Interest is a fee we pay to borrow money.
There are two types of interest you will encounter as a consumer: simple interest and compound interest.
We will learn the difference between both of these as we progress though this section.
Today, we will begin with simple interest.
Simple Interest
Interest is a fee we pay to borrow money.
The more we borrow, the more we pay when the term of the loan is up.
The longer we borrow, the more we pay when the term of the loan is up.
The higher the rate, the more we pay when the term of the loan is up.
For simple interest, all of this can be summed up by a single formula: the simple interest formula.
The Simple Interest Formula
The simple interest formula is $$I=Prt.$$ $I$ is the amount of interest paid when the term of the loan is up.
$P$ is the principal amount of the loan. This is the amount borrowed.
$r$ is the interest rate of the loan, the percentage of the principal to be paid back over time.
$t$ is the term of the loan, that is, how long the money is borrowed.
Simple Interest
Short-term loans are often simple-interest loans.
Here are some places you may encounter simple-interest loans:
- Purchasing a used car.
- Pay-day loans (not recommended).
- Certificates of Deposit (CDs).
- Bonds.
Simple Interest: Calculating Interest $I$
Example
Sleazy P.'s restaurant needs to purchase new equipment and has taken out a $9$-year loan of $\$45990$ at an interest rate of $2.8\%.$
How much interest will Sleazy P. be paying?
Use the simple interest rate formula with values $I=\mbox{Unknown},$ $P=45990,$ $r=0.028,$ and $t=9.$
Then $$ \begin{array}{lll} \displaystyle I&\displaystyle=Prt &\mbox{}\\ \displaystyle &\displaystyle=(45990)(0.028)(9) &\mbox{}\\ \displaystyle &\displaystyle=11589.48 &\mbox{}\\ \end{array} $$ So, Sleazy P. will be paying $\$11589.48$ in interest.
Then $$ \begin{array}{lll} \displaystyle I&\displaystyle=Prt &\mbox{}\\ \displaystyle &\displaystyle=(45990)(0.028)(9) &\mbox{}\\ \displaystyle &\displaystyle=11589.48 &\mbox{}\\ \end{array} $$ So, Sleazy P. will be paying $\$11589.48$ in interest.
Simple Interest: Calculating Principal Amount $P$
Example Sleazy P. is looking to go legit by starting a wealth management company. To secure the funds, a loan shark named Mr. Holt has given Sleazy P. a $3.25$-year loan at an interest rate of $3.2\%.$
If Sleazy P. pays Mr. Holt $\$28210$ in interest after $3.25$ years, what is the principal amount of the loan?
Again, we use the simple interest rate formula.
This time we use the values $I=28210,$ $P=\mbox{Unknown},$ $r=0.032,$ and $t=3.25.$
Then $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 28210=P\cdot (0.032)(3.25)&\mbox{ $P$ is unknown}\\ \implies &\displaystyle 28210=P\cdot 0.104&\mbox{}\\ \implies &\displaystyle \frac{28210}{\color{magenta}{0.104}}=\frac{P\cdot 0.104}{\color{magenta}{0.104}}&\mbox{}\\ \implies &\displaystyle 271250=P&\mbox{}\\ \end{array} $$ So, the principal amount of the loan was $\$\mbox{271250}.$
This time we use the values $I=28210,$ $P=\mbox{Unknown},$ $r=0.032,$ and $t=3.25.$
Then $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 28210=P\cdot (0.032)(3.25)&\mbox{ $P$ is unknown}\\ \implies &\displaystyle 28210=P\cdot 0.104&\mbox{}\\ \implies &\displaystyle \frac{28210}{\color{magenta}{0.104}}=\frac{P\cdot 0.104}{\color{magenta}{0.104}}&\mbox{}\\ \implies &\displaystyle 271250=P&\mbox{}\\ \end{array} $$ So, the principal amount of the loan was $\$\mbox{271250}.$
Simple Interest: Calculating Interest Rate $r$
Example
Sleazy P. is looking to go legit by starting a wealth management company. To secure the funds, a loan shark named Mr. Holt has given Sleazy P. a $4.25$-year loan of $\$236450.$
If Sleazy P. pays Mr. Holt $\$158776.17$ in interest after $4.25$ years, what is the interest rate of the loan?
