Download Section 8.1 Percent, Sales Tax, and Income Tax

Section 8.1
Percent, Sales Tax, and Income Tax
1.
2.
3.
4.
5.
6.
7.
Objectives
Express a fraction as a percent.
Express a decimal as a percent.
Express a percent as a decimal.
Solve applied problems involving sales tax and
discounts.
Compute income tax.
Determine percent increase or decrease.
Investigate some of the ways percent can be abused.
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Section 8.1
1
Basics of Percent
•
•
Percents are the result of expressing numbers as a
part of 100.
The word percent means per hundred.
Expressing a fraction as a percent:
1. Divide the numerator by the denominator.
2. Multiply the quotient by 100. This is done by moving the
decimal point in the quotient two places to the right.
3. Add a percent sign.
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Section 8.1
2
Expressing a Fraction as a Percent
5
Example: Express
as a percent.
8
Solution:
Step 1. Divide the numerator by the denominator.
5 ÷ 8 = 0.625
Step 2. Multiply the quotient by 100.
0.625 x 100 = 62.5
Step 3. Add a percent sign.
62.5%
5
Thus, = 62.5%.
8
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Section 8.1
3
Expressing a Decimal as a Percent
To express a decimal as a percent:
• Move the decimal point two places to the right.
• Attach a percent sign.
Example: Express 0.47 as a percent.
Solution:
Thus, 0.47 = 47%.
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Section 8.1
4
Expressing a Percent as a Decimal Number
To express a percent as a decimal number:
1. Move the decimal point two places to the left.
2. Remove the percent sign.
Example: Express each percent as a decimal:
a. 19%
b. ¼%
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Section 8.1
5
Expressing a Percent as a Decimal Number
Example Continued
Solution: Use the two steps.
a.
Thus, 19% = 0.19.
b. We change the fraction into a decimal, then use the
steps
Thus, ¼% = 0.0025.
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Section 8.1
6
Percent, Sales Tax, & Discounts
Many applications involving percent are based on the
following formula:
Note that “of” implies multiplication.
• We use this formula to determine sales tax collected by
states, counties, cities on sales items to customers.
Sales tax amount = tax rate x item’s cost
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Section 8.1
7
Percent and Sales Tax
Example: Suppose that the local sales tax rate is 7.5% and
you purchase a bicycle for $394.
a. How much tax is paid?
b. What is the bicycle’s total cost?
Solution:
a. Sales tax amount = tax rate x item’s cost
The tax paid is $29.55.
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Section 8.1
8
Percent and Sales Tax
Example Continued
b. The bicycle’s total cost is the purchase price, $394, plus
the sales tax, $29.55.
Total Cost = $394.00 + $29.55 = $423.55
The bicycle’s total cost is $423.55.
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Section 8.1
9
Percent and Sales Price
•
•
Businesses reduce prices, or discount, to attract
customers and to reduce inventory.
The discount rate is a percent of the original price.
Discount price = discount rate x original price.
Example: A computer with an original price of $1460 is on
sale at 15% off.
a. What is the discount amount?
b. What is the computer’s sale price?
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Section 8.1
10
Percent and Sales Price
Example Continued
Solution:
a. Discount amount = discount rate x original price
The discount amount is $219.
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Section 8.1
11
Percent and Sales Price
Example Continued
b. A computer’s sale price is the original price, $1460,
minus the discount amount, $219.
Sale price = $1460 - $219 = $1241
The computer’s sale price is $1241.
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Section 8.1
12
Percent and Income Tax
Calculating Income Tax:
1. Determine your adjusted gross income:
2. Determine your taxable income:
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Section 8.1
13
Percent and Income Tax
3. Determine the income tax:
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Section 8.1
14
Percent and Income Tax
• The table shows tax rates, standard deductions, and
exceptions for the four filing status categories.
Tax Rate
Single
Married Filing
Separately
Married Filing
Jointly
Head of
Household
10%
up to $7300
up to $7300
up to $14,600
up to $10,450
15%
$7301 to $29,700
$7301 to $29,700
$14,601 to
$59,400
$10,451 to
$39,800
25%
$29,701 to $71,950
$29,701 to
$59,975
$59,401 to
$119,950
$39,801 to
$102,800
28%
$71,951 to $150,150
$59,976 to
$91,400
$119,951 to
$182,800
$102,801 to
$166,450
33%
$150,151 to
$326,450
$91,401 to
$163,225
$182,801 to
$326,450
$166,451 to
$326,450
35%
more than $326,450
more than
$163,225
more than
$326,450
more than
$326,450
Std. Ded.
$5000
$5000
$10,000
$7300
Exmt/pers
$3200
$3200
$3200
$3200
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Section 8.1
15
Percent and Income Tax
Example: Calculate the income tax owed by a single
woman with no dependents whose gross income,
adjustments, deductions, and credits are given. Use the
marginal tax rates from the table.
Single Women with no dependents:
Gross income: $52,000
Adjustments: $4,000
Tax credit: $500
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Deductions:
• $7500: mortgage interest
• $2200: property taxes
• $2400: charitable contributions
• $1500: medical expenses not
covered the insurance
Section 8.1
16
Percent and Income Tax
Example Continued
Solution:
Step 1. Determine the adjusted gross income.
