Six fully worked problems, each with its own live calculator pre-filled with the example's numbers. Change any field to see the result update instantly.
A $4,200 certificate of deposit earning 6.2% annually grows to $7,269.87 after 9 years of annual compounding. The compound interest formula A = P(1 + r)t applies directly.
The deposit earns 3,017.22 in interest over 9 years. See the calculator for other compounding frequencies to try daily, monthly, or continuous compounding on the same principal.
A 320-cell culture with an 18-minute generation time reaches 20,480 cells after 108 minutes. The bacterial growth formula N(t) = N0 × 2t/g applies directly.
Six whole doublings land on an exact result here. See the generation-time tool for durations that fall between doublings.
A 1,200-count radioactive sample losing 2.6% of its activity per year falls to 714.99 counts after 20 years. The exponential decay formula x(t) = x0(1 − r)t applies directly.
The half-life of 26.31 years means the sample crosses 600 counts somewhere before year 20. See the remaining-amount tool to find that point directly.
A subscriber list growing 4.5% per month doubles in 15.75 months. The doubling time formula t2 = ln(2) / ln(1 + r) gives the exact answer.
The quick Rule-of-70 estimate gives 15.40 months, 0.35 months off the exact figure. See the exact-vs-Rule-of-70 tool for the full accuracy comparison across rates.
A metric that rose from 1,800 to 2,950 over 7 periods grew at an implied rate of 7.32% per period. The growth rate formula r = (x(t)/x0)1/t − 1 solves this backward from observed data.
Entering 7.31% back into the main exponential growth calculator with the same starting value and period count reproduces 2,950, confirming the solve.
A population of 30 growing at 35% per year toward a habitat carrying capacity of 900 reaches about 727 individuals after 15 years, well short of a pure exponential projection. The logistic growth formula accounts for the slowdown a plain exponential model misses.
A pure exponential projection at the same 35% rate would predict 30 × 1.3515 ≈ 2,705, far beyond the habitat's capacity of 900, which is exactly the unrealistic result the carrying-capacity model is built to avoid.
This page covers six fully worked exponential growth and decay problems spanning finance, biology, and general mathematics. Each example includes a small live calculator pre-filled with that problem's numbers plus a complete step-by-step written solution.
Yes, every example's calculator is fully live, so changing any input recalculates that example's result instantly. The written solution below each calculator explains the original numbers; edit the fields to see how a different scenario plays out using the same formula.
Each example uses the formula from its own topic page. Compound interest, bacterial doubling, exponential decay, doubling time, rate-solving, and logistic growth are all represented. Every example links to its dedicated calculator page for a deeper look at that specific formula.
Different fields conventionally describe the same exponential mathematics with different variable names. A bank calls it an interest rate, a biologist calls it a generation time, an ecologist calls it a carrying capacity. Underneath, every example on this page is the same growth or decay formula used throughout this site.
The six examples were chosen to cover every major category on this site at least once, finance, population biology, radioactive-style decay, doubling time, algebraic rate-solving, and constrained logistic growth. Together they show the same handful of formulas applied across very different real-world framings.
No, the logistic example includes a carrying capacity that slows the growth rate as the quantity approaches it, unlike the plain exponential examples on this page. See the dedicated S-curve calculator for the full formula and explanation.
Yes, each example's calculator accepts any values you enter, so you can substitute your own scenario's numbers directly into the same formula. For repeated use with only that one topic's inputs, the dedicated calculator page for each example is a cleaner starting point.