Percentage Growth and Compound Interest

Divide the percentage by 100 and add 1 for growth. A rate of 3% becomes r=0.03 and the factor 1.03. For a 3% decrease, use 1−0.03=0.97. These factors tell you what fraction of the previous amount remains after one step. Do not enter 3 directly when the formula expects a decimal r. A quick check is that a growth factor exceeds 1, while a decrease factor lies between 0 and 1. The initial amount precedes the first step, so n=0 returns exactly G₀.

In every period, the current amount is multiplied again by 1+r. After one period the factor appears once; after two it appears twice, and after n periods n times. Thus G_n=G₀(1+r)ⁿ. For €1,000 at 3% per year, two years give 1,000·1.03²=€1,060.90. A common mistake is to increase the starting amount once by n·r; that describes simple linear interest and ignores growth on previous growth. The rate is specified per step; an annual rate cannot be used monthly without adjustment. Compare matching period lengths.

For a decrease of p%, the factor is 1−p/100. If an amount falls by 8% per year, multiply by 0.92 each year. After three years, G₃=G₀·0.92³, not G₀·(1−0.24), because each decrease applies to the current amount. The factor must stay positive, so the per-step decrease is below 100%. Examples include depreciation or a declining population. Check whether the model remains reasonable for the full time span; real rates can change. For a decrease close to 100%, the amount becomes very small but does not turn negative while the factor stays positive.

With simple interest, each period earns interest only on the original principal; the interest per period stays fixed and the balance grows linearly. With compound interest, credited interest becomes part of the new principal, so later interest also applies to earlier interest. For annual compounding use K_n=K₀(1+r)ⁿ. For monthly or other subannual compounding, adjust the rate per period and the number of periods. These are different models and their formulas should not be confused. Measure total change relative to G₀; do not simply add the rates from individual periods.

If you know the initial amount, final amount, and growth factor, solve G_n=G₀(1+r)ⁿ with logarithms: n=ln(G_n/G₀)/ln(1+r). This requires positive amounts and r≠0. With €1,000 initially, 3% growth, and a final amount near €1,159.27, n=ln(1.15927)/ln(1.03)≈5 years. Round only at the end and check that n fits the whole periods in the context. A calculator does not replace checking the time unit and model assumptions. With subannual crediting, the periodic rate and number of steps must match the fraction of a year. Check that periods have equal length.