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:
There are two basic flavors for interest: simple and compound.
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, $$\cssId{s53}{\text{simple interest} = P\,r\,t}$$ 
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.
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 notverymeaningful calculation:
the time units don't cancel out,
and therefore stick around in the answer:
$$
\begin{align}
\cssId{s65}{\text{simple interest}}\
&\cssId{s66}{= P\,r\,t} \cr
&\cssId{s67}{= \$1000\cdot\frac{1.5\%}{\text{yr}}\cdot 20\text{ months }}\cr
&\cssId{s68}{= (1000)(1.5)(0.01)(20) \frac{\$\,\text{months}}{\text{yr}}}\cr
&\cssId{s69}{= 300 \frac{\$\,\text{months}}{\text{yr}}}
\end{align}
$$
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:
$$
\begin{align}
\cssId{s75}{\text{simple interest}}\
&\cssId{s76}{= P\,r\,t} \cr
&\cssId{s77}{= \$1000\cdot\frac{1.5\%}{\text{yr}\kern 10px{/}}\cdot 20\text{ months}\kern 30px{/}\kern 30px \cdot \overbrace{\frac{1 \text{ yr}\kern 10px{/}}{12 \text{ months}\kern 30px{/}\kern 30px}}^{\text{conversion factor}}}\cr
&\cssId{s78}{= \frac{(\$1000)(0.015)(20)}{12}}\cr
&\cssId{s79}{= \$25}
\end{align}
$$
Or, just replace one year with twelve months in the initial calculation, and achieve the same result:
$$
\begin{align}
\cssId{s81}{\text{simple interest}}\
&\cssId{s82}{= P\,r\,t} \cr
&\cssId{s83}{= \$1000\cdot\frac{1.5\%}{12 \text{ months}\kern 30px{/}\kern 30px }\cdot 20\text{ months}\kern 30px{/}\kern 30px}\cr
&\cssId{s84}{= \frac{(\$1000)(0.015)(20)}{12}}\cr
&\cssId{s85}{= \$25}
\end{align}
$$
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 inanutshell as ‘interest on interest’.
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 socalled continuous compounding, which is the subject of the next section.
It's another beautiful appearance of the irrational number $\,\text{e}\,$!
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
