Quick answer
Exponential decay: P = P₀(1 − r)ᵗ. Starting at 1,000 and losing 5% per period for 10 periods leaves 598.74, a change of −401.26.
Enter the initial amount, decay rate per period and number of periods. The calculator returns the final amount and change.
Related tools: Exponential Growth Calculator: Exponential growth over time at a constant percentage rate. Slope Calculator: Slope, angle, distance and line equation through two points. LCM Calculator: Least common multiple of two or more numbers.
How to use the Exponential Decay Calculator
- Enter initial amount (0 or more).
- Enter decay rate per period (%) (between 0 and 100).
- Enter periods (between 0 and 10000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Initial amount | A number | ≥ 0 |
| Decay rate per period | A number, in % | ≥ 0, ≤ 100 |
| Periods | A number | ≥ 0, ≤ 10000 |
Formula and method
Formula
P = P₀(1 − r)ᵗ
IEEE-754 double precision; shown to at most 6 decimal places.
Discrete compounding per period.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Initial amount = 1000; Decay rate per period = 5; Periods = 10.
| Result | Value |
|---|---|
| Final amount | 598.736939 |
| Change | -401.263061 |
Example 2
Inputs: Initial amount = 80; Decay rate per period = 50; Periods = 3.
| Result | Value |
|---|---|
| Final amount | 10 |
| Change | -70 |
The first line is the headline result (final amount). Below it the exponential Decay Calculator lists change, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Compounding loss
Each period removes 5% of what is left: 1,000 × 0.95 = 950, then 902.5, and so on. After 10 periods 0.95^10 = 0.5987. The half-life is ln 0.5 ÷ ln 0.95 ≈ 13.5 periods.
Applications
Radioactive decay, depreciation, drug elimination and cooling toward ambient temperature follow this pattern. Overflow is rejected.
How the result changes with initial amount
Holding every other input at the first worked example, this table changes only initial amount. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Initial amount | Final amount | Change |
|---|---|---|
| 500 | 299.36847 | -200.63153 |
| 1000 | 598.736939 | -401.263061 |
| 2000 | 1,197.473878 | -802.526122 |
| 4000 | 2,394.947757 | -1,605.052243 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Decay rate per period: entering "-1" shows "Decay rate per period must be at least 0." beside that field and no result.
Choosing between the Exponential Decay Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Exponential Growth Calculator](/math/exponential-growth/) | initial amount, growth rate per period and periods | Final amount |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
| [Quadratic Formula Calculator](/math/quadratic-formula-calculator/) | a, b and c | Roots |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Overflow rejected
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for decay rate per period to confirm this.
Frequently asked questions
What is the half-life at 5%?
Half-life is ln 0.5 ÷ ln 0.95 ≈ 13.5 periods at a 5% decay per period.
Does it ever reach zero?
No, only approaches it.
How is it different from linear decay?
Linear loses a fixed amount each period.
What does the Exponential Decay Calculator calculate?
Exponential decay over time at a constant percentage rate. It returns Final amount and Change.
What formula does the Exponential Decay Calculator use?
P = P₀(1 − r)ᵗ. Discrete compounding per period.
What do I need to enter in the exponential Decay Calculator?
You enter Initial amount, Decay rate per period (%) and Periods.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Overflow rejected.
What is an example of a exponential Decay result?
With initial amount 1000, decay rate per period 5 and periods 10, the final amount is 598.736939.
Related calculators
- Exponential Growth Calculator — Exponential growth over time at a constant percentage rate.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
- Log Calculator — Logarithm of a number to any base.
- Sum of Series Calculator — Sum of the first n integers, squares and cubes.
- Dot Product Calculator — Dot product of two vectors.
- Cross Product Calculator — Cross product of two 3-D vectors.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.