LOANS
ACOS Objective Algebra 1:
Use algebraic techniques to make
financial and economic decisions.
What You’ll Learn
Section 8-1
Section 8-2
Section 8-3
Compute the maturity value and interest
rate of a single-payment loan.
Calculate the amount financed on an
installment loan.
Figure out the monthly payment, total
amount repaid, and finance charge on an
installment loan.
Section 8-4
Work out the payment to interest, payment
to principal, and the new balance.
Section 8-5
Compute the final payment when paying off
a simple interest installment loan.
Section 8-6
Use a table to find the annual percentage
rate of a loan.
Why are loans important?
How would life be different if loans didn’t exist?
What types of things would be more difficult or
impossible for the average person to purchase?
When Will You Ever Use This?
Some day you may want to earn a college
degree, buy a car, or purchase a home.
Taking out a loan is a common way to borrow
the money now and repay it later.
Where can you apply for a loan?
Most lenders require you to go to a bank or
credit union. However, online lenders are
gaining popularity.
Single-Payment Loans
single-payment loan - A loan that has to be repaid with one
payment after a specified period of time.
promissory note - A written promise to pay a certain sum of
money on a certain date in the future.
maturity value - The total amount that must be repaid on a
loan, including the principal borrowed and
the interest owed.
Maturity Value = Principal + Interest Owed
Key Words to Know
term - The amount of time for which a loan is granted before it
has to be repaid.
ordinary interest - Interest on a loan calculated by basing the
time of the loan on a 360-day year.
exact interest - Interest on a loan calculated by basing the time
of the loan on a 365-day year.
FORMULAS TO KNOW
 Interest = Principal x Rate x Time
 Ordinary Interest = Principal × Rate × Time ÷ 360
 Exact Interest = Principal × Rate × Time ÷ 365
Example 1
Anita Sloane’s bank granted her a single-payment
loan of $72,000 for 91 days at 12 percent ordinary
interest.
What is the maturity value of the loan?
Step 1: Find the ordinary interest owed.
Principal × Rate × Time
$7,200.00 × 12% × 91/360 = $218.40
Step 2: Find the maturity value.
Principal + Interest Owed
$7,200.00 + $218.40 = $7,418.40
Example 2
Anita Sloane’s bank granted her a single-payment
loan of $72,000 for 91 days at 12 percent exact
interest.
What is the maturity value of the loan?
Step 1: Find the exact interest owed.
Principal × Rate × Time
$7,200 × 12% × 91/365 = $215.408 or $215.41
Step 2: Find the maturity value.
Principal + Interest Owed
$7,200.00 + $215.41 = $7,415.41
Practice 1
Single payment loan of $2,750.
Interest rate of 11 percent.
Exact day of interest: 50.
What is the interest owed?
$41.44
Practice 2
Emily Andrews borrowed $16,382. Her bank
granted her a single-payment loan for 286 days at
11.5 percent ordinary interest.
What is the interest owed?
What is the maturity value of her loan?
Interest owed: $1,496.68
Maturity value of loan: $17,878.68
Installment Loans
installment loan - A loan repaid in several equal
payments over a specified period of time.
down payment - The amount you must pay before
financing the rest on credit.
amount financed - The amount that is owed on an
item after making the down payment.
Important Questions
What formula do I use?
How do I calculate the amount
financed?
Amount Financed = Cash Price – Down Payment
How do I find the down
payment?
Down Payment = Amount x Percent
BEFORE THE BANK WILL GIVE YOU
LOAN, WHAT DO YOU HAVE TO DO
FIRST?
 Application
 Credit Check
 Pay Down Payment
BANKS USUALLY MAKE INSTALLMENT LOANS
RATHER THAN SINGLE PAYMENT LOANS. WHY DO
YOU THINK PEOPLE PREFER THIS TYPE OF LOAN
RATHER THAN A SINGLE PAYMENT LOAN?
Example 3
Tasheka Quintero is buying a new refrigerator for
$1,399. Quintero made a down payment of $199
and financed the remainder.
How much did Quintero finance?
Find the amount financed.
Cash Price – Down Payment
$1,399 – $199 = $1,200
Example 4
Rebecca Clay purchased a washer and a dryer for
$1,140. She used the store’s installment credit plan
to pay for the items. She made a down payment
and financed the remaining amount.
What amount did she finance if she made a 20
percent down payment?
Step 1: Find the 20 percent down payment.
$1,140 × 20% = $228
Step 2: Find the amount financed.
Cash Price – Down Payment
$1,140 – $228 = $912
Practice 1
Matt Yokohama purchased a fountain for his newly
landscaped backyard. The fountain cost $677. He
made a down payment of 15 percent.
What amount did he finance?
$575.45
Practice 2
Janelle Lewis purchased a new dishwasher for
$425. She made a 20 percent down payment and
financed the remaining amount.
Find the down payment and the amount financed.
Down payment: $85
Amount financed: $340
Finding Monthly Payments
simple interest installment loan - A loan repaid with
equal monthly payments.
annual percentage rate - An index showing the cost
of borrowing money on a yearly basis, expressed as
a percent.
Important Questions
How do I find the
monthly payment?
How do I calculate the
total amount paid?
How do I find the
finance charge?
What formula do I use?
Monthly Payment
Amount of Loan
Online
Calculators
MonthlyPay
ment for a $100 Loan
$100
Total Amount Repaid = Number of Payments x Monthly Payments
Finance Charge = Total Amount Repaid – Amount Financed
Example 5
Carla Hunt obtained an installment loan of
$1,800.00 to purchase some new furniture. The
annual percentage rate is 8 percent. She must
repay the loan in 18 months.
What is the finance charge?
Example 5 Answer: Step 1
Find the monthly payment.
Amount of Loan ÷ $100 × Monthly Payments for a $100 Loan
$1,800.00 ÷ $100.00 × $5.91 = $106.38
Example 5 Answer: Step 2
Find the total amount repaid.
Number of Payments × Monthly Payment
18 × $106.38 = $1,914.84
Example 5 Answer: Step 3
Find the finance charge.
Total Amount Repaid – Amount Financed
$1,914.84 – $1,800.00 = $114.84
Example 6
Tulio and Lupe Fernandez purchase a refrigerator
with an installment loan that has an APR of 12
percent. The refrigerator sells for $1,399.99. The
store financing requires a 10 percent down
payment and 12 monthly payments.
What is the finance charge?
Example 6 Answer: Step 1
Find the amount financed.
Selling Price – Down Payment
$1,399.99 – (0.10 × $1,399.99)
$1,399.99 – $140.00 = $1,259.99
Example 6 Answer: Step 2
Find the monthly payment. (Refer to the Monthly Payment
on a Simple Interest Installment Loan of $100 on page 799
in your textbook).
Amount of Loan ÷ $100 × Monthly Payment for a $100 Loan
$1,259.99 ÷ $100.00 × $8.88 = $111.887 or $111.89
Example 6 Answer: Step 3
Find the total amount repaid.
Number of Payments × Monthly Payment
12 × $111.89 = $1,342.68
Example 6 Answer: Step 4
Find the finance charge.
Total Amount Repaid – Amount Financed
$1,342.68 – $1,259.99 = $82.69
Practice 1
A new heating and air conditioner will cost the
Sangjun family $4,800. They make a down payment
of 20 percent and finance the remaining amount.
They obtain an installment loan for 36 months at
an APR of 14 percent.
a. What is the down payment?
$960
b. What is the amount of the loan?
$3,840
c. What are the monthly payments?
$131.33
d. What is the finance charge?
$887.88
Key Words to Know
repayment schedule - A schedule showing the
distribution of interest and principal payments on a
loan over the life of the loan.
Payment to Principal = Monthly Payment – Interest
Why is it important to get a copy of your
repayment schedule?
Example 7
The Coles obtained the loan of $1,800 at 8 percent for 6
months shown in Figure 8.1 on page 294. Show the
calculation for the first payment.
What is the interest?
What is the payment to principal?
What is the new principal?
Example 7 Answer: Step 1
Find the interest.
Principal × Rate × Time
$1,800.00 × 8% × 1/12 = $12.00
Example 7 Answer: Step 2
Find the payment to principal.
Monthly Payment – Interest
$307.08 – $12.00 = $295.08
Example 7 Answer: Step 3
Find the new principal.
