Simple & Compound Interest

Put 10000 in a bank and the bank pays you a little thank-you every year: interest. How is it calculated? Two rules exist, and after 30 years the gap between them will make you sit up straight.

Simple interest: the same slice every year

Simple interest is computed on the original deposit only. Call the principal P (the money you start with), the yearly rate r, and the time t in years. The interest is:

$I = P \times r \times t$

With 10000 at 5% per year, each year earns $10000 \times 0.05 = 500$ , year after year — 1500 after three years. Year 1 pays 500, year 10 still pays 500. Draw the total and you get a straight line, exactly the $y = mx + b$ shape from Equation of a Straight Line.

Compound interest: the interest starts earning too

Compound interest changes one rule, and that changes everything: each year the interest joins the principal, and the next year it earns interest too. After t years the total is:

$A = P \times (1 + r)^t$

Same 10000 at 5%: after year one it is 10500; year two is computed on 10500, giving $10500 \times 1.05 = 11025$ ... the money starts rolling like a snowball. Working with letter formulas is old news if you met Variables & the Function Machine.

InteractiveCompound Curve

Principal10000Annual rate 5%5Years10

Simple

15,000

Compound

16,289

Compound earns extra: +1,289

Try it: the first two years

10000 at 5% for two years. Simple: $10000 + 500 \times 2 = 11000$ . Compound: $10000 \times 1.05^2 = 11025$ . Only 25 apart? Do not dismiss it — those 25 are the interest earned by the first 500 of interest. Every snowball starts small.

The Rule of 72: years to double

A mental-math gem: divide 72 by the yearly rate (drop the percent sign) and you get roughly the number of years it takes to double.

At 8%: $72 \div 8 = 9$ years to double;
At 6%: about 12 years;
At 3%: about 24 years.

It is an approximation, most accurate between 4% and 12%. For a quick verdict on whether money is growing fast enough, it is all you need.

Why starting early wins

Same 10000, same 5%, left to compound:

Left 10 years: about 16289, a gain of 6289;
Left 30 years: about 43219, a gain of 33219.

Under simple interest those 30 years would yield $10000 + 500 \times 30 = 25000$ — about 18000 less than compound. The late game is where compound shines: the first decade earns 6289, the last decade about 16686 — more than double. Time feeds the snowball, which is why every early year counts.

Compound interest cuts both ways

When you save, it works for you; when you owe, it works for the bank. Credit card debt often runs near 20% a year — by the Rule of 72 it doubles in about 3.6 years. In debt, compound interest sits on the other side of the table.

Check yourself

Quick quiz

1. 10000 sits at simple interest, 5% a year. How much interest after 3 years?
2. By the Rule of 72, how long does money take to double at 8%?
3. Same 10000 at the same 5%. After 30 years, how do simple and compound compare?