Again, we use the simple interest rate formula.
This time we use the values $I=158776.17,$ $P=236450,$ $r=\mbox{Unknown},$ and $t=4.25.$
Then $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 158776.17=236450\cdot r \cdot 4.25&\mbox{ $r$ is unknown}\\ \implies &\displaystyle 158776.17=236450 \cdot 4.25\cdot r&\mbox{ }\\ \implies &\displaystyle 158776.17=1004912.5r&\mbox{ }\\ \implies &\displaystyle \frac{158776.17}{\color{magenta}{1004912.5}}=\frac{1004912.5r}{\color{magenta}{1004912.5}}&\mbox{ }\\ \implies &\displaystyle 0.157999995=r&\mbox{ }\\ \end{array} $$ So, the interest rate of the loan was about $15.8\%.$
This time we use the values $I=158776.17,$ $P=236450,$ $r=\mbox{Unknown},$ and $t=4.25.$
Then $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 158776.17=236450\cdot r \cdot 4.25&\mbox{ $r$ is unknown}\\ \implies &\displaystyle 158776.17=236450 \cdot 4.25\cdot r&\mbox{ }\\ \implies &\displaystyle 158776.17=1004912.5r&\mbox{ }\\ \implies &\displaystyle \frac{158776.17}{\color{magenta}{1004912.5}}=\frac{1004912.5r}{\color{magenta}{1004912.5}}&\mbox{ }\\ \implies &\displaystyle 0.157999995=r&\mbox{ }\\ \end{array} $$ So, the interest rate of the loan was about $15.8\%.$
Simple Interest: Calculating the Term $t$
Example
Mr. Holt purchased a new dining-room set and has taken out a loan of $\$2005$ at an interest rate of $3.7\%.$
If at the end of the loan period Mr. Holt paid $\$649.12$ in interest, what was the duration, or term, of the loan?
Again, we use the simple interest rate formula.
This time we use the values $I=649.12,$ $P=2005,$ $r=0.037,$ and $t=\mbox{Unknown}.$
Then $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 649.12=(2005)(0.037)\cdot t&\mbox{$t$ is unknown}\\ \implies &\displaystyle 649.12=74.185t&\mbox{ }\\ \implies &\displaystyle \frac{649.12}{\color{magenta}{74.185}}=\frac{74.185t}{\color{magenta}{74.185}}&\mbox{ }\\ \implies &\displaystyle 8.75001685=t&\mbox{ }\\ \end{array} $$ So, the term of the loan was about $8.75$ years.
This time we use the values $I=649.12,$ $P=2005,$ $r=0.037,$ and $t=\mbox{Unknown}.$
Then $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 649.12=(2005)(0.037)\cdot t&\mbox{$t$ is unknown}\\ \implies &\displaystyle 649.12=74.185t&\mbox{ }\\ \implies &\displaystyle \frac{649.12}{\color{magenta}{74.185}}=\frac{74.185t}{\color{magenta}{74.185}}&\mbox{ }\\ \implies &\displaystyle 8.75001685=t&\mbox{ }\\ \end{array} $$ So, the term of the loan was about $8.75$ years.
Short-Term Simple-Interest Loans
When the term of a loan is measured in days rather than years, the loan can be considered a short-term loan.
In this case we determine the number of years by dividing the number of days of the loan's term by $365.$
That is $$t=\frac{\mbox{# of days of the loan}}{{365}}.$$
Example
Sleazy P. has taken out an emergency $90$-day business loan of $\$5430$ at an interest rate of $5.8\%.$ How much interest will Sleazy P. be paying back?
