Math101 · Independent Ontario learning library
Math101learn.math101.caCompound Interest
Compound interest applies growth to a changing balance, creating exponential saving and borrowing models.
Compound interest is interest on the current balance, including previously earned or charged interest.
The compound-growth model
For principal $P$, nominal annual rate $r$, $n$ compounding periods per year, and time $t$ years,
The interest is $I=A-P$. The periodic growth factor is $1+r/n$, and the number of periods is $nt$.
Why compounding is exponential
The balance is multiplied by the same factor each period. If $1000$ grows by $5\%$ annually, balances begin $1000$, $1050$, $1102.50$, and $1157.63$. The dollar increase grows because the rate applies to a larger balance.
Simple interest adds a constant amount; compound interest multiplies by a constant factor.
Annual compounding
Do not round each year unless the context requires it. Keep calculator precision and round currency at the final step.
More frequent compounding
For $6\%$ compounded monthly, the periodic rate is $0.06/12=0.005$. Three years contain $12(3)=36$ periods:
Both divisions and exponent changes are required. Using annual rate with monthly period count overstates growth dramatically.
Reading the formula structurally
Separate the calculation into four questions:
- What is the starting amount $P$?
- What decimal rate applies each period, $r/n$?
- What is the periodic multiplier, $1+r/n$?
- How many periods occur, $nt$?
This structure transfers to population growth, inflation, depreciation, and repeated percent change.
Depreciation
A repeated decrease uses factor below $1$. If an item loses $18\%$ of value annually,
The item loses $18\%$ of its current value each year, not $18\%$ of the original value.
Solving for time
When time is unknown, logarithms isolate the exponent. For annual compounding,
gives
Before using the formula, decide whether the result should be rounded up to a complete payment or compounding period.
Comparing rates
Compounding frequency affects actual annual growth. The effective annual rate is
Two products with the same nominal rate can therefore produce slightly different results. Actual loans and investments may also include fees, payment timing, variable rates, and tax effects that this basic model does not include.
Common mistakes
Using percent form directly. $4\%$ becomes $0.04$.
Dividing the exponent by $n$. Frequency increases periods: use $nt$.
Forgetting to divide rate by $n$. Each period receives $r/n$.
Subtracting $r$ for growth. Growth factor is $1+r/n$; depreciation uses subtraction.
Calling the final amount “interest.” Interest is $A-P$.
Quick self-check
- What is the compounding period?
- Did I use decimal rate per period and total number of periods?
- Is this growth or depreciation?
- Does the question ask for balance, interest, or time?
- Have I delayed rounding until the end?
Related topics
Explore the idea
Sequence explorer
Change one quantity at a time and connect what moves to Compound Interest.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Which expression gives the balance of $3000 at 4% compounded annually for 5 years?
- The annual growth factor is 1.04.
- There are 5 annual periods.
- The model is 3000(1.04)⁵.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