Adjusted gross income = Gross income – Adjustments
= $52,000 - $4000
= $48,000
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Section 8.1
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Percent and Income Tax
Example Continued
Step 2. Determine taxable income.
We substitute $12,100 for deductions in the formula for
taxable income.
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Section 8.1
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Percent and Income Tax
Example Continued
Taxable income = Adjusted gross income – (Exemptions + Deductions)
= $48,000
– ($3200 + $12,100)
= $48,000 – $15,300
= $32,700
Step 3. Determine the income tax.
Income tax = Tax computation – Tax credits
= Tax Computation - $500
Since the single woman taxable income is $32,700 she
falls in the tax bracket of 25%.
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Section 8.1
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Percent and Income Tax
Example Continued
• This means that she owes:
10% on the first $7300 of her taxable income,
15% on her taxable income between $7301 to
$29,700, inclusive,
and 25% on her taxable income above $29,701.
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Section 8.1
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Percent and Income Tax
Example Continued
= 0.10 x $7300 + 0.15 x $22,400 + 0.25 x $3000
= $730 + $3360 + $750
= $4840
We substitute $4840 for the tax computation in the formula for
income tax.
Income tax = Tax computation – Tax credits
= $4840 - $500
= $4340
Thus, the income tax owed is $4340.
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Section 8.1
21
Percent and Change
If a quantity changes, its percent increase or its percent
decrease can be found as follows:
1. Find the fraction for the percent increase or decrease:
amount of increase
amount of decrease
or
.
original amount
original amount
2. Find the percent increase or decrease by expressing
the fraction in step 1 as a percent.
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Section 8.1
22
Percent and Change
Example: In 2000, world population was approximately 6
billion. The data are from United Nations Family
Planning Program and are based on optimistic or
pessimistic expectations for successful control of human
population growth.
a. Find the percent increase in
world population from 2000 to
2150 using the high projection
data.
b. Find the percent decrease in
world population form 2000 to
2150 using the low projection
data.
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Section 8.1
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Percent and Change
Example Continued
Solution:
• Use the data shown on the blue, high-projection, graph.
amount of increase
Percent increase =
original amount
30 − 6 24
=
=
= 4 = 400%
6
4
The projected percent increase in world population is
400%.
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Section 8.1
24
Percent and Change
Example Continued
•
Use the data shown on the green, low-projection, graph.
amount of decrease
Percent decrease =
original amount
6−4 2 1
1
=
= = = 33 %
6
6 3
3
•
The projected percent decrease in world population is
33⅓%.
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Section 8.1
25
Abuses of Percents
Example: John Tesh, while he was still co-anchoring
Entertainment Tonight, reported that the PBS series The
Civil War had an audience of 13% versus the usual 4%
PBS audience, “an increase of more than 300%.” Did
Test report the percent increase correctly?
Solution: We begin by finding the actual percent increase.
amount of increase
original amount
13 − 4 9
=
= = 2.25 = 225%
4
4
Percent increase =
The percent increase for PBS was 225%. This is not
more than 300%, so Tesh did not report the percent
increase correctly.
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Section 8.1
26
Section 8.2
Simple Interest
Objectives
1. Calculate simple interest.
2. Use the future value formula.
3. Use the simple interest formula on discounted loans.
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Section 8.2
1
Simple Interest
• Interest is the dollar amount that we get paid for lending
money or pay for borrowing money.
• The amount of money that we deposit or borrow is called
the principal.
• The amount of interest depends on the principal, the
interest rate, which is given as a percent and varies from
bank to bank, and the length of time which the money is
deposited.
• Simple interest involves interest calculated only on the
principal.
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Section 8.2
2
Simple Interest
To calculate the simple interest:
Interest = principal x rate x time
I = Prt
The rate r, is expressed as a decimal when calculating
simple interest.
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Section 8.2
3
Simple Interest
Example: You deposit $2000 in a savings account at
Hometown Bank, which has a rate of 6%. Find the
interest at the end of the first year.
Solution: To find the interest at the end of the first year, we
use the simple interest formula.
At the end of the first year, the interest is $120.
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Section 8.2
4
Simple Interest
Example: A student took out a simple interest loan for
$1800 for two years at a rate of 8% to purchase a new
car. Find the interest of the loan.
Solution: To find the interest of the loan, we use the simple
interest formula.
The interest on the loan is $288.
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Section 8.2
5
Future Value: Principal Plus Interest
The future value, A, of P dollars at simple interest rate r
(as a decimal) for t years is given by
A = P(1 + rt).
• P is also known as the loan’s present value.
Example: A loan of $1060 has been made at 6.5% for three
months. Find the loan’s future value.
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Section 8.2
6
Future Value: Principal Plus Interest
Example Continued
Solution: The variables are given as follows:
P = $1060
r = 6.5% or 0.065
t = ¼ = 0.25 since 3 months is
3 of
12
a year.
Now, we use the future value formula:
A = P(1 + rt)
A = 1060(1 + 0.065 · 0.25)
A ≈ $1077.23
Rounded to the nearest cent, the loan’s future value is $1077.23.
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Section 8.2
7
Future Value: Principal Plus Interest
Example: You borrow $2500 from a friend and promise to
pay back $2655 in six months. What simple interest will
you pay?
Solution: We substitute $2655 for A, $2500 for P, and ½ or
0.5 for t since 6 months is ½ of a year. Then solve for the
simple interest r.
