Compound Interest Formula — A = P(1 + r/n)^nt Explained

Compound interest grows a balance from its updated value each period, while simple interest grows only from the original principal. That single difference is why compound interest uses an exponent and grows faster over time.

The compound interest formula gives the final amount after a balance grows at a fixed annual rate with interest added at regular intervals:

Here $P$ is the starting principal, $r$ is the annual rate as a decimal, $n$ is the number of compounding periods per year, and $t$ is time in years. The result $A$ is the total amount after interest; for interest alone, subtract the principal: $A - P$.

Compound Vs. Simple Interest, Side By Side

                  Simple interest          Compound interest
Grows from        original principal only  updated balance each period
Model type        linear                   exponential
Formula           A = P(1 + rt)            A = P(1 + r/n)^(nt)
Later interest    on principal only        on principal plus past interest

Because simple interest never earns interest on interest, it uses a linear model $A = P(1 + rt)$. Compound interest feeds each period's interest back into the balance, so it uses an exponent. If interest is added back after each period, the exponential model is the right one.

How To Read The Formula

The expression

is the growth factor for one compounding period. If the annual rate is $8\%$ and compounding is quarterly, each quarter multiplies the balance by

The exponent $nt$ counts how many times that growth happens. For $2$ years of quarterly compounding, the balance is multiplied $4 \cdot 2 = 8$ times. That is the key idea: the balance keeps getting multiplied by the same period-by-period factor, so later interest is earned on earlier interest too.

Each variable in detail:

• $P$ is the principal, or starting amount.
• $r$ is the annual rate in decimal form. For example, $8\% = 0.08$.
• $n$ is compounding frequency: $n = 1$ yearly, $n = 2$ semiannual, $n = 4$ quarterly, $n = 12$ monthly.
• $t$ is time in years. If the rate is annual, $18$ months should be written as $1.5$ years.

When To Use This Formula

Use $A = P(1 + r/n)^{nt}$ only when the annual rate stays fixed, the compounding schedule is known, and there are no extra deposits or withdrawals during the period. It is a fixed-rate repeated-growth model, useful because it is simple, but that simplicity depends on the assumptions holding. If money is added every month or the rate changes halfway through, a single use of the formula is not enough.

You see it in savings accounts, certificates of deposit, investment growth examples, and classroom finance problems, and in any setting where a quantity grows by the same percentage over equal time intervals.

Worked Example, Choosing The Right Frequency

Suppose $5{,}000$ is invested at $8\%$ annual interest for $2$ years, compounded quarterly.

Substitute:

Simplify the period growth and the exponent:

Evaluate:

So the final amount is about $5{,}858.30$. For compound interest only, subtract the principal:

This example also shows why frequency matters. With the same principal, rate, and time but yearly compounding, the amount would be $5000(1.08)^2 = 5832$, slightly smaller. More compounding periods means interest is added more often, so quarterly beats yearly here. To see the effect grow, keep $P = 5000$, $r = 0.08$, $t = 2$, but switch to monthly compounding and compare against the quarterly result above; a compound interest calculator makes comparing several frequencies quick.

Mistakes That Change The Answer

Leaving The Rate As A Percent

In the formula, $r$ must be a decimal, so $8\%$ becomes $0.08$, not $8$.

Mixing Up Amount And Interest

The formula gives $A$, the final amount. If the problem asks for compound interest only, subtract $P$.

Using The Wrong Compounding Frequency

Monthly, quarterly, and yearly compounding give different answers. The wording determines $n$.

Forgetting The Time Condition

If $r$ is an annual rate, $t$ must be in years. A mismatch changes the answer.

Using The Formula With Extra Cash Flows

If money is added every month or the rate changes partway, one use of $A = P(1 + r/n)^{nt}$ is not enough.