SIMPLE VERSUS COMPOUND INTEREST

Suppose something is owned.
The thing that is owned is called the ASSET.
The entity that owns the asset is called the OWNER.

Suppose someone else wants to borrow the asset.
The entity that wants to borrow the asset is called the BORROWER.

It is reasonable that the borrower should compensate the owner for use of the asset.
This compensation is called interest.
In other words, interest is the price paid for the use of a borrowed asset.

Here are two common situations involving asset, owner, borrower, and interest:

Interest causes the amount of money involved in a transaction to change:

Principal is the name given to the original amount involved in an investment or a loan, separate from any changes due to interest.

There are two basic flavors for interest: simple and compound.

Simple Interest

By definition, simple interest is interest that is calculated only on the principal amount.
To compute simple interest, you need to know three things:

name typical variable name units example
the amount of principal $\,P\,$ currency \$100
the length of time over which interest is being applied $\,t\,$ time $3$ years
the interest rate:
the fraction of the principal that is to be paid as compensation for borrowing the asset,
per a given period of time

The fraction is typically expressed as a percentage:
for example, $\,\frac{5}{100}\,$ is typically expressed as $\,5\%$
$r$ $\displaystyle\frac{1}{\text{time}}$ $\displaystyle\frac{5\%}{\text{year}}$
Then, $$\text{simple interest} = P\,r\,t$$

BE CAREFUL!
When properly calculated, interest should come out with units of currency only.
All time units should cancel out.
If they don't, then the answer isn't very meaningful, as the next example illustrates.

Example

Suppose you've borrowed \$1000 from a friend for 20 months.
You've agreed to pay simple interest at the rate of $\,1.5\%$ per year.
How much interest do you owe your friend?

Remember that a percent symbol ($\,\%\,$) is just a multiplier of $\,\frac{1}{100}\,$ or $\,0.01\,$.

Here's a not-very-meaningful calculation:
the time units don't cancel out, and therefore stick around in the answer: $$ \begin{align} \text{simple interest} &= P\,r\,t \cr &= \$1000\cdot\frac{1.5\%}{\text{yr}}\cdot 20\text{ months }\cr &= (1000)(1.5)(0.01)(20) \frac{\$\,\text{months}}{\text{yr}}\cr &= 300 \frac{\$\,\text{months}}{\text{yr}} \end{align} $$ Most people don't have good intuition for units of ‘dollar months per year’, so this is NOT a good form of the answer.

Here is a meaningful version of the calculation.
In this calculation, the time units cancel out.
Notice the extra factor of $\,1\,$ that is required for unit conversion: $$ \begin{align} \text{simple interest} &= P\,r\,t \cr &= \$1000\cdot\frac{1.5\%}{\text{yr}\kern -10px{/}}\cdot 20\text{ months}\kern -30px{/}\kern 30px \cdot \overbrace{\frac{1 \text{ yr}\kern -10px{/}}{12 \text{ months}\kern -30px{/}\kern 30px}}^{\text{conversion factor}}\cr &= \frac{(\$1000)(0.015)(20)}{12}\cr &= \$25 \end{align} $$ Or, just replace one year with twelve months in the initial calculation, and achieve the same result: $$ \begin{align} \text{simple interest} &= P\,r\,t \cr &= \$1000\cdot\frac{1.5\%}{12 \text{ months}\kern -30px{/}\kern 30px }\cdot 20\text{ months}\kern -30px{/}\kern 30px\cr &= \frac{(\$1000)(0.015)(20)}{12}\cr &= \$25 \end{align} $$

Compound Interest

By definition, compound interest is interest that is calculated on the principal amount together with accumulated interest.
Since the ‘new and interesting’ part is the interest on the accumulated interest, compound interest is often described in-a-nutshell as ‘interest on interest’.

THE COMPOUND INTEREST FORMULA
Suppose you invest $\,P\,$ dollars at (simple) annual interest rate $\,r\,$,
and add in interest $\,n\,$ times per year (that is, there are $\,n\,$ compounding periods per years).
The amount, $\,A\,$ (principal plus interest), that you have after $\,t\,$ years is given by
the compound interest formula: [beautiful math coming... please be patient] $$A=P{(1+\frac{r}{n})}^{nt}$$

This prior section gives the motivation, derivation, and examples of use of the compound interest formula.

In Precalculus, we now want to think about the result of adding in interest at shorter and shorter intervals—
every hour, every minute, every second, every millisecond, and so on.
That is, we will let $\,n\,$ (the number of times per year that interest is added in) go to infinity.

Letting $\,n\rightarrow\infty\,$ leads to so-called continuous compounding, which is the subject of the next section.
It's another beautiful appearance of the irrational number $\,\text{e}\,$!

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
continuous compounding
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22
AVAILABLE MASTERED IN PROGRESS

(MAX is 22; there are 22 different problem types.)