Compound Interest Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Compound Interest.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Interest calculated on both the initial principal and the accumulated interest from previous periods. The formula A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after tt years, and A=PertA = Pe^{rt} gives the continuously compounded amount.

Simple interest pays you only on your original deposit. Compound interest pays you interest on your interest—your money earns money on the money it already earned. The more frequently you compound, the more you earn, because each tiny interest payment starts earning its own interest sooner. The ultimate limit of compounding more and more frequently is continuous compounding: A=PertA = Pe^{rt}.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Each period's interest is added to the principal so the next period grows a bigger base.

Common stuck point: The procedure for compound interest is the easy part; the trap is using the annual rate rr directly when compounding more than once a year. Asking "Does each period's interest get added to the balance so the next period earns on a larger amount?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does each period's interest get added to the balance so the next period earns on a larger amount?

Worked Examples

Example 1

easy
You invest $5,000\$5{,}000 at 6%6\% annual interest compounded quarterly. Find the amount after 3 years.

Answer

A≈$5,978.09A \approx \$5{,}978.09

First step

1
Use the formula A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt} with P=5000P = 5000, r=0.06r = 0.06, n=4n = 4, t=3t = 3.

Full solution

  1. 2
    Substitute: A=5000(1+0.064)4⋅3=5000(1.015)12A = 5000\left(1 + \frac{0.06}{4}\right)^{4 \cdot 3} = 5000(1.015)^{12}.
  2. 3
    Compute (1.015)12≈1.19562(1.015)^{12} \approx 1.19562, so A≈5000×1.19562≈5978.09A \approx 5000 \times 1.19562 \approx 5978.09.
Compound interest applies the interest rate multiple times per year. The key parameters are the principal PP, annual rate rr, compounding frequency nn, and time in years tt.

Example 2

medium
How long does it take for an investment to double at 8%8\% annual interest compounded monthly?

Example 3

medium
Find the amount when $2,000\$2{,}000 is invested at 5%5\% compounded continuously for 4 years.

Example 4

medium
$2,500\$2{,}500 is invested at 4%4\% compounded semi-annually for 5 years. Find AA.

Example 5

medium
How long does it take $1000 to grow to $2000 at 5%5\% compounded annually?

Example 6

hard
Find the time required for $1,000\$1{,}000 to triple at 5%5\% compounded continuously.

Example 7

hard
Bank A offers 5.1%5.1\% compounded annually; Bank B offers 5%5\% compounded monthly. Which is better?

Example 8

hard
A loan of $5,000\$5{,}000 accrues at 9%9\% compounded monthly. If no payments are made, what is owed after 2 years?

Example 9

challenge
Derive the effective annual rate formula in terms of rr and nn for nn compounding periods.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
You deposit $1,200\$1{,}200 in a savings account at 4%4\% compounded annually. What is the balance after 5 years?

Example 2

hard
What annual interest rate, compounded monthly, is needed to grow $3,000\$3{,}000 to $4,500\$4{,}500 in 6 years?

Example 3

easy
Convert a 5% annual interest rate to the decimal rr used in formulas.

Example 4

easy
For quarterly compounding, what is the value of nn in A=P(1+r/n)ntA=P(1+r/n)^{nt}?

Example 5

easy
For monthly compounding, what is the value of nn?

Example 6

easy
Compute the amount AA for P=$100P=\$100, r=0.10r=0.10, compounded annually (n=1n=1) for t=1t=1 year.

Example 7

easy
After 2 years at 10% compounded annually on $100\$100, what is AA?

Example 8

easy
In A=PertA=Pe^{rt}, what does PP represent?

Example 9

easy
What is the growth factor per period for r=0.08r=0.08 compounded quarterly?

Example 10

easy
Continuous compounding uses which formula?

Example 11

medium
Find AA for P=$1000P=\$1000, r=0.06r=0.06, n=2n=2 (semiannual), t=3t=3 years.

Example 12

medium
How much interest is earned on $500\$500 at 4%4\% compounded annually for 2 years?

Example 13

medium
Find AA for $2000\$2000 at 5%5\% compounded continuously for 4 years. Use e0.2≈1.2214e^{0.2}\approx 1.2214.

Example 14

medium
Which earns more on $1000\$1000 at 8%8\% for 1 year: annual or quarterly compounding? Give both amounts.

Example 15

medium
Find the principal needed to reach $1102.50\$1102.50 in 2 years at 5%5\% compounded annually.

Example 16

medium
A $1500\$1500 deposit grows to $1815\$1815 in 2 years compounded annually. Find rr.

Example 17

medium
Compute the effective annual yield for 6%6\% compounded monthly. Use (1.005)12≈1.06168(1.005)^{12}\approx 1.06168.

Example 18

medium
After 3 years of continuous compounding at r=0.04r=0.04, by what factor has $1\$1 grown? Use e0.12≈1.1275e^{0.12}\approx 1.1275.

Example 19

medium
Find AA for $800\$800 at 10%10\% compounded annually for 3 years.

Example 20

challenge
How long until $1000\$1000 doubles at 7%7\% compounded continuously? Use ln⁡2≈0.693\ln 2\approx 0.693.

Example 21

challenge
Find AA for $5000\$5000 at 4.5%4.5\% compounded daily (n=365n=365) for 1 year. Use (1+0.045/365)365≈1.046025(1+0.045/365)^{365}\approx 1.046025.

Example 22

challenge
An account offers either 6%6\% compounded annually or 5.9%5.9\% compounded continuously. Which has the higher effective yield? Use e0.059≈1.06078e^{0.059}\approx 1.06078.

Example 23

easy
You deposit $500 at 6%6\% annual interest compounded annually. What is the balance after 1 year?

Example 24

easy
For semi-annual compounding, what is nn?

Example 25

easy
For daily compounding, what is the standard value of nn used in most problems?

Example 26

easy
Compute the balance: P=$1000P = \$1000, r=8%r = 8\%, compounded annually, t=2t = 2 years.

Example 27

medium
Find the interest earned (not the total) when $1,500\$1{,}500 is compounded annually at 5%5\% for 4 years.

Example 28

medium
$4,000\$4{,}000 is invested at 6%6\% compounded continuously for 10 years. Find AA.

Example 29

medium
How much must you invest today to have $10,000\$10{,}000 in 8 years at 4%4\% compounded annually?

Example 30

medium
Find the balance after 2 years if $500\$500 is compounded continuously at 7%7\%.

Example 31

medium
Compute the effective annual rate (APY) for a nominal 6%6\% compounded monthly.

Example 32

hard
What annual rate, compounded continuously, doubles your money in 10 years?

Example 33

hard
An account earns 5%5\% compounded quarterly. What is the effective annual rate?

Example 34

hard
How long until $3000 grows to $5000 at 6%6\% compounded monthly?

Example 35

hard
$1000 is split: $600 into Account A (4%4\% annual) and $400 into Account B (5%5\% annual), both compounded annually. Total after 3 years?

Example 36

hard
What principal yields $1500 of interest in 5 years at 6%6\% compounded annually?

Example 37

challenge
An investor wants \$50,000 in 18 years for college. They start with \$10,000. What continuously compounded annual rate is required?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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