THREE WAYS OF USINGPERCENTAGES

1. AS FRACTIONS

58% of the people surveyed reported thatthey 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 girlsmake up what percentage of the class?

2. In a group of 80 people, if 35% of them havebrown eyes, how many of the people have browneyes?

3. 42 is what percent of 80?

4. If you get 82% on a test with 34 questions of equalpoint value, how many questions did you get right?

5. 34 team field goals out of 60 attempted is whatshooting percentage for the team?

6. In a history class of 120 students, 60% are femalesand 25% of the females have blue eyes. Whatpercentage of the history students in the class areblue-eyed females?

7. Assume that to earn a C grade or higher in acollege course you need at least an average of70% on the five exams that are each worth 100points. You have scores of 65, 81, 72, 55 so far inthe semester. What is the lowest score you canreceive on your fifth exam in order to earn a C orhigher?

PERCENTAGES TO DESCRIBE CHANGE

8. In a high school survey of 2500 students, 310students reported that they had used drugs. Thenext year, 382 students reported that they hadused 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 in1997 and 8, 551 in 1998, then:

There were _________ more students at Rick’s in1998 than in 1997.

There were _______% more students at Rick’s in1998 than in 1997.

10. Suppose two companies experience layoffs. Ifcompany A goes from 560 to 520 employees andcompany B goes from 1525 to 1450 employees,which company layed off more people in absoluteterms? In relative terms?

PERCENTAGES OF PERCENTAGES

11. Suppose your bank account interest rate decreasesfrom 5% to 4%. What is the absolute change? What is the relative change?

12. The percentage of all bachelor’s degrees awardedto women increased from 44% in 1972 to 54% in1992. Find the absolute and relative change.

PERCENTAGES FOR COMPARISON

13. Suppose your wage (per hour) is $8.50 and yourfriend’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 inthe U.S., while 209,000 females had the operation. What is the absolute difference and relativedifference 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 salaryis _______% less than Jill’s.

16. The population of Montana is 20% less than thepopulation of New Hampshire, so Montana’spopulation is ________% of New Hampshire’spopulation.

17. You buy a computer in Idaho for $799. Sales tax inIdaho is 5%. What do you pay for the computer?

COMPOUND INTEREST:

General Formula:

A = accumulated balanceP = 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 balanceP = principalAPR = annual percentage rate

(as a decimal) Y = number of years

Interest Paid Multiple Times a Year:

A = accumulated balanceP = principalAPR = annual percentage rateY = number of yearsn = number of compounding

periods per year

USING THE COMPOUND INTEREST FORMULAS

1. Suppose that you invest $2000 in a savingsaccount with a 3.5% APR, where the interest iscompounded annually. How much will theaccount be worth in 5 years?

2. How much would your need to invest in asavings account with a 4% APR, (where theinterest is compounded annually), if you wantthe account to be worth $15,000 in 20 years.

3. Suppose that you invest $10,000 in a 10 yearsavings CD with a 6.8% APR. If the interest iscompounded quarterly, how much money willbe in the CD be worth when it matures in 10years?

Continuously Compounded Interest:

A = accumulated balance

P = principal

APR = annual percentage rate

Y = # years

e . 2.71828 (e is a mathematicalconstant kind of like B. Thereis a button on your calculatorfor e)

CONTINUOUS COMPOUNDINGEXAMPLES

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 theaccount, how much money will be in the account when youretire at age 65?

How much money would you lose by waiting until you were25 years old to invest the same amount of money in thesame account?

2. Tom and Mary have a new baby boy, Hunter. They want toput money now into an interest bearing account in order toset up a college fund of $75, 000 for Hunter when he is 18years old. Tom and Mary want to generate the money witha single investment now. If they put this investment in aCD (savings account) with a fixed 9.2% interest rate, howmuch money would they have to deposit now in order tohave $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 an8.2% 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 thisaccount?

2. Suppose that you invest $3000 in a savings account with an8.2% 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 thisaccount?

(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 an8.2% 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 differentnumbers of compoundings per year?

# of timescompounded per

yearAPR APY

1(annually)

8.2%

12(monthly)

8.2%

365(daily)

8.2%

525, 600(every minute)

8.2%

continuously 8.2%

For a savings account with an 6% APR, how does the APY compare for differentnumbers of compoundings per year?

# of timescompounded per

yearAPR APY

1(annually)

6%

12(monthly)

6%

365(daily)

6%

525, 600(every minute)

6%

continuously 6%

NOTE: The APY only depends on the APR andthe number of compoundings per year, noton the amount invested.

DOUBLING TIME

If I put a lump sum investment of $100 in an annuallycompounded savings account with an APR of 7.6%, howlong will it take me to double my money?

DOUBLING TIME

The time it takes to double your money for agiven investment is called the doubling time. The doubling time does not depend on theamount invested. I depends only on the annualpercentage rate (APR) and on the number oftimes that interest is compounded per year.

What would the doubling time be if the account werecompounded monthly?

What would the doubling time be if the account werecompounded daily?