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Section 8.2
8
Future Value: Principal Plus Interest
Example Continued
A = P( 1 + rt)
2655 = 2500 (1 + r ⋅ 0 .5 )
2655
155
155
1250
r
= 2500 + 1250 r
= 1250 r
1250 r
=
1250
= 0 .124 = 12 .4 %
You will pay a simple interest rate of 12.4%.
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Section 8.2
9
Discounted Loans
•
•
Some lenders collect the interest from the amount of the
loan at the time that the loan is made. This is called a
discounted loan.
The interest that is the deducted from the loan is the
discount.
Example: You borrow $10,000 on a 10% discounted loan
for a period of 8 months.
a. What is the loan’s discount?
b. Determine the net amount of money you receive.
c. What is the loan’s actual interest rate?
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Section 8.2
10
Discounted Loans
Example Continued
Solution:
a. Because the loan’s discount is the deducted interest, we
use the simple interest formula.
The loan’s discount is $666.67.
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Section 8.2
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Discounted Loans
Example Continued
b. The net amount that you receive is the amount of the
loan, $10,000, minus the discount, $666.67:
10,000 – 666.67 = 9333.33.
Thus, you receive $9333.33.
c. We can recalculate the loan’s actual interest rate, rather
than stated 10%, by using the simple interest formula.
I = Prt
⎛2⎞
666.67 = (9333.33)r ⎜ ⎟
⎝3⎠
666.67 = 6222.22r
666.67
≈ 0.107 = 10.7%
r=
6222.22
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Section 8.2
The actual rate of
interest on the 10%
discounted loan is
approximately 10.7%.
12
Section 8.3
Compound Interest
Objectives
1.  Use the compound interest formulas.
2.  Calculate present value.
3.  Understand and compute effective annual yield.
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Section 8.3
1
Compound Interest
•  Compound interest is interest computed on the original
principal as well as on any accumulated interest.
To calculate the compound interest paid once a year we
use
A = P(1 + r)t,
where A is called the account’s future value, the principal
P is called its present value, r is the rate, and t is the
years.
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Section 8.3
2
Compound Interest
Example: You deposit $2000 in a savings account at
Hometown bank, which has a rate of 6%.
a.  Find the amount, A, of money in the account after 3
years subject to compound interest.
b.  Find the interest.
Solution:
a.  Principal P is $2000, r is 6% or 0.06, and t is 3.
Substituting this into the compound interest formula, we
get
A = P(1 + r)t = 2000(1 + 0.06)3 = 2000(1.06)3 ≈ 2382.03
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Section 8.3
3
Compound Interest
Example Continued
Rounded to the nearest cent, the amount in the savings
account after 3 years is $2382.03.
b.  The amount in the account after 3 years is $2382.03.
So, we take the difference of this amount and the
principal to obtain the interest amount.
$2382.03 – $2000 = $382.03
Thus, the interest you make after 3 years is $382.03.
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Section 8.3
4
Compound Interest
To calculate the compound interest paid more than once
a year we use
where A is called the account’s future value, the principal
P is called its present value, r is the rate, n is the number
of times the interest is compounded per year, and t is the
years.
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Section 8.3
5
Compound Interest
Example: You deposit $7500 in a savings account that has
a rate of 6%. The interest is compounded monthly.
a.  How much money will you have after five years?
b.  Find the interest after five years.
Solution:
a. Principal P is $7500, r is 6% or 0.06, t is 5, and n is 12
since interest is being compounded monthly.
Substituting this into the compound interest formula, we
get
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Section 8.3
6
Compound Interest
Example Continued
Rounded to the nearest cent, you will have $10,116.38
after five years.
b. The amount in the account after 5 years is $10,116.38.
So, we take the difference of this amount and the
principal to obtain the interest amount.
$ 10,116.38 – $7500 = $2616.38
Thus, the interest you make after 5 years is $ 2616.38.
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Section 8.3
7
Compound Interest
Continuous Compounding
• 
Some banks use continuous compounding, where
the compounding periods increase infinitely.
After t years, the balance, A, in an account with principal P
and annual interest rate r (in decimal form) is given by
the following formulas:
1.  For n compounding per year:
2.  For continuous compounding: A = Pert.
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Section 8.3
8
Compound Interest
Continuous Compounding
Example: You decide to invest $8000 for 6 years and you
have a choice between two accounts. The first pays 7%
per year, compounded monthly. The second pays 6.85%
per year, compounded continuously. Which is the better
investment?
Solution: The better investment is the one with the greater
balance at the end of 6 years.
7% account:
The balance in this account after 6 years is $12,160.84.
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Section 8.3
9
Compound Interest
Continuous Compounding
Example Continued
6.85% account:
A = Pert = 8000e0.0685(6) ≈ 12,066.60
The balance in this account after 6 years is $12,066.60.
Thus, the better investment is the 7% monthly
compounding option.
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Section 8.3
10
Planning for the Future with Compound Interest
Calculating Present Value
If A dollars are to be accumulated in t years in an account
that pays rate r compounded n times per year, then the
present value P that needs to be invested now is given
by
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Section 8.3
11
Planning for the Future with Compound Interest
Example: How much money should be deposited in an
account today that earns 8% compounded monthly so
that it will accumulate to $20,000 in five years?
Solution: We use the present value formula, where A is
$20,000, r is 8% or 0.08, n is 12, and t is 5 years.
Approximately $13,424.21 should be invested today in
order to accumulate to $20,000 in five years.
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Section 8.3
12
Effective Annual Yield
• 
The effective annual yield, or the effective rate, is the
simple interest rate that produces the same amount of
money in an account at the end of one year as when the
account is subjected to compound interest at a stated
rate.
Example: You deposit $4000 in an account that pays 8%
interest compounded monthly.
a.  Find the future value after one year.
b.  Use the future value formula for simple interest to
determine the effective annual yield.
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Section 8.3
13
Effective Annual Yield
Example Continued
Solution:
a.  We use the compound interest formula to find the
account’s future value after one year.
Rounded to the nearest cent, the future value after one
year is $4332.00.
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Section 8.3
14
Effective Annual Yield
Example Continued
b.  The effective annual yield is the simple interest rate. So,
we use the future value formula for simple interest to
determine rate r.
Thus, the effective annual yield
is 8.3%. This means that an
account that earns 8% interest
compounded monthly has an
equivalent interest rate of 8.3%
compounded annually.
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Section 8.3
15
Section 8.4
Annuities, Stocks, and Bonds
Objectives
1.  Determine the value of an annuity.
2.  Determine regular annuity payments needed to achieve
a financial goal.
3.  Understand stocks and bonds as investments.
4.  Read stock tables.
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Section 8.4
1
Annuities
• 
• 
An annuity is a sequence of equal payments made at
equal time periods.
The value of an annuity is the sum of all deposits plus
all interest paid.
Example: You deposit $1000 into a savings plan at the end
of each year for three years. The interest rate is 8% per
year compounded annually.
a.  Find the value of the annuity after three years.
b.  Find the interest.
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Section 8.4
2
Annuities
Example Continued
Solution:
a.  The value of the annuity after three years is the sum of
all deposits made plus all interest paid over three years.
The value of annuity
at the end of three
years is $3246.40.
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Section 8.4
3
Annuities
Example Continued
b.  You made three payments of $1000 each, depositing a
total of 3 x $1000, or $3000. Because the value of
annuity is $3246.40, the interest is
$3246.40 - $3000 = $246.40.
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Section 8.4
4
Annuity
Interest Compounded Once a Year
If P is the deposit made at the end of each year for an
annuity that pays an annual interest rate r (in decimal
form) compounded once a year, the value, A, of the
annuity after t years is
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Section 8.4
5
Annuity
Interest Compounded Once a Year
Example: To save for retirement, you decide to deposit
$1000 into an IRA at the end of each year for the next
30 years. If you can count on an interest rate 10% per
year compounded annually,
a.  How much will you have from the IRA after 30 years?
b.  Find the interest.
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Section 8.4
6
Annuity
Interest Compounded Once a Year
Example Continued
Solution:
a.  The amount that you will have in the IRA is its value
after 30 years.
After 30 years, you will have
approximately $164,494 from the
IRA.
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Section 8.4
7
Annuity
Interest Compounded Once a Year
Example Continued
b.  You made 30 payments of $1000 each, a total of
$30,000. Because the annuity is $164,494, the interest
is
$164,494 - $30,000 = $134,494.
The interest is nearly 4½ times the amount pf your
payments.
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Section 8.4
8
Annuity
Interest Compounded N Times Per Year
If P is the deposit made at the end of each compounding
period for an annuity that pays an annual interest rate r
(in decimal form) compounded n times per year, the
value, A, of the annuity after t years is
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Section 8.4
9
Annuity
Interest Compounded Once a Year
Example: At age 25, to save for retirement, you decide to
deposit $200 into an IRA at the end of each month at an
interest rate of 7.5% per year compounded monthly,
a.  How much will you have from the IRA when you retire at
age 65?
b.  Find the interest.
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Section 8.4
10
Annuity Interest Compounded
Once a Year Example Continued
Solution:
a.  The amount that you will have in the IRA is its value
after 40 years.
At age 65, you will have
approximately $604,765 from the
IRA.
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Section 8.4
11
Annuity
Interest Compounded Once a Year
Example Continued
b.
The interest is more than 5 times the amount of your
contributions to the IRA.
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Section 8.4
12
Planning for the Future with an Annuity
P, the deposit that must be made at the end of each
compounding period into an annuity that pays an annual
interest rate r (in decimal form) compounded n times per
year in order to achieve a value of A dollars after t years
is
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Section 8.4
13
Planning for the Future with an Annuity
Example: You would like to have $20,000 to use as a down
payment for a home in five years by making regular,
end-of-the-month deposits in an annuity that pays 8%
compounded monthly.
a.  How much should you deposit each month?
b.  How much of the $20,000 down payment comes from
deposits and how much comes from interest?
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Section 8.4
14
Planning for the Future with an Annuity
Example Continued
Solution:
a.
You should deposit $273 each month to be certain of
having $20,000 for a down payment on a home.
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Section 8.4
15
Planning for the Future with an Annuity
Example Continued
b.
We see that $16,380 of the $20,000 comes from your
deposits and the remainder, $3620, comes from interest.
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Section 8.4
16
Investments
•  When depositing money into a bank account, you are
making a cash investment.
•  The account’s interest rate guarantees a certain percent
increase in your investment, called its return.
•  Other kind of investments that are riskier are called
stocks and bonds.
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Section 8.4
17
Stocks
•  Investors purchase stock, shares of ownership in a
company.
For example, if a company has issued a total of 1 million
shares and an investor owns 20,000 of these shares,
that investor owns
of the company.
•  Any investor who owns some percentage of the
company is called a shareholder.
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Section 8.4
18
Stocks
•  Buying or selling stocks is referred to as trading.
•  Stocks are traded on a stock exchange.
There are two ways to make money by investing in stock:
•  You sell shares for more money than you paid for them,
in which case you have a capital gain on the sale of
stock.
•  While you own the stock, the company distributes all or
part of its profits to shareholders as dividends.
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Section 8.4
19
Bonds
•  In order to raise money and dilute the ownership of
current stockholders, companies sell bonds.
•  People who buy a bond are lending money to the
company from which they buy the bond.
•  Bonds are a commitment from a company to pay the
price an investor pays for the bond at the time it was
purchased, called the face value, along with interest
payments at a given rate.
•  A listing of all the investments that a person holds is
called a financial portfolio.
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Section 8.4
20
Bonds
•  A listing of all the investments that a person holds is
called a financial portfolio.
•  Most financial advisors recommend a portfolio with a
mixture of low-risk and high-risk investments, called a
diversified portfolio.
1/17/11
Section 8.4
21
Reading Stock Tables
Daily newspapers and online services give current stock
prices and other information about stocks.
•  52-week high refers to the highest price at which a
company traded during the past 52 weeks.
•  52-week low refers to the lowest price at which a
company traded during the past 52 weeks.
•  Stock refers to the company name.
•  SYM refers to the symbol the company uses for trading.
•  Div refers to dividends paid per share to stockholders
last year.
•  Yld% stands for percent yield.
•  Vol100s stands for sales volume in hundreds.
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Section 8.4
22
Reading Stock Tables
•  Hi stands for the highest price at which the company’s
stock traded yesterday.
•  Low stands for the lowest price at which the company’s
stock traded yesterday.
•  Close stands for the price at which shares traded when
the stock exchanged closed yesterday.
•  Net Chg stands for net change.
•  PE stands for the price-to-earnings ratio.
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Section 8.4
23
Reading Stock Tables
Example: Use the stock table for Disney to answer the
following questions:
a.  What were the high and low prices for the past 52
weeks?
b.  If you owned 3000 shares of Disney stock last year,
what dividend did you receive?
c.  What is the annual return for dividends alone? How
does this compare to a bank account offering a 3.5%
interest rate?
d.  How many shares of Disney were traded yesterday?
e.  What were the high and low prices for Disney shares
yesterday?
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Section 8.4
24
Reading Stock Tables
Example Continued
f.  What was the price at which Disney shares traded when
the stock exchange closed yesterday?
g.  What does “…” in the net change column mean?
h.  Compute Disney’s annual earnings per share using
52-week
High
Low
Stock
SYM
Div
42.38
22.50
Disney
DIS
.21
1/17/11
Yld
PE
%
.6
43
Section 8.4
Vol
100s
Hi
Lo
Close
Net
Chg
115900
32.50
31.25
32.50
…
25
Reading Stock Tables
Example Continued
Solution:
a.  We look under the heading High and Low. The 52week high was 42.38 or $42.38. The 52-week low was
22.50 or $22.50.
b.  We look under the heading Div for the dividend paid for
a share of Disney stock. So, the annual return of a
share last year was $0.21. If you owned 3000 shares,
then you received a dividend of 3000 x 0.21 = $630.
c.  The heading Yld% is the percentage that give investors
an annual return of their dividends. For Disney, it is
0.6%. This is much lower than a bank account paying
3.5% interest rate.
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Section 8.4
26
Reading Stock Tables
Example Continued
d. We look under Vol100s for the number of shares
Disney traded yesterday. So, Disney traded 115900 x
100 = 11,590,000 shares yesterday.
e.  The high and low prices for Disney shares yesterday
are found by looking under Hi and Lo. Hence, the high
price was $32.50 and the low price was $31.25 per
share yesterday.
f.  When the stock exchange closed yesterday Disney’s
shares were $32.50 per share.
g.  The “…” under Net Chg means that there was no
change in price in Disney stock from the market close
two days ago to yesterday’s market close.
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Section 8.4
27
Reading Stock Tables
Example Continued
h.  We use the formula given along with 43 for the PE ratio
and $32.50 for the closing price per share.
Thus, the annual earnings per share for Disney are $0.76.
•  The PE ratio tells us that yesterdays closing price is 43
times greater than the earning’s per share.
1/17/11
Section 8.4
28
Section 8.5
Installment Buying
Objectives
1.  Determine the amount financed, the installment price,
and the finance charge for a fixed loan.
2.  Determine the APR.
3.  Compute unearned interest and the payoff amount for a
loan paid off early.
4.  Find the interest, the balance due, and the minimum
monthly payment for credit card loans.
5.  Calculate interest on credit cards using three methods.
1/17/11
Section 8.5
1
Fixed Installment Loans
•  The amount financed is what the consumer borrows:
Amount financed = cash price – down payment.
