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Simple and Compound Interest

Interest over time with different compounding periods.

Interest is the price of borrowing money, or the reward for lending it. Simple interest is paid only on the original principal: $I = Prt$ . Compound interest is paid on the principal plus interest already earned, so the balance grows faster and faster: $A = P\left(1 + \frac{r}{n}\right)^{nt}$ for $n$ compounding periods per year.

Over short periods the difference is small; over decades it is enormous. The rule of 72 gives a quick estimate of doubling time: divide 72 by the annual percentage rate. The effective annual rate converts any compounding frequency to an equivalent once-a-year rate for comparison.

Key terms

Compound interest: Balance after t years compounded n times per year.
$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Key ideas

Simple: $A = P(1 + rt)$ .
Compound: $A = P(1 + \frac{r}{n})^{nt}$ ; continuous: $A = Pe^{rt}$ .
Effective annual rate $= (1 + \frac{r}{n})^n - 1$ .
Rule of 72: doubling time $\approx \frac{72}{\text{rate } \%}$ .

Worked example

$1,000 is invested at 6% per year for 10 years. Compare simple interest with annual compounding.

Simple: $1000(1 + 0.06 \times 10) = 1600$ .
Compound: $1000(1.06)^{10} \approx 1790.85$ .
Compounding earns about $190.85 more.
Rule of 72: doubles in about $72/6 = 12$ years.

Answer: $1,600 simple versus $1,790.85 compound

Common mistakes

Using the percentage (6) instead of the decimal (0.06).
Forgetting to divide the rate by $n$ or multiply the time by $n$ .
Applying simple interest formulas to compounding accounts.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Simple interest on $2,000 at 5% for 3 years.
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$300
2.$500 at 4% compounded annually for 2 years. Final amount.
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$540.80
3.Effective annual rate of 12% compounded monthly, as a percent to two decimals.
Show answer
$12.68\%$

Simple and Compound Interest

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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Calculators

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