 100 \cdot 2^{t/3}\). For \(t = 6\): \(100 \cdot 4 = 400\).

In general

\(y = a \cdot b^{t/T}\): \(a\) is the starting value, \(T\) the time for one step. \(b = 2\): doubling, \(b = \frac{1}{2}\): half-life. Caffeine: \(100\) mg, \(T = 5\) h, after \(10\) h: \(25\) mg.

After two half-lives a quarter is left.
After two half-lives a quarter is left.

Careful: After two half-lives not everything is gone, but a quarter is left. The amount never becomes exactly zero.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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Show the worksheet solutions
The worksheet – Page 3
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Try it yourself

Task

A substance weighs \(200\) g and its half-life is \(3\) days. How many grams are left after \(9\) days?

Show solution

\(9 : 3 = 3\) half-lives. \(200 \cdot (\frac{1}{2})^3 = 25\) g.

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Verdopplungszeit und Halbwertszeit (Funktion y = a·b^(t/T)). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.