Compound interest is one of 3 finance formulas in the arithmetic section of this library, and it is used at high school level.
Why compound interest works
Each compounding period the balance is multiplied by 1 plus the period's share of the rate. Doing that over and over is what an exponent records, and nt counts how many times it happens. The interest earned early starts earning its own interest, which is why the graph curves upward instead of climbing in a straight line.
What each symbol means
$P$ is principal, $r$ is annual decimal rate, $n$ is compounds per year, and $t$ is years.
Compound interest: when it holds
$n>0$; the annual rate and time unit must agree.
When it stops applying
The formula assumes the rate and the schedule never change and that nothing is added or withdrawn. Regular monthly deposits need an annuity formula instead, since each deposit would have its own exponent here.
Compound interest: a worked example
$1000$ at $6\%$ monthly for $2$ years gives $1000(1+0.06/12)^{24}\approx1127.16$.
The mistake to avoid
What people do: Students use the full annual rate for each period, writing (1 + 0.06) raised to the number of months.
Why it goes wrong: That pays a whole year's interest twelve times a year. On 1000 dollars for 2 years it reports 4048.93 instead of the true 1127.16, roughly quadrupling the money.
Do this instead: Split the rate across the periods with r/n, then count all the periods with nt. Monthly compounding at 6 percent for 2 years is 1000(1 + 0.06/12)²⁴ = 1127.16.
Compound interest: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $P$ is principal, $r$ is annual decimal rate, $n$ is compounds per year, and $t$ is years.
- Check the conditions before substituting. $n>0$; the annual rate and time unit must agree.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Finance
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where compound interest comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about compound interest
What exactly does n stand for?
The number of times interest is added per year: 12 for monthly, 4 for quarterly, 365 for daily, and 1 for yearly. It appears twice, once dividing the rate and once multiplying the years.
Why is monthly compounding barely better than yearly?
Because the extra interest is earned on very small amounts. 1000 dollars at 6 percent for 2 years grows to 1123.60 yearly and 1127.16 monthly, a difference of about three and a half dollars.
How long until my money doubles?
The rule of 72 gives a quick estimate: divide 72 by the percent rate. At 6 percent that predicts about 12 years, and the exact calculation gives 11.9 years, so the shortcut is close enough for planning.
Does this work for something losing value?
Yes, if you make the rate negative. A car dropping 15 percent a year is modelled with r = −0.15 and n = 1, and the same formula tracks the falling value.