Previous Principal – Payment to Principal
$1,800.00 – $295.08 = $1,504.92
Example 8
Carol Blanco obtained a loan of $6,000 at 8 percent for 36
months. The monthly payment is $187.80. The balance of
the loan after 20 payments is $2,849.08.
What is the interest for the first payment?
What is the interest for the 21st payment?
Why is the interest so different for the two payments?
Example 8 Answer: Step 1
Find the interest for the first payment.
Principal × Rate × Time
$6,000.00 × 8% × 1/12 = $40.00
Example 8 Answer: Step 2
Find the interest for the 21st payment.
Principal × Rate × Time
$2,849.08 × 8% × 1/12 = $18.00
Example 8 Answer
The interest is much greater for the first payment
than the 21st payment because the principal is
much greater.
Practice 1
Cathleen Brooks obtained an 18-month loan for
$3,200. The interest rate is 15 percent. Her
monthly payment is $199.68. The balance of the
loan after 6 payments is $2,341.45.
a. What is the interest for the first payment?
b. What is the interest after the seventh payment?
c. How much more goes toward the principal on
the seventh payment compared to the first
payment?
Practice 1 Answer
a. Interest for the first payment: $40.00
b. Interest after the seventh payment: $29.27
c. Amount more that goes toward the principal on
the seventh payment compared to the first
payment: $10.73
Practice 2
Sam Billings obtained a personal loan for $1,500 at
12 percent for 12 months. The monthly payments
on the loan are $133.20. Find the interest, payment
to principal, and balance for the first three
payments.
Practice 2 (cont.)
a. Interest on first payment?
b. Payment to principal?
c. New principal?
d. Interest after second payment?
e. Payment to principal?
Practice 2 (cont.)
f. New principal?
g. Interest on third payment?
h. Payment to principal?
i. New principal?
Practice 2 Answer
a. Interest on first payment: $15.00
b. Payment to principal: $118.20
c. New principal: $1,381.80
d. Interest after second payment: $13.82
e. Payment to principal: $119.38
Practice 2 Answer (cont.)
f. New principal: $1,262.42
g. Interest on third payment: $12.62
h. Payment to principal: $120.58
i. New principal: $1,141.84
Key Word to Know
final payment - Payment on a simple interest loan
that consists of the remaining balance plus the
current month’s interest.
Final Payment = Previous Balance + Current Month’s Interest
Interest Saved =
Total Payback – (Sum of Previous Payments + Final Payment)
Why might a bank not encourage you to pay
off a loan early?
Example 9
The first 3 months of the repayment schedule for
Doug and Donna Collins’s loan of $1,800 at 12
percent interest for 6 months is shown below.
What is the final payment if they pay the loan off
with the fourth payment?
Example 1 Answer: Step 1
Find the previous balance.
$913.70
Example 1 Answer: Step 2
Find the interest for the fourth month.
Principal × Rate × Time
$913.70 × 12% × 1/12 = $9.137 or $9.14
Example 1 Answer: Step 3
Find the final payment.
Previous Balance + Current Month’s Interest
$913.70 + $9.14 = $922.84
Example 2
How much would the Collins in Example 1 save by
paying off the loan early?
Example 2 Answer
Step: Find the interest saved.
Total Payback – (Sum of Previous Payments + Final Payment)
(6 × $310.50) – [(3 × $310.50) + $922.84] =
$1,863.00 – [$931.50 + $922.84] =
$1,863.00 – $1,854.34 = $8.66
Practice 1
Chanelle Thompson took out a simple interest loan
of $2,200 at 15 percent for 6 months. Her monthly
payment on the loan is $382.80. After 3 payments
the balance is $1,120.72. She pays off the loan with
the fourth payment.
a. What is the interest?
$14.01
b. What is the final payment?
$1,134.73
c. How much is saved by paying
$13.67
off the loan early?
Practice 2
Alicia Fleming took out a simple interest installment loan of
$8,300 at 8 percent for 18 months. The monthly payment is
$490.53. After 5 payments, the balance is $6,094.80.
If she pays off the loan when the next payment is due, what
is the final payment?
How much is saved by paying off the loan early?
Amount due if she pays off the loan with the next payment:
$6,135.43
Amount saved by paying off the loan early: $241.46