LOANS

ACOS Objective Algebra 1:

Use algebraic techniques to make financial and economic decisions.

What You’ll LearnSection 8-1 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 8-3

Section 8-2

Section 8-4 Work out the payment to interest, payment to principal, and the new balance.

Compute the final payment when paying off a simple interest installment loan.

Use a table to find the annual percentage rate of a loan.

Section 8-6

Section 8-5

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.

When Will You Ever Use This?

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?

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 loan - A loan that has to be repaid with onepayment 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

Single-Payment Loans

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.

Key Words to Know

FORMULASTOKNOW

Interest = Principal x Rate x Time

Ordinary Interest = Principal × Rate × Time ÷ 360

Exact Interest = Principal × Rate × Time ÷ 365

Anita Sloane’s bank granted her a single-payment loan of $72,000 for 91 days at 12 percent ordinaryinterest.

What is the maturity value of the loan?

Example 1

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

Anita Sloane’s bank granted her a single-payment loan of $72,000 for 91 days at 12 percent exactinterest.

What is the maturity value of the loan?

Example 2

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

Single payment loan of $2,750.

Interest rate of 11 percent.

Exact day of interest: 50.

What is the interest owed?

Practice 1

$41.44

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?

Practice 2

Interest owed: $1,496.68

Maturity value of loan: $17,878.68

installment loan - A loan repaid in several equalpayments over a specified period of time.

down payment - The amount you must pay beforefinancing 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

Installment Loans

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?

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?

Example 3

Find the amount financed.

Cash Price – Down Payment

$1,399 – $199 = $1,200

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?

Example 4

Step 1: Find the 20 percent down payment.

Step 2: Find the amount financed.

Cash Price – Down Payment

$1,140 × 20% = $228

$1,140 – $228 = $912

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?

Practice 1

$575.45

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.

Practice 2

Down payment: $85

Amount financed: $340

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.

Finding Monthly Payments

Important Questions What formula do I use?

How do I find the monthly payment?

How do I calculate the total amount paid?

Total Amount Repaid = Number of Payments x Monthly Payments

How do I find the finance charge?

Finance Charge = Total Amount Repaid – Amount Financed

Loan $100 afor ment MonthlyPay100$

Loan ofAmount Payment Monthly Online Calculators

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

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 1

Find the total amount repaid.

Number of Payments × Monthly Payment

18 × $106.38 = $1,914.84

Example 5 Answer: Step 2

Find the finance charge.

Total Amount Repaid – Amount Financed

$1,914.84 – $1,800.00 = $114.84

Example 5 Answer: Step 3

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

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 1

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 2

Find the total amount repaid.

Number of Payments × Monthly Payment

12 × $111.89 = $1,342.68

Example 6 Answer: Step 3

Find the finance charge.

Total Amount Repaid – Amount Financed

$1,342.68 – $1,259.99 = $82.69

Example 6 Answer: Step 4

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.

Practice 1

a. What is the down payment?

b. What is the amount of the loan?

c. What are the monthly payments?

d. What is the finance charge?

$960

$3,840

$131.33

$887.88

repayment schedule - A schedule showing the distribution of interest and principal payments on a loan over the life of the loan.

Key Words to Know

Payment to Principal = Monthly Payment – Interest

Why is it important to get a copy of your repayment schedule?

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

Find the interest.

Principal × Rate × Time

$1,800.00 × 8% × 1/12 = $12.00

Example 7 Answer: Step 1

Find the payment to principal.

Monthly Payment – Interest

$307.08 – $12.00 = $295.08

Example 7 Answer: Step 2

Find the new principal.

Previous Principal – Payment to Principal

$1,800.00 – $295.08 = $1,504.92

Example 7 Answer: Step 3

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

Find the interest for the first payment.

Principal × Rate × Time

$6,000.00 × 8% × 1/12 = $40.00

Example 8 Answer: Step 1

Find the interest for the 21st payment.

Principal × Rate × Time

$2,849.08 × 8% × 1/12 = $18.00

Example 8 Answer: Step 2

The interest is much greater for the first payment than the 21st payment because the principal is much greater.

Example 8 Answer

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.

Practice 1

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?

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 1 Answer

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

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 (cont.)

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

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

Practice 2 Answer (cont.)

final payment - Payment on a simple interest loan that consists of the remaining balance plus the current month’s interest.

Key Word to Know

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?

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 9

Find the previous balance.

$913.70

Example 1 Answer: Step 1

Find the interest for the fourth month.

Principal × Rate × Time

$913.70 × 12% × 1/12 = $9.137 or $9.14

Example 1 Answer: Step 2

Find the final payment.

Previous Balance + Current Month’s Interest

$913.70 + $9.14 = $922.84

Example 1 Answer: Step 3

How much would the Collins in Example 1 save by paying off the loan early?

Example 2

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

Example 2 Answer

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.

Practice 1

$14.01

$1,134.73

$13.67

a. What is the interest?

b. What is the final payment?

c. How much is saved by paying

off the loan early?

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?

Practice 2

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