Here,
$$
t=\frac{\mbox{# of days of the loan}}{{365}}=\frac{90}{365}
$$
So,
$$
\begin{array}{lll}
\displaystyle I&\displaystyle=Prt &\mbox{}\\
\displaystyle &\displaystyle=(5430)(0.058) \cdot \frac{90}{365} &\mbox{}\\
\displaystyle &\displaystyle=77.65643836 &\mbox{}\\
\end{array}
$$
That is, Sleazy P. will be paying $\$77.66$ in interest.
Short-Term Simple-Interest Loans & The Banker's Rule
Some lenders determine the term $t$ of a short-term loan by using what is called The Banker's Rule. $$t=\frac{\mbox{# of days of the loan}}{{360}}.$$ Before computers were a thing, this rule actually made sense. Nowadays it's a way for unscrupulous lenders to make a little more money.
Example
Sleazy P. has taken out an emergency $90$-day business loan of $\$5430$ at an interest rate of $5.8\%.$
If the lender uses the Banker's Rule to compute interest, how much more interest will Sleazy P. be paying back than if the lender computed interest using a $365$-day year?
Using the Banker's Rule,
$$
t=\frac{\mbox{# of days of the loan}}{{360}}=\frac{90}{360}=0.25
$$
So,
$$
\begin{array}{lll}
\displaystyle I&\displaystyle=Prt &\mbox{}\\
\displaystyle &\displaystyle=(5430)(0.058)(0.25) &\mbox{}\\
\displaystyle &\displaystyle=78.735 &\mbox{}\\
\end{array}
$$
That is, Sleazy P. will be paying $\$78.74$ in interest.
This lender made an extra $\$1.08$ off of Sleazy P. by using the Banker's Rule.
This lender made an extra $\$1.08$ off of Sleazy P. by using the Banker's Rule.
Future Value of a Simple-Interest Loan
The future value of a loan is the amount the lender receives when the term of the loan is up.
That is, the lender gets back the principal amount loaned plus the interest. This can be summed up by the following formula: $$A=P+I$$ $A$ is the future value, that is the total amount paid back
$P$ is principal of the loan
$I$ is interest made on the loan
Future Value of a Simple-Interest Loan
An alternative formula for the future value of a loan is the following $$A=P(1+rt)$$ $A$ is the future value, that is the total amount paid back
$P$ is the principal amount of the loan. This is the amount borrowed.
$r$ is the interest rate of the loan, the percentage of the principal to be paid back over time.
$t$ is the term of the loan, that is, how long the money is borrowed.
Example: Future Value of a Simple-Interest Loan
Sleazy P. Martini needs emergency funds and has taken out loan from a dangerous loan shark named Mr. Holt.
The amount of the loan is $\$2815$ at an interest rate of $8.5\%,$ and Sleazy P. has $90$ days to pay the loan back... or else. What is the total amount of money (future value) Sleazy P. will have to pay Mr. Holt back when the loan is due?
The amount Sleazy P. will be paying back is the future value, $A,$ of the loan.
Then, $$ \begin{array}{lll} \displaystyle A&\displaystyle=P(1+rt) &\mbox{}\\ \displaystyle &\displaystyle=2815\left(1+0.085\cdot\frac{90}{365}\right) &\mbox{}\\ \displaystyle &\displaystyle=2873.999315 &\mbox{}\\ \end{array} $$ That is, Sleazy P. will pay $\$2874$ when the loan is due.
Then, $$ \begin{array}{lll} \displaystyle A&\displaystyle=P(1+rt) &\mbox{}\\ \displaystyle &\displaystyle=2815\left(1+0.085\cdot\frac{90}{365}\right) &\mbox{}\\ \displaystyle &\displaystyle=2873.999315 &\mbox{}\\ \end{array} $$ That is, Sleazy P. will pay $\$2874$ when the loan is due.
Buying in Installments
When we make moderately substantial purchases, a car for example, we don't pay all the money back at once. We buy in instalments. That is, the future amount of the loan is divided up into smaller payments to be made over the term of the loan. Usually these are monthly payments.
Example: Mr. Holt purchased a new set of furniture and has taken out a $8.25$-year loan of $\$3445$ at an interest rate of $4.6\%.$ If Mr. Holt repays the loan in equal monthly instalments over the course of $8.25$ years, what is the monthly payment?
We break up the future value of the loan into monthly instalments over $8.25$ years, or
$12\cdot 8.25=99$ months.
The future value is $$ \begin{array}{lll} \displaystyle A&\displaystyle=P(1+rt) &\mbox{}\\ \displaystyle &\displaystyle=3445\left(1+0.046\cdot 8.25\right) &\mbox{}\\ \displaystyle &\displaystyle=4752.3775 &\mbox{}\\ \end{array} $$ So, the future value of the loan is $4752.38.$
Therefore, each monthly instalment is about $$ \frac{4752.38}{99}=48.00383838 $$ or, $\$48$ per month.
The future value is $$ \begin{array}{lll} \displaystyle A&\displaystyle=P(1+rt) &\mbox{}\\ \displaystyle &\displaystyle=3445\left(1+0.046\cdot 8.25\right) &\mbox{}\\ \displaystyle &\displaystyle=4752.3775 &\mbox{}\\ \end{array} $$ So, the future value of the loan is $4752.38.$
Therefore, each monthly instalment is about $$ \frac{4752.38}{99}=48.00383838 $$ or, $\$48$ per month.
Bonus Example
Mr. Holt would like to buy a new computer and is offered financing with $36$ eaaaassy payments of $\$60.4.$ If the out-the-door price of the computer is $\$1783.63,$ what is the interest rate of the loan?
Since Mr. Holt will be making $36$ payments of $\$60.40,$ the future value of the loan is
$$
36\cdot 60.40=2174.4
$$
That is, $A=2174.40.$
The out-the-door price is $P=\$1783.63,$ which is the principle amount of the loan.
Since the loan takes place over $36$ months, the term of the loan is $$ \frac{36}{12}=3 $$ That is, $t=3$ years.
We now have everything we need to recover the interest rate. $$ \begin{array}{lll} &\displaystyle A=P(1+rt)&\mbox{}\\ \implies &\displaystyle 2174.40 = 1783.63(1+r\cdot 3) &\mbox{$r$ is unknown}\\ \implies &\displaystyle 2174.40 = 1783.63(1+3r) &\mbox{}\\ \implies &\displaystyle 2174.40 = 1783.63+1783.63\cdot 3r &\mbox{distribute}\\ \implies &\displaystyle 2174.40 = 1783.63+5350.89r &\mbox{simplify}\\ \implies &\displaystyle 2174.40 \color{magenta}{-1783.63}= 1783.63+5350.89r\color{magenta}{-1783.63} &\mbox{}\\ \implies &\displaystyle 390.77=5350.89r &\mbox{}\\ \implies &\displaystyle \frac{390.77}{\color{magenta}{5350.89}}=\frac{5350.89r}{\color{magenta}{5350.89}} &\mbox{}\\ \implies &\displaystyle 0.07302897275=r &\mbox{}\\ \end{array} $$ The interest rate of the loan is $7.3\%.$
The out-the-door price is $P=\$1783.63,$ which is the principle amount of the loan.
Since the loan takes place over $36$ months, the term of the loan is $$ \frac{36}{12}=3 $$ That is, $t=3$ years.
We now have everything we need to recover the interest rate. $$ \begin{array}{lll} &\displaystyle A=P(1+rt)&\mbox{}\\ \implies &\displaystyle 2174.40 = 1783.63(1+r\cdot 3) &\mbox{$r$ is unknown}\\ \implies &\displaystyle 2174.40 = 1783.63(1+3r) &\mbox{}\\ \implies &\displaystyle 2174.40 = 1783.63+1783.63\cdot 3r &\mbox{distribute}\\ \implies &\displaystyle 2174.40 = 1783.63+5350.89r &\mbox{simplify}\\ \implies &\displaystyle 2174.40 \color{magenta}{-1783.63}= 1783.63+5350.89r\color{magenta}{-1783.63} &\mbox{}\\ \implies &\displaystyle 390.77=5350.89r &\mbox{}\\ \implies &\displaystyle \frac{390.77}{\color{magenta}{5350.89}}=\frac{5350.89r}{\color{magenta}{5350.89}} &\mbox{}\\ \implies &\displaystyle 0.07302897275=r &\mbox{}\\ \end{array} $$ The interest rate of the loan is $7.3\%.$
Discounted Loans
Discounted loans are another brilliant banking scheme that is not-so-brilliant if you're the borrower.
When a borrower takes out a discounted loan, they repay the interest upfront by subtracting it from the principal.
This sounds GREAT! until you think about it: the borrower gets to use less money while paying the same amount of interest.
As we shall see in the next example, discounted loans are anything but a discount! (Bankers like them though.)
Discounted Loans
Sleazy P. Martini has taken out a discounted $2$-year loan of $\$41905$ at an interest rate of $9.7\%.$
(a) What is the discount? (This is the interest that is subtracted from the principal.)
(b) How much money did Sleazy P. actually receive?
(c) What is the true interest rate of the loan? (The answer in part (b) is the new principal.)
(a) The interest to be repaid up front is
$$
\begin{array}{lll}
\displaystyle I&\displaystyle=Prt &\mbox{}\\
\displaystyle &\displaystyle=(41905)(0.097)(2) &\mbox{}\\
\displaystyle &\displaystyle=8129.57 &\mbox{}\\
\end{array}
$$
Thus, the discount is $\$8129.57.$
(b) The amount that Sleazy P. actually receives is $$ \mbox{Principal}-\mbox{Interest}=41905-8129.57=33775.43 $$ (c) The new principal amount is $P=33775.43.$
From the new principal, we get a new interest rate $r.$ $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 8129.57=33775.43\cdot r \cdot 2&\mbox{the new interest rate $r$ is unknown}\\ \implies &\displaystyle 8129.57=33775.43\cdot 2\cdot r &\mbox{}\\ \implies &\displaystyle 8129.57=67550.86r &\mbox{}\\ \implies &\displaystyle \frac{8129.57}{\color{magenta}{67550.86}}=\frac{67550.86r}{\color{magenta}{67550.86}} &\mbox{}\\ \implies &\displaystyle 0.1203473945=r &\mbox{}\\ \end{array} $$ That is, the new interest rate is about $12\%.$
Again, the reason the interest rate is higher is that Sleazy P. actually borrowed less while paying the same amount of interest.
(b) The amount that Sleazy P. actually receives is $$ \mbox{Principal}-\mbox{Interest}=41905-8129.57=33775.43 $$ (c) The new principal amount is $P=33775.43.$
From the new principal, we get a new interest rate $r.$ $$ \begin{array}{lll} &\displaystyle I=Prt&\mbox{}\\ \implies &\displaystyle 8129.57=33775.43\cdot r \cdot 2&\mbox{the new interest rate $r$ is unknown}\\ \implies &\displaystyle 8129.57=33775.43\cdot 2\cdot r &\mbox{}\\ \implies &\displaystyle 8129.57=67550.86r &\mbox{}\\ \implies &\displaystyle \frac{8129.57}{\color{magenta}{67550.86}}=\frac{67550.86r}{\color{magenta}{67550.86}} &\mbox{}\\ \implies &\displaystyle 0.1203473945=r &\mbox{}\\ \end{array} $$ That is, the new interest rate is about $12\%.$
Again, the reason the interest rate is higher is that Sleazy P. actually borrowed less while paying the same amount of interest.