•  The total installment price is the sum of all monthly
payments plus the down payment:
Total Installment Price = Total of all monthly payments + down payment.
•  The finance charge is the interest on the installment
loan:
Finance charge = Total installment price - Cash price.
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Section 8.5
2
Fixed Installment Loans
Example: The cost of a used pick-up truck is $9345. We
can finance the truck by paying $300 down and $194.38
per month for 60 months.
a.  Determine the amount financed.
b.  Determine the total installment price.
c.  Determine the finance charge.
1/17/11
Section 8.5
3
Fixed Installment Loans
Example Continued
Solution:
•  Amount financed = cash price – down payment
= $9345 - $300
= $9045
The amount financed is $9045.
b.
Total Installment Price = Total of all monthly payments + down payment
= $194.37 x 60 + $300
= $11,662.80 + $300
= $11,962.80
The total installment price is $11,962.80.
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Section 8.5
4
Fixed Installment Loans
Example Continued
c.  Finance charge = Total installment price - Cash price
= $11,962.80 - $9345
= $2617.80
The finance charge for the loan is $2617.80.
1/17/11
Section 8.5
5
Annual Percentage Rate
APR
Steps in using the APR table below:
1.  Compute the finance charge per $100 financed:
2.  Look in the row (in table below) corresponding to the
number of payments to be made and find the entry
closest to the value in step 1.
3.  Find the APR at the top of the column in which the entry
from step 2 is found. This is the APR rounded to the
nearest ½%.
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Section 8.5
6
Annual Percentage Rate
APR Table
1/17/11
Section 8.5
7
Annual Percentage Rate
APR
Example: In the previous example, we found that the
amount financed for the truck was $9045 and the finance
charge was $2617.80. The borrower financed the truck
with 60 monthly payments. Use the table to determine
the APR.
1/17/11
Section 8.5
8
Annual Percentage Rate
Example Continued
Solution:
Step 1. Find the finance charge per $100 of the amount
financed.
This means that the borrower pays $28.94 interest for each
$100 being financed.
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Section 8.5
9
Annual Percentage Rate
Example Continued
Step 2. We look for 60 in the left column since we make 60
monthly payments. Move to the right until
you find the amount closest to the finance
charge of $28.94.
Step 3. Find the APR. Look at the corresponding APR for
$28.96 at the top of the column.
Thus, the APR for the fixed installment truck loan is
approximately 10.5%.
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Section 8.5
10
Methods for Computing Unearned Interest
The total amount due on the day that a loan is paid off early
is called the payoff amount.
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Section 8.5
11
Methods for Computing Unearned Interest
Example: You got a big bonus at work—and a raise.
Instead of making the thirty-sixth payment on your truck,
you decide to pay the remaining balance and terminate
the loan.
a.  Use the actuarial method to determine how much
interest will be saved (the unearned interest, u) by
repaying the loan early.
b.  Find the payoff amount.
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Section 8.5
12
Methods for Computing Unearned Interest
Example Continued
Solution:
a.  We use the formula
to find u, the unearned
interest. We need to find k, R, and V.
k: k=60 – 36 = 24, since the current payment is number 36.
R: R = $194.38
V: V = 11.30, from the APR table with 24 monthly payments
left.
Thus,
.
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Section 8.5
13
Methods for Computing Unearned Interest
Example Continued
Approximately $473.64 will be saved in interest by
terminating the loan early using the actuarial method.
b.  The pay off amount, the amount due on the day of the
loan’s termination is determined as follows:
The payment needed to terminate the loan at the end of 36
months is $4385.86.
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Section 8.5
14
Methods for Computing Unearned Interest
Example: The loan in the previous example is paid off at
the time of the thirty-sixth monthly payment. Use the
rule of 78 to find
a.  the unearned interest.
b.  the payoff amount.
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Section 8.5
15
Methods for Computing Unearned Interest
Example Continued
Solution:
a.  We use the formula
to find u, the unearned interest.
k: k=60 – 36 = 24, since the current payment is number 36.
n: n = 60, the original number of payments.
F: F = $2617.80, the finance charge (interest).
1/17/11
Section 8.5
16
Methods for Computing Unearned Interest
Example Continued
Now,
Thus, by the rule of 78, the borrower will save $429.15 in
interest.
b.  Next, we determine the payoff amount.
Thus, the payoff amount is $4430.35.
1/17/11
Section 8.5
17
Open-end Installment Loans
•  Using a credit card is an example of an open-end
installment loan, commonly called revolving credit.
•  Customers receive a statement, called an itemized
billing.
•  One method for calculating interest on credit cards is the
unpaid balance method.
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Section 8.5
18
Open-end Installment Loans
Example: A particular VISA card calculates interest using
the unpaid balance method. The monthly interest rate is
1.3% on the unpaid balance on the first day of the billing
period less payments and credits. Here are some of the
details in the May 1–May 31 itemized billing:
May 1 Unpaid Balance: $1350
Payment Received May 8: $250
Purchases Charged to the VISA Account: Airline tickets: $375, Books: $57,
Meals: $65
Last Day of the Billing Period: May 31
Payment Due Date: June 9
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Section 8.5
19
Open-end Installment Loans
Example Continued
a. Find the interest due on the payment due date.
b. Find the total balance owed on the last day of the billing
period.
c. This credit card requires a $10 minimum monthly
payment if the total balance owed on the last day of the
billing period is less than $360. Otherwise, the minimum
monthly payment is
of the balance owed on the last
day of the billing period, rounded to the nearest whole
dollar. What is the minimum monthly payment due by
June 9?
