Math in Our World
Section 8.4
Installment Buying
Learning Objectives
 Find amount financed, total installment price, and
finance charge for a fixed installment loan.
 Use a table to find APR for a loan.
 Compute unearned interest and payoff amount
for a loan paid off early.
 Compute credit card finance charges using the
unpaid balance method.
 Compute credit card finance charges using the
average daily balance method.
Installment Buying
Installment buying is when an item is
purchased and the buyer pays for it by
making periodic partial payments, or
installments.
Fixed Installment Loans
A fixed installment loan is a loan that is
repaid in equal payments.
Sometimes the buyer will pay part of the cost
at the time of purchase. This is known as a
down payment.
Fixed Installment Loans
The amount financed is the amount a borrower will
pay interest on.
Amount financed = Price of item – Down payment
The total installment price is the total amount of
money the buyer will ultimately pay.
Total installment price = Sum of all payments + Down pmt
The finance charge is the interest charged for
borrowing the amount financed.
Finance charge = Total installment price – Price of item
EXAMPLE 1
Calculating Information About
a Car Loan
Cat bought a 2-year old Santa Fe for $12,260. Her
down payment was $3,000, and she will have to
pay $231.50 for 48 months. Find the amount
financed, the total installment price, and the
finance charge.
SOLUTION
Using the formulas previously shown:
Amount financed = Cash price – Down payment
= $12,260 – $3,000
= $9,260
EXAMPLE 1
Calculating Information About
a Car Loan
SOLUTION
Since she paid $231.50 for 48 months and her down
payment was $3,000,
Total installment price = Total of monthly payments + Down pmt
= (48 x $231.50) + $3,000
= $14,112.00
Now we can find the finance charge:
Finance charge = Total installment price – Cash price
= $14,112.00 – $12,260.00
= $1,852.00
The amount financed was $9,260.00; the total installment
price was $14,112.00, and the finance charge was $1,852.00.
EXAMPLE 2
Computing a Monthly Payment
After a big promotion, a young couple bought
$9,000 worth of furniture. The down payment was
$1,000. The balance was financed for 3 years at
8% simple interest per year.
(a) Find the amount financed.
(b) Find the finance charge (interest).
(c) Find the total installment price.
(d) Find the monthly payment.
EXAMPLE 2
Computing a Monthly Payment
SOLUTION
(a) Amount financed = Price of item – Down payment
= $9,000 – $1,000 = $8,000
(b) To find the finance charge, we use the simple
interest formula: I = Prt
= $8,000 x 0.08 x 3
= $1,920
(c) In this case, the total installment price is simply the
cost of the furniture plus the finance charge:
Total installment price = $9,000 + $1,920 = $10,920
EXAMPLE 2
Computing a Monthly Payment
SOLUTION
(d) To calculate the monthly payment, divide the
amount financed plus the finance charge
($8,000 + $1,920) by the number of payments:
Monthly payment = $9,920 ÷ 36 = $275.56
In summary, the amount financed is $8,000, the
finance charge is $1,920, the total installment price is
$10,920, and the monthly payment is $275.56.
Annual Percentage Rate (APR)
Lenders are required by law to disclose an annual
percentage rate, or APR, that reflects the true
interest charged. This allows consumers to compare
loans with different terms.
This is a
partial APR
table. See
Text for a
more
complete
table.
Using the APR Table
Step 1 Find the finance charge per $100 borrowed
using the formula
Finance Charge
 $100
Amount Financed
Step 2 Find the row in the table marked with the
number of payments and move to the right until
you find the amount closest to the number from
Step 1.
Step 3 The APR (to the nearest half percent) is at
the top of the corresponding column.
EXAMPLE 3
Finding APR
Burk Carter purchased a color laser printer for
$600.00. He made a down payment of $50.00 and
financed the rest for 2 years with a monthly
payment of $24.75. Find the APR.
SOLUTION
Find the finance charge per $100.00. The total amount he will
pay is $24.75 per month x 24 payments, or $594.00. Since he
financed $550.00, the finance charge is $594.00 – $550.00 = $44.
Finance Charge
Finance charge per $100 =
 $100
Amount Financed
$44
=
 $100  $8.00
$550
EXAMPLE 3
Finding APR
SOLUTION
Step 2 Find the row for 24
payments and move across
the row until you find the
number closest to $8.00. In
this case, it is exactly $8.00.
Step 3 Move to the top of the
column to get the APR.
It is 7.5%.
Unearned Interest
One way to save money on a fixed installment
loan is to pay it off early. This will allow a buyer
to avoid paying the entire finance charge.
The amount of the finance charge that is saved
when a loan is paid off early is called unearned
interest.
There are two methods for calculating unearned
interest, the actuarial method and the rule of 78.
Actuarial Method
kRh
u
100  h
where u = unearned interest
k = number of payments remaining,
excluding the current one
R = monthly payment
h = finance charge per $100 for a loan
with the same APR and k monthly
payments
EXAMPLE 4
Using the Actuarial Method
Our friend Burk from Example 3 decides to use part
of his tax refund to pay off the full amount of his
laser printer with his 12th payment. Find the
unearned interest and the payoff amount.
SOLUTION
To use the formula for the actuarial method, we’ll need
values for k, R, and h.
Half of the original 24 payments will remain, so k = 12.
From Example 3, the monthly payment is $24.75 and the
APR is 7.5%.
