6.3: Understanding Loans

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).

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.

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.

Exercises

1. 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?
   
   Answer (click to Show/Hide)

   This problem requires us to calculate the total amount paid over the life of the loan and then find the difference between that amount and the initial purchase price to determine the finance charge.

   Step 1: Calculate the total cost of the washer and dryer.
   The total cost is the sum of all the monthly payments. We can find this by multiplying the monthly payment amount by the number of months.

   Monthly Payment: $137.27

   Number of Months: 12

   $$\text{Total Cost}=137.27\times 12=1647.24$$

   Step 2: Calculate the cost of the loan (finance charge).
   The cost of the loan is the difference between the total amount paid and the original purchase price.

   Total Cost: $1647.24

   Purchase Price: $1357.98

   $$\text{Cost of Loan}=1647.24-1357.98=289.26$$

   Answer: The total cost for the washer and dryer is $1,647.24, and the cost of the loan was $289.26.

2. 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?
   
   Answer (click to Show/Hide)

   This problem requires us to calculate the total amount paid for the furnace and AC unit and then determine the finance charge (the cost of the loan) by comparing it to the original price.

   Step 1: Calculate the total cost of the furnace and AC unit.
   The total cost is the sum of all the monthly payments. We find this by multiplying the monthly payment by the number of months.

   Monthly Payment: $496.60

   Number of Months: 18

   $$\text{Total Cost}=496.60\times 18=8938.80$$

   Step 2: Calculate the cost of the loan (finance charge).
   The cost of the loan is the difference between the total amount paid and the original purchase price.

   Total Cost: $8938.80

   Purchase Price: $7981.00

   $$\text{Cost of Loan}=8938.80-7981.00=957.80$$

   Answer: The total cost for the furnace and AC unit is $8,938.80, and the cost of the loan was $957.80.

3. 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.
   
   Answer (click to Show/Hide)

   This problem requires us to find the total amount paid over the life of the loan and then determine the cost of the loan (the total interest paid).

   Step 1: Calculate the total amount paid for the loan.
   First, we need to determine the total number of payments.

   Number of Payments = 20 years × 12 months/year = 240 payments.
   Next, we multiply the total number of payments by the monthly payment amount.

   Monthly Payment: $337

   Number of Payments: 240

   $$\text{Total Paid}=337\times 240=80880$$

   Step 2: Calculate the cost of the loan (finance charge).
   The cost of the loan is the difference between the total amount paid and the original loan amount.

   Total Paid: $80,880

   Loan Amount: $60,000

   $$\text{Cost of Loan}=80880-60000=20880$$

   Answer: The student paid a total of $80,880 for the loan, and the cost of the loan was $20,880.

4. 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.
   
   Answer (click to Show/Hide)

   To compare the cost of the loans, we need to calculate the total amount paid and the finance charge (cost of the loan) for both scenarios.

   Part 1: 1980s Loan (18% Rate)

   Step 1: Calculate the total amount paid.
   The loan term is 30 years, so we first find the total number of payments.

   Total Payments = 30 years × 12 months/year = 360 payments.
   Now, multiply the number of payments by the monthly payment amount.

   Monthly Payment: $1,507

   $$\text{Total Paid}=1507\times 360=542520$$

   Step 2: Calculate the cost of the loan.
   The cost is the difference between the total paid and the original loan amount.

   Loan Amount: $100,000

   $$\text{Cost of Loan}=542520-100000=442520$$
   The cost of the loan in the 1980s was $442,520.

   Part 2: 2021 Loan (3% Rate)

   Step 1: Calculate the total amount paid.
   The loan term is also 30 years, so there are 360 payments.

   Monthly Payment: $422

   $$\text{Total Paid}=422\times 360=151920$$

   Step 2: Calculate the cost of the loan.

   Loan Amount: $100,000

   $$\text{Cost of Loan}=151920-100000=51920$$
   The cost of the loan in 2021 was $51,920.

   Part 3: Comparison

   To compare the costs, we find the difference between the two loan costs.
   $$\text{Difference}=442520-51920=390600$$

   Answer: The cost of the loan in the 1980s was $442,520, while the cost of the loan in 2021 was $51,920. The loan from the 1980s cost $390,600 more in interest over its 30-year term.

5. 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?
   
   Answer (click to Show/Hide)

   This problem requires us to first find the total purchase price including tax, then calculate the total amount paid through the financing plan to find the cost of the loan.

   Step 1: Calculate the total purchase price (the amount financed).
   First, we find the sales tax amount and add it to the cost of the TV.

   Cost of TV: $699.99

   Sales Tax Rate: 8.7% or 0.087

   Sales Tax = $699.99\times 0.087\approx 60.90$

   Total Purchase Price = $699.99+60.90=760.89$
   The amount financed is $760.89.

   Step 2: Calculate the total amount paid.
   We multiply the monthly payment amount by the total number of payments.

   Monthly Payment: $64.27

   Number of Payments: 12

   $$\text{Total Paid}=64.27\times 12=771.24$$
   The total cost of the TV purchase is $771.24.

   Step 3: Calculate the cost of the financing.
   The cost of financing is the difference between the total amount paid and the purchase price.

   Total Paid: $771.24

   Purchase Price: $760.89

   $$\text{Cost of Financing}=771.24-760.89=10.35$$

   Answer: The cost of the financing was $10.35, and the total cost of the TV purchase was $771.24.

6. 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).
   
   Answer (click to Show/Hide)

   This problem requires us to calculate the down payment, the total amount paid over the life of the loan, and then use those values to find the total cost of the home and the total interest paid.

   Step 1: Calculate the down payment and the loan amount.
   The down payment is 20% of the purchase price.

   Purchase Price: $195,000

   Down Payment = $195000\times 0.20=39000$
   The loan amount is the purchase price minus the down payment.

   Loan Amount = $195000-39000=156000$

   Step 2: Calculate the total amount paid for the loan.
   First, we find the total number of payments over the 30-year term.

   Total Payments = 30 years × 12 months/year = 360 payments.
   Next, we multiply the total number of payments by the monthly payment amount.

   Monthly Payment: $876

   $$\text{Total Paid on Loan}=876\times 360=315360$$

   Step 3: Calculate the total cost for the home.
   The total cost is the sum of the down payment and the total payments made on the loan.

   Down Payment: $39,000

   Total Paid on Loan: $315,360

   $$\text{Total Cost for Home}=39000+315360=354360$$

   Step 4: Calculate the cost of the loan (interest).
   The cost of the loan is the difference between the total paid on the loan and the original loan amount.

   Total Paid on Loan: $315,360

   Loan Amount: $156,000

   $$\text{Cost of Loan}=315360-156000=159360$$

   Answer: The total cost for the home was $354,360, and the cost of the loan (total interest paid) was $159,360.

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.
• Portions of the exercise solutions in this answer key were generated with the assistance of Gemini, a large language model from Google.