Exponential decay calculator
Enter a starting value, the decrease per period in percent and the number of periods. You get the end value, the decay factor, the half-life and the value after every period.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
The formula
N(t) = N₀ · gᵗ with the decay factor g = 1 − percentage ÷ 100. A car worth 20,000 that loses 15% of its value per year is worth 20,000 × 0.85⁵ ≈ 8,874 after five years.
Half-life
The half-life is the number of periods after which half is left: log(0.5) ÷ log(g). At 15% per year that is about 4.3 years. Radioactive decay, medicine leaving the body and depreciation all follow this pattern.
The decrease must be between 0% and 100%. For growth, use the exponential growth calculator.
Half-life by decrease per period
| Decrease per period | Half-life |
|---|---|
| 1% | 69.0 periods |
| 5% | 13.5 periods |
| 10% | 6.6 periods |
| 15% | 4.3 periods |
| 20% | 3.1 periods |
| 50% | 1 period |
Caffeine in the body
Caffeine has a half-life of about 5 hours. Half of it is left after 5 hours, a quarter after 10 hours and about 3.6% after 24 hours, since 0.5^(24/5) ≈ 0.036. The same pattern describes medicines in the blood, the cooling of a hot drink towards room temperature and the discharge of a capacitor.
The calculator's example
1,000 with a decrease of 12% per period is 880 after one period, 774.40 after two and 527.73 after five. The half-life is 5.42 periods: after 5.42 periods exactly 500 is left. For growth instead of decay, use the exponential growth calculator, and for oscillations that die out, see simple harmonic motion.
Frequently asked questions
How do I enter a half-life instead of a percentage?
Convert it first: the decrease per period is (1 − 0.5^(1/half-life)) × 100. For a half-life of 5 periods that is 12.94%. Enter that percentage and the calculator returns a half-life of 5.
Does the value ever reach zero?
No. Each period removes a fraction of what is left, so the value gets closer to zero without reaching it. The decrease must be between 0% and 100%.
Is this the same as depreciation?
Declining-balance depreciation follows the same curve: a fixed percentage of the remaining value is lost each period. Straight-line depreciation loses the same amount every period and is not exponential.
How do I find the decay rate from two values?
Divide the end value by the start value, take the t-th root and subtract the result from 1. From 1,000 to 527.73 in 5 periods the factor is 0.88, so 12% per period.