THREE WAYS OF USING
PERCENTAGES
1. AS FRACTIONS
58% of the people surveyed reported that
they get less than 8 hours of sleep a night.
2. TO DESCRIBE CHANGE
Enrollment in the armed forces increased 8%
last year.
3. FOR COMPARISONS
The Mercedes in the advertisement costs 25%
more than the Cadillac.
PERCENTAGES AS FRACTIONS
1.
If there are 15 girls in a class of 27, then girls
make up what percentage of the class?
2.
In a group of 80 people, if 35% of them have
brown eyes, how many of the people have brown
eyes?
3.
42 is what percent of 80?
4.
If you get 82% on a test with 34 questions of equal
point value, how many questions did you get right?
5.
34 team field goals out of 60 attempted is what
shooting percentage for the team?
6.
In a history class of 120 students, 60% are females
and 25% of the females have blue eyes. What
percentage of the history students in the class are
blue-eyed females?
7.
Assume that to earn a C grade or higher in a
college course you need at least an average of
70% on the five exams that are each worth 100
points. You have scores of 65, 81, 72, 55 so far in
the semester. What is the lowest score you can
receive on your fifth exam in order to earn a C or
higher?
PERCENTAGES TO DESCRIBE CHANGE
8.
In a high school survey of 2500 students, 310
students reported that they had used drugs. The
next year, 382 students reported that they had
used drugs.
How much did drug use increase?
What was the absolute change?
By what percent did drug use increase?
9.
If the enrollment at Rick’s College was 8, 277 in
1997 and 8, 551 in 1998, then:
There were _________ more students at Rick’s in
1998 than in 1997.
There were _______% more students at Rick’s in
1998 than in 1997.
10. Suppose two companies experience layoffs. If
company A goes from 560 to 520 employees and
company B goes from 1525 to 1450 employees,
which company layed off more people in absolute
terms? In relative terms?
PERCENTAGES OF PERCENTAGES
11. Suppose your bank account interest rate decreases
from 5% to 4%. What is the absolute change?
What is the relative change?
12. The percentage of all bachelor’s degrees awarded
to women increased from 44% in 1972 to 54% in
1992. Find the absolute and relative change.
PERCENTAGES FOR COMPARISON
13. Suppose your wage (per hour) is $8.50 and your
friend’s wage is $7.35.
How much more (per hour) do you make than your
friend?
You make _____% more than your friend does.
How much less (per hour) does your friend make than
you?
Your friend makes _____% less than you do.
14. In 1995, 504,000 males had open heart surgery in
the U.S., while 209,000 females had the operation.
What is the absolute difference and relative
difference between men and women.
OF VERSUS MORE THAN
15. If Joe’s salary is 125% of Bill’s, then Joes salary is
________% more than Bill’s.
If Mary’s salary is 88% of Jill’s, then Mary’s salary
is _______% less than Jill’s.
16. The population of Montana is 20% less than the
population of New Hampshire, so Montana’s
population is ________% of New Hampshire’s
population.
17. You buy a computer in Idaho for $799. Sales tax in
Idaho is 5%. What do you pay for the computer?
COMPOUND INTEREST:
General Formula:
A = accumulated balance
P = principal (amount on
which interest is paid)
i = interest rate (as a decimal)
N = number of pay periods
Interest Paid Once a Year:
A = accumulated balance
P = principal
APR = annual percentage rate
(as a decimal)
Y = number of years
Interest Paid Multiple Times a Year:
A = accumulated balance
P = principal
APR = annual percentage rate
Y = number of years
n = number of compounding
periods per year
USING THE
COMPOUND INTEREST FORMULAS
1. Suppose that you invest $2000 in a savings
account with a 3.5% APR, where the interest is
compounded annually. How much will the
account be worth in 5 years?
2. How much would your need to invest in a
savings account with a 4% APR, (where the
interest is compounded annually), if you want
the account to be worth $15,000 in 20 years.
3. Suppose that you invest $10,000 in a 10 year
savings CD with a 6.8% APR. If the interest is
compounded quarterly, how much money will
be in the CD be worth when it matures in 10
years?
Continuously Compounded Interest:
A = accumulated balance
P = principal
APR = annual percentage rate
Y = # years
e . 2.71828 (e is a mathematical
constant kind of like B. There
is a button on your calculator
for e)
CONTINUOUS COMPOUNDING
EXAMPLES
1.
Suppose that you are 21 years old and that you invest
$3000 this year in a savings account with a 7.3% APR
(compounded continuously). If you don’t touch the
account, how much money will be in the account when you
retire at age 65?
How much money would you lose by waiting until you were
25 years old to invest the same amount of money in the
same account?
2.
Tom and Mary have a new baby boy, Hunter. They want to
put money now into an interest bearing account in order to
set up a college fund of $75, 000 for Hunter when he is 18
years old. Tom and Mary want to generate the money with
a single investment now. If they put this investment in a
CD (savings account) with a fixed 9.2% interest rate, how
much money would they have to deposit now in order to
have $75, 000 for Hunter when he is 18 years old?
