Compound interest is interest earned on both your original money and on interest you have already earned. Each time interest is added to your balance, the next round of interest is calculated on the bigger balance, so growth speeds up over time. Simple interest, by contrast, is calculated on the original amount only, forever. That one difference is why time matters more than the interest rate.

Work through the example year by year. Put 1,000 dollars in an account paying 5 percent a year, compounded annually, and leave it alone. After year one: 1,000 x 1.05 = 1,050 dollars. After year two: 1,050 x 1.05 = 1,102.50 dollars. After year three: 1,102.50 x 1.05 = 1,157.63 dollars. Notice the interest itself grows: 50 dollars in year one, 52.50 in year two, 55.13 in year three. After ten years the balance is 1,000 x 1.05 to the power of 10 = 1,628.89 dollars. Simple interest over the same ten years would give only 1,000 + (10 x 50) = 1,500 dollars. Compounding added an extra 128.89 dollars for doing nothing but waiting.

The formula is A = P(1 + r) to the power of n, where P is the starting amount, r is the rate per period as a decimal, and n is the number of periods. A shortcut for doubling time is the Rule of 72: divide 72 by the interest rate. At 5 percent, 72 divided by 5 = 14.4 years to double your money. At 7 percent it is about 10.3 years. The rule is an estimate, not exact, but it is close enough for planning.

Regular contributions show the effect even better. Add 100 dollars at the end of every month to an account earning 5 percent a year compounded monthly, for ten years. You pay in 100 x 120 = 12,000 dollars, and the balance grows to 15,528.23 dollars. That is 3,528.23 dollars of interest on top of your own money, and the interest portion grows larger every year because each month's interest starts earning interest of its own the very next month.

The gap most beginner guides leave is the reverse side: compound interest works against you on debt exactly as it works for you on savings. A credit card balance left unpaid grows the same way, with interest charged on interest, which is why minimum payments barely shrink a balance. The other missing piece is inflation: 1,628.89 dollars in ten years will not buy what it buys today, so compounding must be compared against rising prices, not judged in isolation.

Three things decide how much compounding helps you: the rate, how often interest is added, and above all time. Monthly compounding beats annual compounding by a small amount at the same rate, because each month's interest starts earning sooner. But starting ten years earlier beats a slightly higher rate started late, because the growth curve bends upward in the later years. That is the practical lesson: start with whatever amount you have, add regularly, and let the interest earn its own interest.