How compound interest grows

The repeated operation

With annual compounding at 10%, multiply the balance by 1.10 once each year. The rate stays fixed in this example, while the dollar amount earned grows.

The principal

Principal means the starting amount. Reinvested interest becomes part of the balance used to calculate later interest.

An illustrative model

We assume no deposits, withdrawals, taxes, fees, or inflation. A constant 10% annual rate is an arithmetic example, not a promise of investment returns.

Year 1: $100 of interest

$1,000 × 10% = $100. Add it to the principal: $1,000 + $100 = $1,100.

Year 2: $110 of interest

$1,100 × 10% = $110. Of that, $100 comes from the original principal and $10 comes from the first year's interest.

Year 3: $121 of interest

$1,210 × 10% = $121, taking the balance to $1,331. The growing interest amount is the effect of compounding.

Start with the balance already carried

The preceding year turned $1,000 into $1,100. The long measuring bed contains that whole $1,100: $1,000 principal and $100 earlier interest.

The instrument embodies a fraction

A proportional divider uses two pivoted arms. Here the distance from pivot to long tip is exactly ten times the distance to short tip. Its short span therefore measures one tenth of the long span: $110, at a common $10-per-unit scale.

Copy, then carry forward

The small bed holds the $110 increment. The separate addition bed places a copy beside the current $1,100 to make $1,210. This is an analogy for a measuring operation, not a machine that generates money or a literal account mechanism.

A compact formula

For annual compounding, final balance = principal × (1 + annual rate)^years. Write 10% as 0.10, so the yearly multiplier is 1.10.

What the exponent means

The exponent counts how many times the multiplier is applied. Ten years means multiplying by 1.10 ten times, rather than adding 10% of the original amount ten times.

A more general version

For a fixed nominal annual rate r compounded m times per year over t years, the model is P × (1 + r/m)^(m × t). Rates must be compared on a consistent basis; an effective annual rate already includes compounding.

The same starting point

Both examples begin with $1,000 and use a stated 10% annual rate. The difference is which balance the rate applies to after the first year.

The compounding difference

The compound balance is about $593.74 larger after ten years. That extra amount comes from earning interest on earlier interest.

Interest can work against a borrower

The same mechanism can increase debt when unpaid interest is added to the balance. Real loan terms, payment schedules, and fees determine the actual cost.

Returns may vary

Market investments do not generally deliver the same return every year. Gains and losses multiply: a 10% gain followed by a 10% loss leaves $1,000 at $990, not $1,000.

Costs change the path

Fees and taxes can reduce the amount left to compound. Inflation reduces what a future dollar can buy, so a larger dollar balance does not by itself measure purchasing-power growth.

Compare like with like

Check whether a quoted rate is nominal or effective, how often interest compounds, and whether the example includes fees and contributions. More frequent compounding at the same positive nominal annual rate produces a higher effective annual rate.

The next year uses the current $525 balance: $525 × 0.05 = $26.25. The balance would become $551.25.

The base changes; the rule does not

Apply each year’s rate to the balance at the beginning of that year. The first year earned $25, leaving $525; the second earns $26.25. We assume annual reinvestment and no deposits, withdrawals, costs or inflation.

One portable rule

Compounding is repeated percentage change. Interest, investment returns, price changes and some debt balances can use similar arithmetic, but the relevant rates and assumptions differ.

The compact annual model

For a fixed annual rate r and annual compounding, final balance = principal × (1+r)^years. This is a model under explicit assumptions, not a prediction about a real account or investment.