Model exponential growth and decay with discrete A₀·(1+r)ᵗ or continuous A₀·eʳᵗ formulas. Find doubling time, growth factor, and timeline.
Exponential growth describes a process where the rate of increase is proportional to the current value, producing the familiar curve that starts slowly and then skyrockets. This pattern appears everywhere — from bacteria doubling every hour and savings accounts compounding annually to viral social-media posts and nuclear chain reactions. Conversely, exponential decay models radioactive half-lives, depreciating assets, and cooling liquids. This calculator supports both the discrete model A = A₀·(1 + r/n)^(n·t), used in finance and population studies, and the continuous model A = A₀·e^(r·t), preferred in physics and biology. Enter your initial amount, growth or decay rate, and time horizon to instantly see the final value, total change, percentage change, doubling or halving time, and a complete growth timeline with proportional visual bars. Eight presets cover classic scenarios — bacterial doubling, compound savings, population growth, viral spread, inflation, and more. A model-comparison table highlights the differences between discrete and continuous compounding so you can choose the right formula for your situation. Whether you are studying algebra, planning investments, or building a biological model, this tool makes exponential growth and decay intuitive and concrete.
Exponential Growth Calculator helps you solve exponential growth problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Initial amount (A₀), Rate r (%), Time (t) once and immediately inspect Final Amount, Growth Factor, Total Change to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Discrete: A = A₀ · (1 + r/n)^(n·t) Continuous: A = A₀ · e^(r·t) Doubling time (discrete) = ln(2) / ln(1 + r) Doubling time (continuous) = ln(2) / r Growth factor = A / A₀
Result: Final Amount shown by the calculator
Using the preset "Bacteria 2×/hr", the calculator evaluates the exponential growth setup, applies the selected algebra rules, and reports Final Amount with supporting checks so you can verify each transformation.
This calculator takes Initial amount (A₀), Rate r (%), Time (t), Compounding periods per unit time (n) and applies the relevant exponential growth relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Final Amount, Growth Factor, Total Change, Percent Change to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Exponential growth occurs when a quantity increases at a rate proportional to its current value. The larger it gets, the faster it grows, producing a J-shaped curve.
Discrete growth compounds at fixed intervals (e.g., yearly). Continuous growth compounds every instant, using the natural base e. Continuous always yields slightly more.
Doubling time is the time required for a quantity to double. For continuous growth it is ln(2)/r; for discrete growth it is ln(2)/ln(1+r).
Yes. Enter a negative rate percentage. The calculator will compute the halving time and show the declining timeline.
The Rule of 72 is a quick estimate: divide 72 by the annual percentage rate to approximate how many years it takes to double. For example, 72 ÷ 6% ≈ 12 years.
Real populations face carrying capacity, resource limits, and stochastic events. Pure exponential models assume unlimited resources and work best for short-to-medium horizons.