MISSION · Algebra
Exponential Growth and Decay
What happens when the rate of change depends on the amount
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
Linear growth adds the same amount each step; exponential growth multiplies by the same factor each step, and that difference eventually dwarfs everything.
\[y = a \cdot b^x\]The two numbers
\(a\) is the starting amount, the value when \(x = 0\). \(b\) is the growth factor — what you multiply by for each step.
- \(b > 1\) means growth. \(b = 1.05\) is a 5% rise per step.
- \(0 < b < 1\) means decay. \(b = 0.8\) means keeping 80% each step, a 20% fall.
- \(b = 1\) means nothing changes.
Note that decay is still multiplication. Halving repeatedly gets smaller and smaller but never reaches zero, which is why radioactive material has a half-life rather than an expiry date.
Linear loses, eventually and then completely
Start a linear job at £1000 a day, and an exponential one at 1p doubling daily. After ten days the linear job has paid £10,000 and the exponential one £10.24. After twenty, it’s £20,000 against £10,485. After thirty days the exponential job has paid over £10 million.
Nothing about the early figures warns you. Exponential growth looks flat, then looks modest, then looks impossible — and the transition happens much faster than intuition allows for. Every argument about pandemics, compound interest, or Moore’s law is a version of this one comparison.
Where it comes from
Exponential behaviour appears whenever the change depends on the current amount rather than on time:
- Interest. The interest earned depends on the balance, which includes past interest.
- Population. More rabbits make more rabbits.
- Cooling and decay. The amount lost each minute depends on how much is left.
That’s the mark to look for. “Grows by 3 per year” is linear. “Grows by 3% per year” is exponential, because the 3% is measured against an amount that keeps changing.
Doubling time and half-life
Since the factor is constant, the time to double is constant too — it doesn’t matter where you start. A population growing 7% a year doubles about every ten years, whether it’s at a thousand or a million.
A useful estimate, the rule of 70: doubling time is roughly \(70 \div\) the percentage growth rate. At 2% a year, doubling takes about 35 years. At 10%, about 7.
Decay works identically. A half-life of 5 days means half is gone after 5 days, half of the remainder after 10, and so on — after 20 days a sixteenth is left, not nothing.
Common mix-ups
- Confusing \(x^2\) with \(2^x\). In the first the exponent is fixed; in the second the exponent is what’s growing, and it wins in the long run every time.
- Adding percentages across periods. 10% growth for 3 years is \(\times 1.1^3 = 1.331\), not 30%.
- Thinking decay reaches zero. It approaches zero without arriving.
- Judging the shape from a short stretch. Early exponential growth is easily mistaken for linear, and that’s precisely what makes it dangerous.
Check yourself
£500 is invested at 6% a year. Write the formula, and estimate how long it takes to double.
\(V = 500 \times 1.06^t\). By the rule of 70, doubling takes about \(70 \div 6 \approx 12\) years — the exact figure is 11.9.
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES