Thinking Mathematically Chapter 8 Consumer Math. Thinking Mathematically Section 1 Percent

Thinking Thinking MathematicallyMathematically

Chapter 8Chapter 8

Consumer MathConsumer Math


Section 1Section 1

PercentPercent

What is a percent?What is a percent?

1.1. A A percentpercent, such as 12%, represents a , such as 12%, represents a fraction of 100. So 12% is the same as fraction of 100. So 12% is the same as 12/10012/100

2.2. Much of what we work with in consumer Much of what we work with in consumer mathematics is based on percents: interest mathematics is based on percents: interest rates on loans and credit cards, for rates on loans and credit cards, for example.example.

Expressing a Fraction as a PercentExpressing a Fraction as a Percent

1.1. Divide the numerator by the denominator.Divide the numerator by the denominator.

2.2. Multiply the quotient by 100. Multiply the quotient by 100. Equivalently, move the decimal point in Equivalently, move the decimal point in the quotient two places to the right.the quotient two places to the right.

3.3. Add a percent sign.Add a percent sign.

Multiplying and dividing by Multiplying and dividing by 100100

• Whenever we multiply a decimal by 100, Whenever we multiply a decimal by 100, we merely move the decimal point two we merely move the decimal point two places to the right. When we divide, we places to the right. When we divide, we move two places to the left.move two places to the left.

• We fill with zeros as necessary.We fill with zeros as necessary.

multiply .12 by 100multiply .12 by 100 1212..multiply 1.2 by 100multiply 1.2 by 100 1 21 2.. 00

divide 12 by 100divide 12 by 100 1212..

Example 1: Expressing a Fraction Example 1: Expressing a Fraction as a Percentas a Percent

Express 5/8 as a percent.Express 5/8 as a percent.

SolutionSolution

Step 1Step 1 Divide the numerator by the Divide the numerator by the denominator.denominator.

5 5 8 = 0.625 8 = 0.625

Step 2Step 2 Multiply the quotient by 100. Multiply the quotient by 100.

0.625 x 100 = 62.50.625 x 100 = 62.5

Step 3Step 3 Add a percent sign. Add a percent sign.

62.5%62.5%

Expressing a Decimal Number as a Expressing a Decimal Number as a PercentPercent

Same as expressing fractions, except it's Same as expressing fractions, except it's already in decimal format.already in decimal format.

a)a) Move the decimal point two places to the Move the decimal point two places to the right.right.

b)b) Add a percent sign.Add a percent sign.

Example 2: Expressing a Decimal Example 2: Expressing a Decimal Number as a PercentNumber as a Percent

Express 0.47 as a percent.Express 0.47 as a percent.

SolutionSolution

Step 1Step 1 Move the decimal point two places to Move the decimal point two places to the right.the right.

0.47 0.47 47 47

Step 2Step 2 Add a percent sign. Add a percent sign.

47 47 47% 47%

Expressing a Percent as a DecimalExpressing a Percent as a Decimal

Just go the other way:Just go the other way:1. Move the decimal point two places to the left.1. Move the decimal point two places to the left.

2. Remove the percent sign2. Remove the percent sign

Example 3: Expressing Percents Example 3: Expressing Percents as Decimalsas Decimals

Express 180% as a decimal.Express 180% as a decimal.

SolutionSolution

Step 1Step 1 Move the decimal point two places to Move the decimal point two places to the left.the left.

180% 180% 1.80% 1.80%

Step 2Step 2 Remove the percent sign Remove the percent sign

1.80% 1.80% 1.80 or 1.8 1.80 or 1.8

ReviewReview

Fraction Fraction DecimalDecimal PercentPercent

5/85/8 .625.625 66 22 55.. %%

400%400%44 00 00 %%..

5%5%55%%..00

Finding Percent IncreaseFinding Percent Increase

• Use subtraction to find the amount of increase.Use subtraction to find the amount of increase.• Find the fraction for the percent increase, Find the fraction for the percent increase,

using using

• Find the percent increase by expressing the Find the percent increase by expressing the fraction in step 2 as a percent.fraction in step 2 as a percent.