(Assume interest is compounded continuously in this CD).
UNDERSTANDING THE
ANNUAL PERCENTAGE YIELD (APY)
1.
Suppose that you invest $3000 in a savings account with an
8.2% APR, where the interest is compounded annually.
How much money will be in the account after 1 year?
By what percent does your money grow per year in this
account?
2.
Suppose that you invest $3000 in a savings account with an
8.2% APR, where the interest is compounded monthly.
How much money will be in the account after 1 year?
By what percent does your money grow per year in this
account?
(HINT: For the second question, find the amount accumulated in one year and
then use our formula for relative change.)
3.
Suppose that you invest $3000 in a savings account with an
8.2% APR, where the interest is compounded daily.
What is the Annual Percent Yield (APY) for this account?
For a savings account with an 8.2% APR,
how does the APY compare for different
numbers of compoundings per year?
# of times
compounded per
year
APR
1
(annually)
8.2%
12
(monthly)
8.2%
365
(daily)
8.2%
525, 600
(every minute)
8.2%
continuously
8.2%
APY
For a savings account with an 6% APR,
how does the APY compare for different
numbers of compoundings per year?
# of times
compounded per
year
APR
1
(annually)
6%
12
(monthly)
6%
365
(daily)
6%
525, 600
(every minute)
6%
continuously
6%
APY
NOTE: The APY only depends on the APR and
the number of compoundings per year, not
on the amount invested.
DOUBLING TIME
If I put a lump sum investment of $100 in an annually
compounded savings account with an APR of 7.6%, how
long will it take me to double my money?
DOUBLING TIME
The time it takes to double your money for a
given investment is called the doubling time.
The doubling time does not depend on the
amount invested. I depends only on the annual
percentage rate (APR) and on the number of
times that interest is compounded per year.
What would the doubling time be if the account were
compounded monthly?
What would the doubling time be if the account were
compounded daily?
What would the doubling time be if the account were
compounded continuously?
FINANCIAL GROWTH PRACTICE
Initial
# of Times
Investment Compounded
$1000
APR
12
8.5%
365
7%
APY
11.21 yrs
4
continuously
Amount in
25 Years
$10,000
12
$1000
Doubling
Time
$3,000
8%
9%
$100, 000
INFLATION AND BUYING POWER
1. Assuming a 3.5% average annual inflation rate,
how much buying power would $1000 have in
25 years?
2. Assuming a 3% average annual inflation rate,
how much buying power would $1000 have in
25 years?
3. Assuming a 3.5% average annual rate of
inflation, how much should we expect gas to
cost in 5 years if it costs $1.57 per gallon right
now?
4. Estimate the average annual inflation rate over
the last 27 years given that a home that sold for
$60,000 in 1975 went on the market in 2002 for
$132,000.
5. If grandma put $100 dollars under her mattress
in 1952, how much buying power does her
$100 have today, assuming an average annual
inflation rate of 3%?
UNDERSTANDING SAVINGS PLANS
Suppose that you deposit $200 at the end of each month into an account that
earns an APR of 12%. Fill in the chart below to see how your money grows in
the first few months. (Assume that the interest is compounded monthly.)
END OF
PRIOR
INTEREST ON PRIOR
MONTH
BALANCE
BALANCE
1
$0
$0
2
$200
3
4
5
DEPOSIT
$200
NEW
BALANCE
$200
Savings Plan Formula:
A = accumulated balance
PMT = regular payment (deposit) amount
n = number of payment periods per year
APR = annual percentage rate
Y = # years
SAVINGS PLAN EXAMPLES
1.
At age 22, Gina starts an IRA to save for retirement. She deposits
$150 at the end of each month. If she can count on a constant APR of
7.8%, how much will she have when she retires at age 65?
How much of her ending balance came from deposits? How much
came from interest?
2.
You want to purchase a car in 5 years and expect the car to cost
$12,000. Your bank offers a plan with a guaranteed 6.5% APR if you
make regular monthly deposits. How much should you deposit each
month to end up with $12,000 in 3 years?
What percent of your ending $12,000 balance will come from interest?
3.
Juan starts out a savings account with a deposit of $1000. He then
adds $200 per month for the next 10 years. In a similar account, Maria
deposits $2500 at the end of each year for 10 years.
If both Maria and Juan have accounts with a 6% APR, compare their
balances after 10 years.
Who deposits more money over the ten years? Who comes out ahead
in the end?
4.
If you have $2000 in an account at the start and wish to add $300 to it
monthly, how much will you have in 40 years if you assume an 8%
consistent growth rate compounded monthly?
What percent of your ending balance comes from your own deposits?
TOTAL AND ANNUAL RETURN
Say I invest in a particular stock and my money grows
from $1000 to $1900 in 7 years. How much has my
money grown over the 7 year period?
We can answer this by looking at total and annual return.