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Section 8.5
20
Open-end Installment Loans
Example Continued
Solution:
a.  The monthly interest rate is 1.3% on the unpaid balance
on the first day of the billing period, $1350, less
payments and credits, $250. The interest due is
computed using I = Prt.
The interest due on the payment due date is $14.30.
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Section 8.5
21
Open-end Installment Loans
Example Continued
b.  The total balance owed on the last day of the billing
cycle is determine as follows:
The total balance owed on the last day of the billing period
is $1611.30.
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Section 8.5
22
Open-end Installment Loans
Example Continued
c. Because the balance owed on the last day of the billing
period, $1611.30, exceeds $360, the customer must pay
a minimum of
of the balance owed.
Rounded to the nearest whole dollar, the minimum monthly
payment due by June 9 is $45.
1/17/11
Section 8.5
23
Methods for Calculating Interest on Credit Cards
For all three methods, I = Prt, where r is the monthly rate
and t is one month.
Unpaid balance method: The principal, P, is the balance
on the first day of the billing period less payments and
credits.
Previous balance method: The principal, P, is the unpaid
balance on the first day of the billing period.
Average daily balance method: The principle, P, is the
average daily balance. This is determined by adding
the unpaid balances for each day in the billing period
and dividing by the number of days in the billing period.
1/17/11
Section 8.5
24
Methods for Calculating Interest on Credit Cards
Example: A credit card has a monthly rate of 1.75%. In the
January 1-31 itemized billing, the January 1 unpaid
balance is $2500. A payment of $1000 was received on
January 8. There are no purchases or cash advances in
this billing period. The payment due date is February 9.
Find the interest due on this date using each of the three
methods for calculating credit card interest.
Solution: Because the monthly rate is 1.75%, for all three
methods I = Prt = P x 0.0175 x 1 month. The principal, P,
is different for each method.
1/17/11
Section 8.5
25
Methods for Calculating Interest on Credit Cards
Example Continued
a. The Unpaid Balance Method
The principal, P, is the unpaid balance on the first day of
the billing period, $2500, less payments and credits,
$1000. The interest is
I = Prt = ($2500 - $1000) x 0.0175 x 1 = $1500 x 0.0175 x 1 = $26.25.
The interest due on the payment due date is $26.25.
1/17/11
Section 8.5
26
Methods for Calculating Interest on Credit Cards
Example Continued
b. The Previous Balance Method
The principal, P, is the unpaid balance on the first day of
the billing period, $2500. The interest is
I = Prt = $2500 x 0.0175 x 1 = $43.75.
The interest due on the payment due date is $43.75.
1/17/11
Section 8.5
27
Methods for Calculating Interest on Credit Cards
Example Continued
c. The Average Daily Balance Method
The principal, P, is the average daily balance. Add the
unpaid balances for each day in the billing period and
divide by the number of days in the billing period, 31.
The unpaid balance on the first day of the billing period
is $2500, and a $1000 payment is recorded on January
8. The sum of the balances owed for each day of the
billing period is $2500 added for each of the first 7 days,
expressed as $2500 x 7, plus $1500 added for each of
the remaining 24 days, expressed as $1500 x 24.
1/17/11
Section 8.5
28
Methods for Calculating Interest on Credit Cards
Example Continued
Average daily balance
The average daily balance serves as the principal.
1/17/11
Section 8.5
29
Methods for Calculating Interest on Credit Cards
Example Continued
The interest is
I = Prt = $1725.81 x 0.0175 x 1 ≈ $30.20.
The interest due on the payment due date is $30.20.
1/17/11
Section 8.5
30
Section 8.6
Amortization and the Cost of Home Ownership
Objectives
1.  Understand mortgage options.
2.  Compute the monthly payment and interest costs for a
mortgage.
3.  Prepare a partial loan amortization schedule.
4.  Compute payments and interest for other kinds of
installment loans.
1/17/11
Section 8.6
1
Mortgages
•  A mortgage is a long-term loan for the purpose of buying a
home.
•  The down payment is the portion of the sale price of the
home that the buyer initially pays to the seller.
•  The amount of the mortgage is the difference between the
sale price and the down payment.
•  Some companies, called mortgage brokers, offer to find you
a mortgage lender willing to make you a loan.
•  Fixed-rate mortgages have the same monthly payment
during the entire time of the loan.
•  Variable-rate mortgages known as adjustable-rate
mortgages (ARMs), have payment amounts that change from
time to time depending on changes in the interest rate.
1/17/11
Section 8.6
2
Computation Involved with Buying a Home
•  Most lending institutions require the buyer to pay one or
more points at the time of closing—that is, the time at
which the mortgage begins.
•  A point is a onetime charge that equals 1% of the loan
amount.
For example, two points means that the buyer must pay 2%
of the loan amount at closing.
•  A document, called the Truth-in-Lending Disclosure
Statement, shows the buyer the APR for the mortgage.
•  In addition, lending institutions can require that part of
the monthly payment be deposited into an escrow
account, an account used by the lender to pay real
estate taxes and insurance.
