Simple and compound interest

Simple interest as an arithmetic sequence and linear function, compound interest as a geometric sequence and exponential function, compounding periods, comparing the two, and depreciation.

Simple interest

$$I = PRn \qquad A = P(1 + Rn)$$

Here $P$ is the principal, $R$ the interest rate per period as a decimal, and $n$ the number of periods. The amounts grow by the same amount each period: an AP, and a linear function of $n$.

Example 1●

$\$5000$ is invested at $4\%$ p.a. simple interest for 6 years. Find the final amount.

$I = 5000 \times 0.04 \times 6 = \$1200$, so $A = \$6200$.

Compound interest

$$A = P(1 + R)^n$$

Interest is added to the balance each period, so the amounts form a GP with ratio $1 + R$: an exponential function of $n$.

Example 2●●

$\$5000$ is invested at $4\%$ p.a. for 6 years. Find the amount if interest compounds (a) annually and (b) monthly.

$5000 \times 1.04^6 \approx \$6326.60$

Annually

The monthly rate is $\tfrac{0.04}{12}$, over $72$ months: $5000\left(1 + \tfrac{0.04}{12}\right)^{72} \approx \$6353.71$.

Monthly

Simple versus compound

Simple interest (orange) grows in a straight line. Compound interest (blue) grows exponentially and pulls further ahead each year.

Depreciation

When a value falls by a fixed percentage each year, $A = P(1 - R)^n$.

Example 3●●

A $\$12\,000$ coffee machine depreciates at $15\%$ p.a. Find its value after 5 years.

$12\,000 \times 0.85^5 \approx \$5324.46$

Flashcards

Simple interest

$I = PRn$, an AP.

Compound interest

$A = P(1 + R)^n$, a GP.

Depreciation

$A = P(1 - R)^n$

Practice

$\$5000$ at $4\%$ p.a. simple interest for 6 years. Find the final amount.
$\$5000$ at $4\%$ p.a. compounded annually for 6 years. Find the amount to the nearest cent.
The same, but compounded monthly. Find the amount to the nearest cent.
A $\$12\,000$ machine depreciates at $15\%$ p.a. Find its value after 5 years, to the nearest cent.
Simple interest amounts form:
$6\%$ p.a. compounded quarterly for 3 years. The rate and number of periods are:
At $5\%$ p.a. compound interest, after how many whole years does money first double?
Find the simple interest on $\$10\,000$ at $3\%$ p.a. for 3 years.
Over a long time, compared with simple interest at the same rate, compound interest gives: