Example 1

You purchase a solar panel system with installation for $20,150 plus sales tax at the contractor rate in
Tucson of 5.66%. The solar company offers an installment loan that has payments of $376.14 per month for
the next five years. What is the cost of the loan if all the payments are made over the entire five years?

Solution

The cost of the loan is the difference between the total of all payments and principal of the loan. To find
the total payments take the monthly payments of $376.14 times 12 months to get an annual amount paid and
then multiply by 5 to get the total over all five years.

$$\text{Total Payments }=376.14\cdot \left(12\right)\cdot \left(5\right)=\$22,568.40$$

The principal for the loan would be the price for the solar panal with installation plus the sales tax
charged.

$$\text{Principal }=\$20,150+\$20,150\left(0.0566\right)=\$21,290.49$$

Take the difference to find the total cost of the loan:

$$\text{Cost of the Loan }=\$22,568.40-\$21,290.49=\$1,277.91$$

The loan cost a total of $1,277.91 and can be referred to as the interest paid on the loan as well.

The table below shows the total cost of the loan for both the 15 year and 30 year term.

We can see the 30 year term loan was a little more than double in how much the loan costs.

Example 3

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest.  How much interest will
you earn?

Solution

$\begin{array}{ll}P=\$\text{3}00\hfill & \text{the principal}\hfill \\ r=0.0\text{3} \hfill & \text{3}\%\text{ rate}\hfill \\ I=\$\text{3}00\left(0.0\text{3}\right)=\$\text{9}\text{.}\hfill & \text{You will earn }\$\text{9 interest}.\hfill \end{array}$

Video Solution(2 mins 23 secs – CC)

Example 4

Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a
$1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?

Solution

Each year, you would earn 5% interest:  $1000(0.05) = $50 in interest.  So over the course of five
years, you would earn a total of $250 in interest.  When the bond matures, you would receive back the
$1,000 you originally paid, leaving you with a total of $1,250.

Video Solution (2 mins 23 secs – CC)

Example 5

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses.  Suppose you
obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years.  How much
interest will you earn?

Solution

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2%
payments.

$\begin{array}{ll}P=\$1000\hfill & \text{the principal}\hfill \\ r=0.02 \hfill & \text{2}\%\text{ rate per half}-\text{year}\hfill \\ t=8 \hfill & \text{4 years}=\text{ 8 half}-\text{years}\hfill \\ I=\$1000\left(0.02\right)\left(8\right)=\$160. \hfill & \hfill \end{array}$

Video Solution (2 mins 2 secs – CC)

Example 6

A freshman college student needs to borrow money to pay for furniture in a new apartment. They agree to pay the bank $1,200 in 2 years at 6.0% simple interest. Find the Maturity Value, Discount, and the Proceeds for this loan.

Solution

The maturity value is the amount of money that is agree to be repaid to the lender. In this case that would be the $1,200 they agreed to pay back after two years.

The discount is found by finding the amount of simple interest for 2 years on a $1,200 loan at a rate of 6.0%

$D=1200\left(.06\right)\left(2\right)=144$

The discount was found to be $144. This means the proceeds for the loan (the amount given to the student) will be $1,056.

Example 7

Francisco borrows $5000 to pay off some bills and catch up on car payments. The bank offers him a discount
loan with an advertised annual rate of 7.5% (simple interest) for a period of 2 years. Find the amount
Francisco receives (proceeds) and then use that amount to determine a loans simple interest rate that ends up giving a total cost of the loan to be $5000.

Solution

The discount is found by finding the amount of simple interest for 2 years on a $5000 loan at a rate of 7.5%

$D=5000\left(.075\right)\left(2\right)=750$

The discount was found to be $750. This means Francisco will receive after going through the loan process
$4250 to use. He will need to pay back the $5000 by the end of two years (so the bank is making $750 on the
loan).

If we now reverse the process we can find what an equivalent interest rate would be for a initial principal
of $4250 to grow to a value of $5000 over two years.

$\begin{array}{l}\mathrm{FV}=P\left(1+rt\right)\\ 5000=4250\left(1+r\cdot 2\right)\\ \frac{5000}{4250}=1+2r\\ \frac{5000}{4250}-1=2r\\ \frac{1}{2}\left(\frac{5000}{4250}-1\right)=r\\ .0882=r\end{array}$

What we find is that the interest rate that would be charged in a simple interest rate structure is
approximately 8.82%. This is a little over 1% above the advertised rate for a discount loan, but represent the
equivalent rate amount if comparing to a simple interest rate loan instead.

Video Example (8 min 21 secs – CC)

Example 8

Samantha purchases new furniture for her first apartment. The furniture costs $1350 (including the tax and
delivery). If the store is offering an annual rate of 8.95% to finance the purchase with a simple interest
add-on loan with a term of 2 years, then what would her payments be?

Solution

We first calculate the Interest owed where the principal is $1350, rate of 8.95%, and time of 2 years

$$\begin{array}{l}I=P\cdot r\cdot t\\ I=\left(\$1350\right)\cdot \left(0.0895\right)\cdot \left(2\right)\\ I=\$241.65\end{array}$$

Add that interest to the principal to find the future value or total amount owed

$$\begin{array}{l}FV=P+I\\ FV=\$1350+\$241.65\\ FV=\$1591.6\end{array}$$

Next divide that total amount owed by 24 since there are 24 months in 2 years.

$$\begin{array}{l}\text{pymts}=\frac{FV}{n}\\ \text{pymts}=\frac{\$1591.65}{24}\\ \text{pymts}=\$66.32\end{array}$$

The monthly payments will be $66.32 for the purchase. Now the very last payments may not be exactly that
total as the payment amount was rounded.