1/17/11
Section 8.6
3
Computation Involved with Buying a Home
Loan Payment Formula for Installment Loans
The regular payment amount, PMT, required to repay a
loan of P dollars paid n times per years over t years at an
annual rate r is given by
1/17/11
Section 8.6
4
Computation Involved with Buying a Home
Example: The price of a home is $195,000. The bank
requires a 10% down payment and two points at the
time of closing. The cost of the home is financed with a
30-year fixed rate mortgage at 7.5%.
a. 
b. 
c. 
d. 
Find the required down payment.
Find the amount of the mortgage.
How much must be paid for the two points at closing?
Find the monthly payment (excluding escrowed taxes
and insurance).
e.  Find the total interest paid over 30 years.
1/17/11
Section 8.6
5
Computation Involved with Buying a Home
Example Continued
Solution:
a.  The required down payment is 10% of $195,000 or
0.10 x $195,000 = $19,500.
b.  The amount of the mortgage is the difference between
the price of the home and the down payment.
1/17/11
Section 8.6
6
Computation Involved with Buying a Home
Example Continued
c.  To find the cost of two points on a mortgage of
$175,500, find 2% of $175,500.
0.02 x $175,000 = $3510
The down payment ($19,500) is paid to the seller and
the cost of two points ($3510) is paid to the lending
institution.
1/17/11
Section 8.6
7
Computation Involved with Buying a Home
Example Continued
d.  We need to find the monthly mortgage payment for
$175,000 at 7.5% for 30 years. We use the loan
payment formula for installment loans.
The monthly mortgage payment for principal and interest is
approximately $1227.00.
1/17/11
Section 8.6
8
Computation Involved with Buying a Home
Example Continued
e. The total cost of interest over 30 years is equal to the
difference between the total of all monthly payments and
the amount of the mortgage. The total of all monthly
payments is equal to the amount of the monthly payment
multiplied by the number of payments. We found the
amount of each monthly payment in (d): $1227. The
number of payments is equal to the number of months in
a year, 12, multiplied by the number of years in the
mortgage, 30: 12 x 30 = 360. Thus, the total of all
monthly payments = $1227 x 360.
1/17/11
Section 8.6
9
Computation Involved with Buying a Home
Example Continued
Now we calculate the interest over 30 years.
The total interest paid over 30 years is approximately
$266,220.
1/17/11
Section 8.6
10
Loan Amortization Schedules
•  When a loan is paid off through a series of
regular payments, it is said to be amortized,
which literally means “killed off.”
•  Although each payment is the same, with each
successive payment the interest portion decreases
and the principal portion increases.
•  A document showing important information about the
status of the mortgage is called a loan amortization
schedule.
1/17/11
Section 8.6
11
Loan Amortization Schedules
Example: Prepare a loan amortization schedule for the first
two months of the mortgage loan shown in the following
table:
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Section 8.6
12
Loan Amortization Schedules
Example Continued
Solution: We begin with payment number 1.
Interest for the month = Prt = $130,000 x 0.095 x
≈ $1029.17
Principal payment = Monthly payment - Interest payment
= $1357.50 - $1029.17
= $328.33
Balance of loan = Principal balance - Principal payment
= $130,000 - $328.33
= $129,671.67
1/17/11
Section 8.6
13
Loan Amortization Schedules
Example Continued
Now, starting with a loan balance of $129,671.67, we repeat
these computations for the second month.
Interest for the month = Prt = $129,671.67 x 0.095 x
= $1026.57
Principal payment = Monthly payment – Interest payment
= $1357.50 – $1026.57
= $330.93
Balance of loan = Principal balance – Principal payment
= $129,671.67 – $330.93
= $129,340.74
1/17/11
Section 8.6
14
Loan Amortization Schedules
Example Continued
1/17/11
Section 8.6
15
Monthly Payments and Interest Costs for
Other Kinds of Installment Loans
Example: You decide to take a $20,000 loan for a new car.
You can select one of the following loans, each
requiring regular monthly payments:
Installment Loan A: 3-year loan at 7%.
Installment Loan A: 5-year loan at 9%.
a.  Find the monthly payments and the total interest for
Loan A.
c.  Find the monthly payments and the total interest for
Loan B.
d.  Compare the monthly payments and total interest for
the two loans.
1/17/11
Section 8.6
16
Monthly Payments and Interest Costs for
Other Kinds of Installment Loans
Solution:
a.  We first determine monthly payments and total interest
for Loan A.
The monthly payments are approximately $618.
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Section 8.6
17
Monthly Payments and Interest Costs for
Other Kinds of Installment Loans
Now we calculate the interest paid over 3 years or 36
months.
The interest paid over 3 years is approximately $2248.
1/17/11
Section 8.6
18
Monthly Payments and Interest Costs for
Other Kinds of Installment Loans
b.  Next, we determine monthly payment and total interest
for Loan B.
The monthly payments are approximately $415.
1/17/11
Section 8.6
19
Monthly Payments and Interest Costs for
Other Kinds of Installment Loans
Now we calculate the interest over 5 years, or 60 months.
The total interest paid over 5 years is approximately $4900.
1/17/11
Section 8.6
20
Monthly Payments and Interest Costs for
Other Kinds of Installment Loans
c. The monthly payments and total interest for the two
loans is given in the table below.
1/17/11
Section 8.6
21