6.3 Understanding Loans
6.3: Understanding Loans
Learning Objectives
Upon completion of this section, you should be able to
- Determine total cost of a loan.
Basics of Loans
In chapter 6.1 we defined interest as the cost of using money over time and looked over scenarios involving simple interest including
When someone borrows money (gets a loan) they usually do not get it for free (as we mentioned in chapter 6.1). The typical way people pay for this borrowing is by paying a percentage of the amount borrowed plus some fees back to the lender. The amount borrowed is typically called the loan proceeds or principal amount. The amount over the proceeds that is paid back to the lender is the cost of the loan or finance charge (if we ignore any fees associated with the loan it is called the interest).
Loan Terminology
The principal is the amount to be borrowed. This is also called the loan proceeds.
The term for a loan is the total length of time before the loan must be paid back in full.
The total amount repaid (or total payments for short) is the sum of all payments made over the life of the loan.
The cost of the loan is the total amount repaid to the lender minus the principal. Sometimes simply referred to as the interest for the loan if there are no fees.
In our scenarios we will ignore any fees related to the cost of a loan, so in essence when we find the cost of the loan we also identified the interest paid on the loan.
Example 1
Madeline borrows $10,000 for a used car. The loan terms are $220 per month for 5 years. What is the cost of this loan?
Solution
The cost of the loan is the total amount of repaid to the lender minus the principal. Start with calculating the total amount repaid by taking the monthly payments multiplied by 12 to find the total paid over one year and then multiply by 5 as the loan has a term of five years:
We found that over five years Madeline paid $13,200 to the lender. To find the cost of the loan subtract off the principal from the total payments:
The cost of the loan is $3,200 (also referred to as the interest paid on the loan).
Example 2
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.
The principal for the loan would be the price for the solar panel with installation plus the sales tax charged.
Take the difference of the total payments and principal to find the total cost of the loan:
The loan costs (interest paid) a total of $1,277.91.
Example 3
Daryl finds a home and estimates that a loan of $190,000 will be needed to purchase the house. When talking to a loan officer at a bank two options were given for the loan: a 15 year loan with payments $1312.11 and a 30 year loan with payments $801.04. Compare the cost of the two loan options.
Solution
The table below shows the total cost of the loan for both the 15 year and 30 year term that both have a principal of $190,000.
| Term | Payments | Total Payments | Total Cost |
|---|---|---|---|
| 15 | $1312.11 | ||
| 30 | $801.04 |
We can see the 30 year term loan was a little more than double in how much the loan costs.
In many cases when buying a home the loan principal is reduced by a down payment made on the house. If a house was for sale at $200,000, but the bank required a 20% down payment you may be looking at a loan for $160,000 as the principal. In the Try it Now you will need to factor in the down payment, but we will ignore any other fees and costs that get added to a home loan.
Try it Now 1
A house was purchased at a price of $180,000, but the loan requires a down payment of 15%. The payments were calculated at $1242.04 per month for 15 years. Answer the following?
- What was the principal of the loan?
- What is the total cost of the home?
- What is the cost of the loan (the interest paid)?
Hint 1 (click to Show/Hide)
For part : To find the principal of the loan you need to remove the down payment from the purchase price of the home. Take 15% away from the purchase price of $180,000.
For part : The total cost of the home can be though of us as the purchase price plus the interest, but it can be found as the sum of all the payments plus the down payment as well.
For part : The total cost of the loan is the sum of all payments minus the principal.
Answer (click to Show/Hide)
- The principal for the loan is the initial purchase price minus the down payment:
The principal is found to be $153,000 after taking away the down payment of $27,000 from the home price.
- The total cost of the home will be the sum of all payments plus the down payment.
- The total cost of the loan is the sum of payments minus the original loan principal.
Notice in the work above we are not using the total cost of the home to find the interest (we are using just the sum of all payments). The total cost of the loan is $70,7474.20 (also called the interest).
Exercises
- A new washer and dryer is purchased for $1357.98 (including sales tax and delivery). Financing was used for the full purchase price and the payments are $137.27 per month for 12 months. What is the total cost for the washer and dryer and what was the cost of the loan?
- A new furnace and AC unit was purchased for a home. The cost for the unit (including sales tax and installation) was $7,981. If the payments $496.60 for 18 months, then what is the total cost of the loan and total cost for the furnace and AC unit purchase?
- A student took out a $60,000 loan to finish a graduate program. If the payments of the loan are $337 per month for 20 years, then what is the cost of the loan and what did the students pay in total for the loan.
- In the early 1980’s the average mortgage rates hit a peak at approximately 18%. For a $100,000 loan the monthly payment for a 30 year loan was approximately $1,507. In 2021 the mortgage rates were hitting new lows with 30 year loans for those with good credit around 3%. A payment for a $100,000 loan with that rate is approximately $422. Compare the cost of the loan from 1980s to the 2022 example.
- A TV is purchased and financed through an electronics store. The cost of the TV is $699.99 and there is a local sales tax of 8.7%. The entire purchase is financed and has a monthly payment of $64.27 for 12 months. What is the cost of the financing and what is the total cost of the TV purchase?
- A home is being purchased for $195,000. The bank requires a down payment of 20%. If the monthly payments for the home are $876, then what is the total cost for the home and how much of that was interest on the loan (the cost of the loan).
Attributions
This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.