TOTAL AND ANNUAL RETURN
My total return is the % my money grows over
the entire investment. (It’s just a relative
change.)
My annual return is the average annual % that
my money grows by over the investment. (It’s
the constant APY that would have given me the
same result.)
What is my total return in the example above?
What is my annual return in the example above?
Suppose Dustin buys a home for $125,000 and sells it 3
years later for $106,000. What is his total return? His
annual return?
Suppose that you invested $10, 000 (lump sum) in a diversified mutual fund portfolio and
that the portfolio earned the rates given in the table for each respective fund.
If you decided at the beginning of the year to contribute 10% of your investment to each
fund except the Stock Index and the International stock (in which you place 20%), how
much was your balance by the end of the year?
FUND
Money
Market
Fixed
Income
Bond
Index
Stock
Index
Value
Stock
Growth
Stock
Small
Company
International
Stock
RETURN
5.19%
6.70%
8.30%
28.24%
14.28%
- 0.65%
- 7.30%
14.65%
CONTRIBUTION
10%
10%
10%
20%
10%
10%
10%
20%
(Notice that the contributions total up to 100% as they should. Also, remember that a
negative return means that you gain no interest, but in fact lose that percentage of your
principal amount.)
What was your total return overall?
What was your annual return?
PAYING OFF LOANS
1. Suppose that your student loans (totaling
$9,000) begin to accumulate interest at an APR
of 9.2% when you graduate. What would your
monthly payments have to be in order to pay off
the loans in 5 years? In 10 years?
2. Suppose you find a used car that you want to
buy for $8,600. The car will cost you $9,030
(after you add in sales tax.) If you pay a $2,000
down payment on the car and take out a loan for
the rest, how much will your monthly payment
be if you are able to finance a 5 year loan at a
7.8% interest rate?
An AMORTIZATION SCHEDULE is a table that shows how much of your payment
goes toward paying off the principal and how much goes toward interest over the lifetime
of a loan.
The first few months of an amortization schedule are shown below. The schedule is for a
$121,806 mortgage with a 7.8% interest rate. Fill in the blanks in the schedule.
DATE
PAYMENT
INTEREST
PRINCIPAL
LOAN BALANCE
$121, 806.00
1/2002
$924.04
$791.74
$132.30
2/2002
$924.04
$790.87
$133.16
3/2002
4/2002
$121,673.70
AMORTIZATION SCHEDULE
(LAST FEW MONTHS)
MONTH
PAYMENT
INTEREST
PRINCIPAL
BALANCE
295
$924.04
$35.22
$888.82
$4,529.51
296
$924.04
$29.44
$894.60
$3,634.92
297
$924.04
$23.63
$900.41
$2,734.50
298
$924.04
$17.77
$906.27
$1,828.24
299
$924.04
$11.88
$912.16
$916.08
CREDIT CARD DEBT
1. Suppose that Bob has a credit card balance of
$1,900.00. If his credit card has an APR of
20.99%, how much would Bob need to pay in
order to pay off his balance in 15 months
(assuming that he will not be charging anymore
items on his card)?
If Bob were to make payments of $100 per
month instead, how long would it take him to pay
off his debt?
2. Assume VISA has sent you a bill in which they
charge 17.8% for their APR and you have an
outstanding credit card balance of $1,456.38.
The minimum monthly payment they ask for is
$25.00. Also assume that you will not be
charging anymore items on your card as you
pay off your debt.
If you actually paid $25.00 every month, about
how many months would it take to pay off your
debt?
How much interest would this add up to over
that period of time?
TAX CALCULATION EXAMPLES
1.
In 2000, Dianne was single with no dependants. Her
adjusted gross income was $65,000. She paid $6,500
in tithing over the year, but has no other deductions or
tax credits. Calculate her taxable income and her tax
owed.
2.
Bill is a head of household in 2000 with two
dependant children and a taxable income of
$85,000. Assuming that Bill can not take any tax
credits, find Bill’s income tax.
3.
In 2000, Joe’s taxable income is $80,000. Since Joe
is single, this puts him in the 31% tax bracket.
If Joe itemizes deductions, how much will an
additional $1000 contribution to charity save him in
taxes? How much will an additional $1000 tax credit
save him?
If Joe takes the standard deduction, how much will an
additional $1000 contribution to charity save him in
taxes? How much will an additional $1000 tax credit
save him?
In general, which saves more money, a tax deduction
or a tax credit of the same size?
4.
In 2000, Kiersten and Danny had adjusted gross
incomes of $18,000 and $22,800 respectively. They
had no dependants and they filed jointly as a married
couple, claiming the standard deduction. How much
tax did they owe?
How much would they have owed combined if they’d
both been single?
Notice that the tax they owe as a married couple is
more than the sum of what their individual taxes
would have been. This is called the marriage
penalty. The marriage penalty is the additional tax
they paid over what they would have paid if each had
been single.
Find the amount of Kiersten and Danny’s marriage
penalty.