Finding Percent IncreaseFinding Percent Increase• At the convenience store, the 6-pack that was At the convenience store, the 6-pack that was

$5.00 last week is now $5.50. What is the percent $5.00 last week is now $5.50. What is the percent increase in price?increase in price?

• Use subtraction to find the amount of increase.Use subtraction to find the amount of increase.$5.50 - $5.00 = $0.50$5.50 - $5.00 = $0.50

• Find the fraction for the percent increase, usingFind the fraction for the percent increase, using amount of increaseamount of increase $0.50 $0.50

original amount $5.00original amount $5.00 • Find the percent increase by expressing the Find the percent increase by expressing the

fraction in step 2 as a percent. fraction in step 2 as a percent. • 0.1 = 10%0.1 = 10%

==

Finding Percent DecreaseFinding Percent Decrease

• Use subtraction to find the amount of Use subtraction to find the amount of decrease.decrease.

• Find the fraction for the percent decrease, Find the fraction for the percent decrease,

usingusing

• Find the percent decrease by expressing the Find the percent decrease by expressing the fraction in step 2 as a percent.fraction in step 2 as a percent.

Finding Percent DecreaseFinding Percent Decrease

• At the convenience store, another 6-pack that was $5.00 At the convenience store, another 6-pack that was $5.00 last week is now $4.50. What is the percent decrease in last week is now $4.50. What is the percent decrease in price?price?

• Use subtraction to find the amount of decrease.Use subtraction to find the amount of decrease.

• $5.00 - $4.50 = $0.50$5.00 - $4.50 = $0.50

• Find the fraction for the percent decrease, usingFind the fraction for the percent decrease, using

• Find the percent decrease by expressing the fraction in Find the percent decrease by expressing the fraction in step 2 as a percent. step 2 as a percent.

• 0.1 = 10%0.1 = 10%

Increase and DecreaseIncrease and Decrease

• At the convenience store, one week a 6-pack At the convenience store, one week a 6-pack increases 10% in price. The next week the price increases 10% in price. The next week the price decreases by 10%.decreases by 10%.

• Has the price gone back to the original price?Has the price gone back to the original price?• No. There's a cumulative effect. 10% of the new No. There's a cumulative effect. 10% of the new

price is more money than 10% of the old price. price is more money than 10% of the old price. So the price goes down MORE than it has gone So the price goes down MORE than it has gone up. up.

• It costs less now!It costs less now!• Think about it.Think about it.

Increase and DecreaseIncrease and Decrease

Starting priceStarting price: $5.00: $5.00

$5.00 + 10% of $5.00 = $5.50.$5.00 + 10% of $5.00 = $5.50.

$5.50 – 10% of $5.50 = $5.50 - 55$5.50 – 10% of $5.50 = $5.50 - 55¢¢ = $4.95 = $4.95

Final priceFinal price: $4.95: $4.95


Section 2Section 2

Simple InterestSimple Interest

What is simple interest?What is simple interest?

• Whenever you borrow or lend money, a certain Whenever you borrow or lend money, a certain percentage of the total amount will be paid in percentage of the total amount will be paid in addition to paying back the loan.addition to paying back the loan.

• Why?Why?• Because while you have the money, the lender Because while you have the money, the lender

doesn't and can't invest it and make money off it.doesn't and can't invest it and make money off it.• Simple interestSimple interest is a fixed percentage of the is a fixed percentage of the

amount borrowed that will be paid for each of the amount borrowed that will be paid for each of the years the loan is not paid off.years the loan is not paid off.

What is simple interest?What is simple interest?

• When you borrow or lend money, the When you borrow or lend money, the amount borrowed or loaned is called the amount borrowed or loaned is called the principalprincipal ((PP))..

• The The interestinterest raterate ((rr)) is a percentage of is a percentage of the principal that will be paid back in the principal that will be paid back in addition to the principal.addition to the principal.

• The The interestinterest ((II)) is the amount paid back is the amount paid back in addition to the principal.in addition to the principal.

Calculating Simple InterestCalculating Simple Interest

Interest =(Principal)(Interest Rate)(Time)Interest =(Principal)(Interest Rate)(Time) I = PrtI = Prt

TheThe accumulatedaccumulated amount amount ((AA)) is the total is the total value including the principal.value including the principal.

Accumulated Amount:Accumulated Amount: A = P + IA = P + I

A = P + Prt = PA = P + Prt = P(1(1+rt+rt))

Calculating Simple Interest for a YearCalculating Simple Interest for a Year

You deposit $2000 in a savings account at You deposit $2000 in a savings account at Hometown Bank, which has a rate of 6%. Hometown Bank, which has a rate of 6%. Find the interest at the end of the first Find the interest at the end of the first year.year.

SolutionSolution

The amount deposited, or principal The amount deposited, or principal ((PP),), is is $2000. The rate ($2000. The rate (rr) is 6%, or 0.06. The time ) is 6%, or 0.06. The time of the deposit (of the deposit (t)t) is one year. The interest is: is one year. The interest is:

I = Prt I = Prt = =

At the end of the first year, the interest is At the end of the first year, the interest is $120. You can withdraw the $120 in $120. You can withdraw the $120 in interest, and you still have $2000 in the interest, and you still have $2000 in the savings account. savings account.

($2000)(0.06)(1)($2000)(0.06)(1) = $120= $120.

Calculating Simple InterestCalculating Simple Interest

You deposit $2000 in a savings account at You deposit $2000 in a savings account at Hometown Bank, which has a rate of 6%. Hometown Bank, which has a rate of 6%. Find the interest after 5 years.Find the interest after 5 years.

How much will be accumulated in the How much will be accumulated in the bank after 5 years if you never take out bank after 5 years if you never take out any money?any money?

SolutionSolution

The amount deposited, or principal The amount deposited, or principal ((PP),), is is $2000. The rate ($2000. The rate (rr) is 6%, or 0.06. The ) is 6%, or 0.06. The time of the deposit (time of the deposit (tt) is 5 years. The ) is 5 years. The interest is:interest is:

I = PrtI = Prt = =

At the end of 5 years, the interest is $600.At the end of 5 years, the interest is $600.

Accumulated Amount: Accumulated Amount: A = P + IA = P + I

($2000)(0.06)(5)($2000)(0.06)(5) = $600= $600.

= = $2600$2600

Computing the Interest RateComputing the Interest Rate

You deposit $2000 in a savings account at You deposit $2000 in a savings account at Hometown Bank, and after a year you Hometown Bank, and after a year you discover you now have $2180. Find the discover you now have $2180. Find the interest rate.interest rate.

SolutionSolution

The amount deposited, or principal (The amount deposited, or principal (PP), is ), is $2000. Since you now have $2180, and $2000. Since you now have $2180, and the time is 1 year (the time is 1 year (tt = 1) = 1)

A = PA = P(1+(1+rtrt) = ) = PP(1+(1+rr))

$2180 = ($2000)(1+$2180 = ($2000)(1+rr) = $2000 + 2000) = $2000 + 2000rr

rr = = $180/$2000 = 0.09$180/$2000 = 0.09

0.09 = 9%,0.09 = 9%, which is the interest rate. which is the interest rate.

Discounted LoansDiscounted Loans

Similar to the Simple Interest Loan is Similar to the Simple Interest Loan is something called a something called a Discounted Loan.Discounted Loan.

If the Simple Interest Loan is unrealistic If the Simple Interest Loan is unrealistic and hardly ever seen, the Discounted and hardly ever seen, the Discounted Loan is even rarer.Loan is even rarer.

I have no idea why the book even brings I have no idea why the book even brings it up.it up.


In a In a Discounted LoanDiscounted Loan, the borrower must , the borrower must pay the interest "up front" at the time of pay the interest "up front" at the time of the loan. the loan.

