Simple Interest

Simple interest is calculated on a principal that does not itself increase by accumulated interest. Over equal time intervals at a constant rate, the interest additions are equal. With interest retained until maturity, the principal-plus-interest amount grows linearly under a constant-rate model.

The examples below use hypothetical Canadian-dollar amounts. Each example specifies how time is counted and when principal changes; real agreements may use different conventions.

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Derive interest from a rate and a duration

Let $P$ be principal, $r$ an annual simple-interest rate written as a decimal, and $t$ a duration measured in years. One year’s interest is $Pr$ , so the interest over $t$ years is

I=Prt.

If this interest is paid with the principal at the end, the maturity amount is

A=P+I=P(1+rt).

These formulas assume the same principal and rate apply throughout the interval. OpenStax’s simple-interest treatment develops this structure. They are not automatically an amortizing-loan payment formula.

For $P=1200$ , $r=0.05$ and 18 months, take $t=18/12=1.5$ years. Then

I=1200(0.05)(1.5)=90,\qquad A=1290.

The annual interest is $60. At six, twelve and eighteen months, the accrued amounts are $30, $60 and $90 respectively.

Simple interest keeps the interest base fixed

Interest is shown separately from the principal, and no interim repayment occurs.

Diagram explanation and exact data

Five percent of 1200 dollars is 60 dollars per year. Six months therefore adds 30 dollars. Equal six-month intervals add the same amount rather than interest on prior interest.

Simple interest keeps the interest base fixed: exact reference data
Time (months)	Interest ($)	Principal plus interest ($)
0	0	1200
6	30	1230
12	60	1260
18	90	1290

Match the time unit to the rate

A 5% annual rate is not 5% per month. At the annual simple-interest rate, nine months contributes $9/12$ of a year under a month-based model. Converting 5% to 0.05 is equally important; multiplying by 5 would increase the result by a factor of 100.

For day-based agreements, state the day-count basis. On $2400 at 6% annual simple interest for 73 days, an Actual/365 fixed calculation gives

I=2400(0.06)\frac{73}{365}=28.80\text{ dollars}.

Using an Actual/360 basis with the same 73-day numerator instead gives $29.20. Neither denominator should be selected merely because it produces a preferred answer. The convention belongs to the agreement. A 30/360 method also specifies how dates are counted; it is not simply another name for actual days divided by 360.

Solve for the quantity that is missing

If the other values are known and the relevant denominator is non-zero,

r=\frac{I}{Pt},\qquad t=\frac{I}{Pr},\qquad P=\frac{I}{rt}.

For $75 interest on $1500 over ten months,

r=\frac{75}{1500(10/12)}=0.06,

so the annual simple-interest rate is 6%. Substitution into the original equation confirms the result.

To earn $160 simple interest on $2000 at 4% annually, $t=160/(2000\times0.04)=2$ years. The rate and duration must be interpreted together; two years is not the answer if 4% was a monthly rate.

Discount a future simple-interest amount consistently

If a stipulated $1200 maturity amount is due in two years at 5% annual simple interest, its modelled principal is

P=\frac{1200}{1+0.05(2)}=\frac{12000}{11}\approx1090.91.

Subtracting 10% of the future amount would give $1080, which uses the wrong reference. Interest was calculated on the unknown principal, not on the maturity amount.

This simple-interest discount relation is also different from compound discounting. Use the same interest model to move an amount forward and backwards in time.

Split the calculation when principal changes

Suppose $1200 is outstanding at 6% annual simple interest. After six months the borrower pays the $36 accrued interest and repays $600 of principal. For the next six months, only $600 remains outstanding.

The two interest amounts are

I_1=1200(0.06)(0.5)=36,

I_2=600(0.06)(0.5)=18.

Total interest is $54. At month six the outflow is $636; at month twelve it is $618. These sum to $1254, which reconciles with the $1200 original principal plus $54 interest.

Applying $1200(0.06)(1)=72$ to the whole year would incorrectly ignore the principal repayment. The general piecewise calculation is $\sum_jP_jr_jt_j$ when each interval has a stated constant principal and rate and interest is handled according to the model.

Compare simple growth with compound growth

For $1000 at 10% per year over two years, simple interest gives $1200. Annual compounding gives $1000(1.1)^2=1210$ . The extra $10 is interest on the first year’s interest.

Under simple interest, extending a maturity date does not by itself mean that earlier interest begins earning interest. Under compound interest, the balance is updated at each specified compounding point. The contractual rule, not the everyday word “interest”, determines which calculation applies.

Simple-interest models can still have changing principal, rates or fees. “Simple” describes the interest basis, not a guarantee that every product using the term has only one arithmetic step. FCAC’s loan guidance identifies other terms and costs that need to be read separately.

Check exact calculations before reporting money

Retain sufficient precision until the reporting step. If the agreement rounds each day’s interest, that is a different ledger from a model that retains all digits and rounds once at maturity. A final cent can depend on this choice.

For the algebra, $P,r,t$ may be exact stipulated values. For a real transaction, use the actual dates, balances and contractual rounding rule. Never infer an annual percentage rate or total borrowing cost from a simple-interest formula without checking what that rate includes.

Practice questions

Find the simple interest on $1000 at 4% annually for nine months.
A $2000 principal earns $180 simple interest in 1.5 years. Find the annual rate.
Under Actual/365 fixed, find interest on $500 at 7.3% annually for 20 days.

Check your answers

$I=1000(0.04)(9/12)=30$ dollars. The maturity amount would be $1030 if principal and interest are paid together.
$r=180/(2000\times1.5)=0.06$ , or 6% annually.
$I=500(0.073)(20/365)=2$ dollars. The exact result follows from the stipulated annual rate and day-count basis.