Simple and Compound Interest

Simple interest pays you a fixed percent of the original principal every period, so the balance after t years is P(1 + rt). Nothing from last year's interest earns anything new this year. Compound interest instead multiplies the whole balance, principal plus everything already earned, by the same growth factor each period. That balance is P(1 + r)^t for annual compounding. Because the base keeps growing, each year's interest payment is larger than the last, and the graph bends upward instead of tracing a straight line.

The exponent in the formula counts compounding periods, not years, which is where quarterly or monthly compounding changes the expression. If a rate of r percent compounds quarterly, each period only applies a quarter of the annual rate, but there are four periods per year, so the balance after x years is P(1 + r/4)^4x. Monthly compounding splits the rate into twelfths and multiplies the exponent by 12 instead. Smaller slices of the rate applied more often still add up to more total growth than one lump sum once a year, since interest starts earning interest sooner within the same year.

This topic sits next to percent change as a growth factor: a compound interest formula is just repeated percent increase, the same (1 + rate)^time structure used for population growth or depreciation. Recognizing that shared pattern is what lets a balance problem, a growth problem, and a decay problem all resolve the same way once the growth factor and the number of periods are identified correctly.