Therefore the borrower only gets the Therefore the borrower only gets the desired amount minus the interest charged. desired amount minus the interest charged.

However, since he has already paid the However, since he has already paid the interest, he only has to pay the loan interest, he only has to pay the loan amount when he pays back.amount when he pays back.


For example, you want to borrow $10,000 For example, you want to borrow $10,000 and the lender offers you a Discounted Loan for and the lender offers you a Discounted Loan for one year at the rate of 5%.one year at the rate of 5%.

II = = PrtPrt The lender gives you $10,000 and you The lender gives you $10,000 and you

immediatelyimmediately pay the lender $500. Or, simply put,pay the lender $500. Or, simply put, the lender gives you $9,500 and, after a the lender gives you $9,500 and, after a

year, you have to pay back $10,000year, you have to pay back $10,000

= ($10,000)(.05)(1) = $500= ($10,000)(.05)(1) = $500


The lender gives you $9,500 and, after a year, you The lender gives you $9,500 and, after a year, you have to pay back $10,000.have to pay back $10,000.

The The Effective Interest RateEffective Interest Rate is computed by is computed by

A = PA = P(1+(1+rtrt))

$10,000 = $9,500(1 +$10,000 = $9,500(1 + rr) since ) since tt is 1 year is 1 year

$500 = $9,500$500 = $9,500rr

rr = about 0.0526 which is 5.26% !!!= about 0.0526 which is 5.26% !!!

Sometimes, 5% isn't really 5%!!!Sometimes, 5% isn't really 5%!!!

Quick QuizQuick Quiz A mother puts $1000 in an account for her new-A mother puts $1000 in an account for her new-

born daughter. The account offers 5% simple born daughter. The account offers 5% simple interest. If the account is left alone for the next 20 interest. If the account is left alone for the next 20 years, how much will be in the account when the years, how much will be in the account when the daughter turns 20?daughter turns 20?

AA = = PP((1 + rt1 + rt)), with , with P = P = $1000$1000, , r = r = .05.05 (or (or 1/201/20), and ), and t t = = 2020..

A = A = $1000(1 + .05 $1000(1 + .05 20) = $1000 (1 + 1) 20) = $1000 (1 + 1) A = A = $1000 $1000 2 = $2000. 2 = $2000.


Section 3Section 3

Compound InterestCompound Interest

What is Compound Interest?What is Compound Interest?

Since the lender could have earned money Since the lender could have earned money each year of the loan at some interest rate, the each year of the loan at some interest rate, the amount of principal is recomputed every year amount of principal is recomputed every year (if the loan is "(if the loan is "compounded annuallycompounded annually").").

After one year, the lender could have made After one year, the lender could have made money on the principal. The lender should money on the principal. The lender should compute the amount he or she could make the compute the amount he or she could make the next year on the accumulated amount.next year on the accumulated amount.

Simple Interest vs. Compound Simple Interest vs. Compound InterestInterest

$2000$2000 $120$120 $2120$2120

$2120$2120 127.20127.20 2247.202247.20

2247.202247.20 134.83134.83 2382.032382.03

2382.032382.03 142.92142.92 2524.952524.95

2524.952524.95 151.50151.50 2676.452676.45

$2000$2000 $120$120 $2120$2120

$2000$2000 $120$120 $2240$2240

$2000$2000 $120$120 $2360$2360

$2000$2000 $120$120 $2480$2480

$2000$2000 $120$120 $2600$2600

$2000 at 6%$2000 at 6%

SimpleSimple Compounded AnnuallyCompounded Annually

1

2

3

4

5

YearYear

Calculating the Amount in an Calculating the Amount in an Account for Compound Account for Compound

Interest Paid Once a YearInterest Paid Once a YearIf If PP dollars are deposited at a rate dollars are deposited at a rate rr, in decimal form, subject to , in decimal form, subject to

compound interest, then the amount, compound interest, then the amount, AA, of money in the account , of money in the account after 1 year is given by:after 1 year is given by:

