decay calculator

What Is Exponential Decay?

Exponential decay describes processes where a quantity decreases by the same percentage during equal time intervals. Unlike a linear drop, the amount removed gets smaller over time because it is always taken from the remaining quantity. This pattern appears everywhere: radioactive isotopes, medication concentration in the body, battery degradation, and asset value depreciation.

Core Decay Formulas

1) Decay Rate Per Period

If a value loses a fixed percentage each period, use:

A(t) = A₀ × (1 − r)^t

• A₀ = initial amount
• r = decay rate as a decimal (8% = 0.08)
• t = number of periods

2) Half-Life Model

If the quantity halves every fixed interval, use:

A(t) = A₀ × (1/2)^{t / h}

• h = half-life
• t = elapsed time

This is common in chemistry and nuclear physics where decay is often measured with half-life rather than a direct percent-per-period rate.

How to Use This Decay Calculator

1. Select Decay rate per period or Half-life.
2. Enter your initial amount.
3. Provide either:
   • rate + number of periods, or
   • half-life + elapsed time.
4. Click Calculate Decay to view final amount, amount decayed, and a timeline table.

Worked Examples

Example A: Value Depreciation

Suppose equipment is worth 10,000 units and loses 12% of value per year. After 6 years:
A(6) = 10000 × (0.88)⁶ ≈ 4,640.37

The value did not drop by 12 × 6 = 72% linearly. Instead, each year’s 12% is taken from a progressively smaller base.

Example B: Radioactive Half-Life

A sample starts at 80 grams with a half-life of 3 days. After 9 days:
A(9) = 80 × (1/2)^9/3 = 80 × (1/2)³ = 10 grams

Because 9 days equals 3 half-lives, the sample halves three times: 80 → 40 → 20 → 10.

Practical Use Cases

• Finance: declining balance depreciation models and asset wear.
• Healthcare: medication concentration reduction over time.
• Physics: radioactive decay and isotope tracking.
• Technology: battery health decline and retention curves.
• Marketing/Product: user churn over repeated periods.

Common Mistakes to Avoid

Confusing Percent and Decimal

5% should be entered as 5 in this tool’s rate field (the calculator converts it internally), while formulas often use 0.05.

Mixing Time Units

Keep units consistent. If half-life is in months, elapsed time should also be in months.

Assuming Decay Is Linear

Exponential processes curve downward. The absolute change per period shrinks even if the percentage stays constant.

Final Thoughts

Decay math is simple once the model is clear. If you know a periodic percentage decrease, use the rate formula. If you know how long it takes to halve, use half-life. This calculator supports both approaches and gives you a quick timeline so you can see not just the endpoint, but the path.