The Compound Interest Formula Explained

Each part of the formula

• A — the balance at the end.
• P — the principal, the starting amount.
• r — the annual interest rate as a decimal (4% is 0.04).
• n — how many times per year interest is added (12 for monthly).
• t — the time in years.
• r/n is the rate for each period, and nt is the total number of periods.

A worked example

$2,500 at 4% compounded quarterly for 6 years:

1. Rate per quarter: r/n = 0.04 ÷ 4 = 0.01.
2. Number of quarters: nt = 4 × 6 = 24.
3. Growth factor: 1.01^24 = 1.2697346485.
4. Balance: A = 2,500 × 1.2697346485 = $3,174.34, so the interest is $674.34.

Solving for the starting amount

Rearranged, P = A ÷ (1 + r/n)^(nt). To have $5,000 in 5 years at 4% compounded monthly, the growth factor is (1 + 0.04/12)^60 = 1.2209965939, so the starting amount is $5,000 ÷ 1.2209965939 = $4,095.02. This is often called the present value.

Solving for the time

Taking logarithms gives t = ln(A ÷ P) ÷ (n × ln(1 + r/n)). For a balance to double at 6% compounded monthly: t = ln 2 ÷ (12 × ln 1.005) = 11.58 years.

Common mistakes

• Using the percentage instead of the decimal: 5, not 0.05, gives a wildly wrong answer.
• Mixing periods: with monthly compounding both the rate division and the exponent use 12.
• Rounding r/n early: 0.04 ÷ 12 is 0.003333…, and cutting it to 0.0033 visibly changes a long-term result.
• Forgetting that A includes the principal: interest earned is A − P.