Enter starting value and pick a reliable model. Choose growth, decay, half‑life, or compounding mode. See results instantly, then download a neat report file.
| Scenario | Model | A0 | Rate | t | Expected A(t) |
|---|---|---|---|---|---|
| Investment growth | Discrete (r=8%, n=12) | 1000 | 8% / year | 5 | ≈ 1490.40 |
| Bacteria culture | Doubling time (d=9) | 250 | d = 9 hours | 18 | ≈ 1000 |
| Radioactive decay | Half-life (h=3) | 80 | h = 3 days | 6 | ≈ 20 |
| Continuous growth | Continuous (k=0.08) | 1000 | k = 0.08 | 5 | ≈ 1491.82 |
| Continuous decay | Continuous (k=-0.12) | 500 | k = −0.12 | 4 | ≈ 309.49 |
In continuous models, the rate parameter k describes proportional change per time unit. If k is positive, values rise smoothly; if k is negative, they fall. For example, A0=1000, k=0.08, and t=5 gives A(t)=1000×e^(0.4)=1491.82, a 49.18% increase. This calculator reports k directly and also shows an equivalent percent per unit when you solve for rate.
Some processes update in steps, so discrete compounding is more realistic. With nominal r=8% and n=12, the periodic factor is 1+r/n=1.006666…, and over five years the multiplier is (1.006666…)^(60)=1.49040. The effective annual rate is (1+r/n)^n−1≈8.30%. Use the table points setting to inspect monthly-to-yearly progression without changing the main inputs.
When you know the start and end values, time is found by isolating t with a logarithm. In continuous form, t=ln(A(t)/A0)/k. Using A0=500, A(t)=309.49, and k=−0.12 returns t≈4.00. In discrete form, t=ln(A(t)/A0)/(n×ln(1+r/n)). The calculator validates that A0 and A(t) stay above zero so the log remains defined.
Half-life inputs are common in decay problems. The relationship k=ln(1/2)/h converts a half-life into a continuous rate. If h=3 days, then k≈−0.231049 per day, and after six days A(t)=A0×(1/4). For A0=80, the expected value is 20. The generated PDF includes the chosen model label and a first‑rows preview for quick review.
Doubling time is the mirror of half-life for growth. With d=9 hours, k=ln(2)/9≈0.077016 per hour. After 18 hours, the amount becomes A0×4. If A0=250, that is 1000. Use this mode for capacity forecasts, user growth, or biomass scaling where doubling is easier to measure than percent change.
To verify results, compare the last row of the time series to the headline A(t) value; with 11 points the step is t/10. Increase decimals for audit work, or enable scientific notation for very large or small quantities. Exporting CSV supports side‑by‑side scenario analysis in spreadsheets, while the built‑in PDF provides a compact record suitable for attachments and documentation. A practical check is the ratio A(t)/A0: above 1 indicates growth and below 1 indicates decay. For sensitivity, change k by ±0.01 and observe how the five‑unit multiplier shifts. across your selected horizon.
Use continuous k when change is proportional at every moment, such as population growth or radioactive decay. Use half‑life or doubling time when those measurements are available, because they convert cleanly into k.
Negative k or negative r means decay. The value decreases over time, and the ratio A(t)/A0 is below 1. The calculator labels the direction and still allows solving for time or rate.
Time and rate solutions use logarithms of A(t)/A0. Logarithms require positive inputs, so both amounts must be above zero. If you have a loss that crosses zero, use a different model.
r is the nominal percent per time unit, while n is the number of compounding periods in that unit. The effective rate is (1+r/n)^n−1, which is usually slightly larger than r.
Use 11 points for a quick overview, because it samples from 0 to t in equal steps. Increase points for smoother curves, or lower them for fast checks and smaller downloads.
The CSV includes the computed series and summary values for auditing. The PDF includes key numbers and a preview of the first rows. You can recreate the graph from the series in any plotting tool.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.