Compound Interest Explained (with the Formula and Examples)

Compound interest is interest earned on your original amount and on interest already added. The balance after $t$ years is $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $P$ is the starting amount, $r$ the yearly rate as a decimal and $n$ how many times a year interest is added. Over long periods it grows much faster than simple interest.

Simple vs compound interest

Put 1,000 into an account at 5% a year.

• Simple interest pays 5% of the original 1,000 every year: 50 a year, so 1,500 after 10 years.
• Compound interest pays 5% of the current balance. Year 1 earns 50, year 2 earns 52.50, year 3 earns 55.13, and so on.

The formula

Worked example 1: yearly compounding

1,000 at 5% for 10 years, compounded yearly ($n = 1$):

Worked example 2: monthly compounding

Same deal, compounded monthly ($n = 12$):

More frequent compounding earns a little more, because interest starts earning interest sooner.

Worked example 3: a longer timescale

5,000 at 3.5% for 20 years, yearly:

The money almost doubles.

Continuous compounding

As $n$ grows without limit, the formula approaches

2,500 at 7% for 15 years gives $2500e^{1.05} \approx 7144.13$.

The rule of 72

To estimate how many years it takes money to double, divide 72 by the percentage rate. At 6%, money doubles in about $72 \div 6 = 12$ years. The exact answer comes from logarithms: $t = \frac{\ln 2}{\ln 1.06} \approx 11.9$.

Common mistakes

• Using 5 instead of 0.05 for the rate.
• Forgetting to divide the rate by $n$ and multiply the years by $n$.
• Confusing the final amount $A$ with the interest earned, $A - P$.

Practice questions

1. 2,000 at 4% for 5 years, compounded yearly. Find $A$.
2. How much interest is earned in question 1?
3. Using the rule of 72, roughly how long does money take to double at 8%?

Answers: 1) about 2,433.31 2) about 433.31 3) about 9 years

The compound interest calculator draws the balance growing year by year and compares compounding options. Growth like this is a geometric sequence, and solving for time uses logarithms.

Frequently asked questions

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