What would the doubling time be if the account werecompounded continuously?

FINANCIAL GROWTH PRACTICE

Initial Investment

# of Times Compounded APR APY Doubling

Time Amount in 25 Years

$1000 12 8.5%

365 7%

$10,000

12

11.21 yrs $3,000

$1000 4 8%

continuously 9% $100, 000

INFLATION AND BUYING POWER

1. Assuming a 3.5% average annual inflation rate,how much buying power would $1000 have in25 years?

2. Assuming a 3% average annual inflation rate,how much buying power would $1000 have in25 years?

3. Assuming a 3.5% average annual rate ofinflation, how much should we expect gas tocost in 5 years if it costs $1.57 per gallon rightnow?

4. Estimate the average annual inflation rate overthe 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 mattressin 1952, how much buying power does her$100 have today, assuming an average annualinflation rate of 3%?

UNDERSTANDING SAVINGS PLANS

Suppose that you deposit $200 at the end of each month into an account thatearns an APR of 12%. Fill in the chart below to see how your money grows inthe first few months. (Assume that the interest is compounded monthly.)

END OF

MONTH

PRIOR

BALANCE

INTEREST ON PRIOR

BALANCEDEPOSIT

NEW

BALANCE

1 $0 $0 $200 $200

2 $200

3

4

5

Savings Plan Formula:

A = accumulated balancePMT = regular payment (deposit) amountn = number of payment periods per yearAPR = annual percentage rateY = # 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 of7.8%, how much will she have when she retires at age 65?

How much of her ending balance came from deposits? How muchcame 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 youmake regular monthly deposits. How much should you deposit eachmonth 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 thenadds $200 per month for the next 10 years. In a similar account, Mariadeposits $2500 at the end of each year for 10 years.

If both Maria and Juan have accounts with a 6% APR, compare theirbalances after 10 years.

Who deposits more money over the ten years? Who comes out aheadin the end?

4. If you have $2000 in an account at the start and wish to add $300 to itmonthly, 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 growsfrom $1000 to $1900 in 7 years. How much has mymoney 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 overthe entire investment. (It’s just a relativechange.)

My annual return is the average annual % thatmy money grows by over the investment. (It’sthe constant APY that would have given me thesame 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 3years later for $106,000. What is his total return? Hisannual 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 APRof 9.2% when you graduate. What would yourmonthly payments have to be in order to pay offthe loans in 5 years? In 10 years?

2. Suppose you find a used car that you want tobuy for $8,600. The car will cost you $9,030(after you add in sales tax.) If you pay a $2,000down payment on the car and take out a loan forthe rest, how much will your monthly paymentbe if you are able to finance a 5 year loan at a7.8% interest rate?

An AAMMOORRTTIIZZAATTIIOONN SSCCHHEEDDUULLEE 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 $121,673.70

2/2002 $924.04 $790.87 $133.16

3/2002

4/2002

AAMMOORRTTIIZZAATTIIOONN SSCCHHEEDDUULLEE ((LLAASSTT FFEEWW MMOONNTTHHSS))

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 of20.99%, how much would Bob need to pay inorder to pay off his balance in 15 months(assuming that he will not be charging anymoreitems on his card)?

If Bob were to make payments of $100 permonth instead, how long would it take him to payoff his debt?

2. Assume VISA has sent you a bill in which theycharge 17.8% for their APR and you have anoutstanding credit card balance of $1,456.38.The minimum monthly payment they ask for is$25.00. Also assume that you will not becharging anymore items on your card as youpay off your debt.

If you actually paid $25.00 every month, abouthow many months would it take to pay off yourdebt?

How much interest would this add up to overthat period of time?

TAX CALCULATION EXAMPLES

1. In 2000, Dianne was single with no dependants. Heradjusted gross income was $65,000. She paid $6,500in tithing over the year, but has no other deductions ortax credits. Calculate her taxable income and her taxowed.

2. Bill is a head of household in 2000 with twodependant children and a taxable income of$85,000. Assuming that Bill can not take any taxcredits, find Bill’s income tax.

3. In 2000, Joe’s taxable income is $80,000. Since Joeis single, this puts him in the 31% tax bracket.

If Joe itemizes deductions, how much will anadditional $1000 contribution to charity save him intaxes? How much will an additional $1000 tax creditsave him?

If Joe takes the standard deduction, how much will anadditional $1000 contribution to charity save him intaxes? How much will an additional $1000 tax creditsave him?

In general, which saves more money, a tax deductionor a tax credit of the same size?

4. In 2000, Kiersten and Danny had adjusted grossincomes of $18,000 and $22,800 respectively. Theyhad no dependants and they filed jointly as a marriedcouple, claiming the standard deduction. How muchtax did they owe?

How much would they have owed combined if they’dboth been single?

Notice that the tax they owe as a married couple ismore than the sum of what their individual taxeswould have been. This is called the marriagepenalty. The marriage penalty is the additional taxthey paid over what they would have paid if each hadbeen single.

Find the amount of Kiersten and Danny’s marriagepenalty.