EXAMPLE 4
Using the Actuarial Method
SOLUTION
Using the APR Table, we find
the row for 12 payments and
the column for 7.5%;
the intersection shows $4.11,
so h = $4.11.
Substituting k = 12, R = 24.75
and h = 4.11:
EXAMPLE 4
Using the Actuarial Method
SOLUTION
The unearned interest is $11.72.
The payoff amount is the amount remaining on the loan
minus unearned interest.
At this point, Burk has made 11 payments, so there would
be 13 remaining if he were not paying the loan off early.
Payoff amount = (13 x $24.75) – $11.72 = $310.03
With a payment of $310.03, Burk is the proud owner of a
laser printer.
The Rule of 78
fk (k  1)
u
n(n  1)
where u = unearned interest
f = finance charge
k = number of remaining monthly
payments
n = original number of payments
EXAMPLE 5
Using the Rule of 78
A $5,000 car loan is to be paid off in 36 monthly
installments of $172. The borrower decides to pay
off the loan after 24 payments have been made.
Find the amount of interest saved by paying the
loan off early. Use the rule of 78.
SOLUTION
Find the finance charge (i.e. total interest).
$172 x 36 = $6,192 ($172 x 36 payments)
$6,192 – $5,000 = $1,192 (Total payments – Amount financed)
EXAMPLE 5
Comparing the Effective Rate
of Two Investments
SOLUTION
Substitute into the formula using f = $1,192, n = 36, and
k = 36 – 24 = 12.
By paying off the loan a year early, the borrower saved
$139.60.
Open-Ended Credit
Open-ended credit has no fixed number of
payments or payoff date. By far the most
common example of this is credit cards.
With the unpaid balance method, interest is
charged only on the balance from the
previous month.
EXAMPLE 6
Computing a Credit Card
Finance Charge
For the month of April, Elliot had an unpaid
balance of $356.75 at the beginning of the month
and made purchases of $436.50. A payment of
$200.00 was made during the month. The interest
on the unpaid balance is 1.8% per month. Find the
finance charge and the balance on May 1.
SOLUTION
Step 1 Find the finance charge on the unpaid balance
using the simple interest formula with rate 1.8%. (r = 0.018)
I = Prt
= $356.75 x 0.018 x 1 (1 month, so t = 1)
= $6.42 (rounded)
EXAMPLE 6
Computing a Credit Card
Finance Charge
SOLUTION
The finance charge is $6.42.
Step 2 To the unpaid balance, add the finance charge and
the purchases for the month; then subtract the payment to
get the new balance.
New balance = $356.75 + $6.42 + $436.50 – $200
= $599.67
The new balance as of May 1 is $599.67.
Average Daily Balance Method
When using the average daily balance
method, the balance for each day of the
month is used to compute an average daily
balance, and interest is computed on that
average.
Average Daily Balance Method
Procedure for the ADB Method
Step 1 Find the balance as of each transaction.
Step 2 Find the number of days for each balance.
Step 3 Multiply the balances by the number of days
and find the sum.
Step 4 Divide the sum by the number of days in the
month.
Step 5 Find the finance charge (multiply the average
daily balance by the monthly rate).
Step 6 Find the new balance (add the finance charge
to the balance as of the last transaction).
EXAMPLE 7
Computing a Credit Card
Finance Charge
Betty’s credit card statement showed the following
transactions during the month of August.
August 1 Previous balance $165.50
August 7 Purchases 59.95
August 12 Purchases 23.75
August 18 Payment 75.00
August 24 Purchases 107.43
Find the average daily balance, the finance charge
for the month, and the new balance on September 1.
The interest rate is 1.5% per month on the average
daily balance.
EXAMPLE 7
Computing a Credit Card
Finance Charge
SOLUTION
Step 1 Find the balance as of each transaction.
August 1
$165.50
August 7
$165.50 + $59.95 = $225.45
August 12
$225.45 + $23.75 = $249.20
August 18
$249.20 + $75.00 = $174.20
August 24
$174.20 + $107.43 = $281.63
Step 2 Find the number of days for each balance.
Date
Balance
Days
Calculations
August 1
$165.50
6
(7 – 1 = 6)
August 7
$225.45
5
(12 – 7 = 5)
August 12
$249.20
6
(18 – 12 = 6)
August 18
$174.20
6
(24 – 18 = 6)
August 24
$281.63
8
(31 – 24 + 1 = 8)
EXAMPLE 7
Computing a Credit Card
Finance Charge
SOLUTION
Step 3 Multiply each balance by the number of days, and
add these products.
Date
Balance Days
Calculations
August 1 $165.50 6
$165.50(6) = $993.00
August 7 $225.45 5
$225.45(5) = $1,127.25
August 12 $249.20 6
$249.20(6) = $1,495.20
August 18 $174.20 6
$174.20(6) = $1,045.20
August 24 $281.63 8
$281.63(8) = $2,253.04
31
$6,913.69
Step 4 Divide the total by the number of days in the month
to get the average daily balance.
Average daily balance = $6,913.69/31 ≈ $223.02
EXAMPLE 7
Computing a Credit Card
Finance Charge
SOLUTION
Step 5 Find the finance charge. Multiply the average daily
balance by the rate, which is 1.5%, or 0.015.
Finance charge = $223.02 x 0.015 ≈ $3.35.
Step 6 Find the new balance. Add the finance charge to the
balance as of the last transaction.
New balance: $281.63 + $3.35 = $284.98
The average daily balance is $223.02. The finance charge
is $3.35, and the new balance is $284.98.