A = P(1+r).A = P(1+r).after 2 yearsafter 2 years

AA = = PP(1+(1+rr) (1+) (1+rr) = ) = PP(1+(1+rr))22........

after after tt years years AA = = PP(1+(1+rr)(1+)(1+rr)(1+)(1+rr)(1+)(1+rr) = ) = PP(1+(1+rr))tt..

tt timestimes

Calculating the Amount in an Calculating the Amount in an Account for Compound Account for Compound

Interest Paid Once a YearInterest Paid Once a YearIf If PP dollars are deposited at a rate dollars are deposited at a rate rr, in , in decimal form, subject to compound interest, decimal form, subject to compound interest, then the amount, then the amount, AA, of money in the account , of money in the account after after tt years is given by: years is given by:

AA = = PP(1+(1+rr))tt..

Example: Using the Compound Example: Using the Compound Interest FormulaInterest Formula

You deposit You deposit PP = = $2000 $2000 in a savings in a savings account at Hometown Bank, which has account at Hometown Bank, which has a rate of 6% (a rate of 6% (rr = = .06.06). ).

a.a. Find the amount, Find the amount, AA, of money in the , of money in the account after 3 years subject to account after 3 years subject to compound interest.compound interest.

b.b. Find the interest.Find the interest.

SolutionSolutionThe amount deposited, or principal The amount deposited, or principal ((PP),),

is $2000. The rate (is $2000. The rate (rr) is 6%, or 0.06. ) is 6%, or 0.06. The time of the deposit (The time of the deposit (tt) is three ) is three years. The amount in the account years. The amount in the account after three years is:after three years is:AA = = PP(1+(1+rr))tt = = $2000(1+0.06)$2000(1+0.06)33 = = $2000(1.06)$2000(1.06)33

= $2382.03= $2382.03

Rounded to the nearest cent, the amount in the Rounded to the nearest cent, the amount in the savings account after three years is savings account after three years is $2382.03$2382.03Because the amount in the account is Because the amount in the account is $2382.03 $2382.03 and the original principal is and the original principal is $2000$2000, the interest , the interest is is $2382.03 - $2000 = $382.03$2382.03 - $2000 = $382.03

Other compound interest loansOther compound interest loans

Loans are not usually compounded annually. Loans are not usually compounded annually. More typical are loans compounded More typical are loans compounded quarterly quarterly (4 times a year), (4 times a year), monthlymonthly [such as car loans and [such as car loans and mortgages] (12 times a year) or even mortgages] (12 times a year) or even dailydaily [such [such as credit cards] (360 times a year).as credit cards] (360 times a year).

This causes us to modify our formula.This causes us to modify our formula.

Calculating the Amount in an Account Calculating the Amount in an Account for Compound Interest Paid n Times a for Compound Interest Paid n Times a

YearYearIf If PP dollars are deposited at rate dollars are deposited at rate rr, in , in decimal form, subject to compound decimal form, subject to compound interest paid interest paid nn times per year, then the times per year, then the amount,amount, A A, of money in the account after , of money in the account after tt years is given by: years is given by:

AA PP 11rr

nn

ntnt

nn: semi-annually: 2: semi-annually: 2n: quarterly: 4n: daily: 365nn: monthly: 12: monthly: 12


You take out a loan for You take out a loan for $4,000 $4,000 at an annual at an annual interest rate of interest rate of 5.25%5.25% compounded compounded monthlymonthly. If . If you pay the loan back 10 years from now, how you pay the loan back 10 years from now, how much will you owe?much will you owe?

We use the formula from the previous slide, We use the formula from the previous slide, with with PP = = $4000$4000, , rr = = .0525.0525, , tt = = 1010 and and nn = = 1212


We use the formula from the previous slide, with We use the formula from the previous slide, with P P = = $4000$4000, , rr = = .0525.0525, , tt = = 1010 and and nn = = 12 12

On your calculator,On your calculator,

1.1. Enter .0525Enter .0525

2.2. Divide by 12Divide by 12

3.3. Add 1Add 1

4.4. Raise the result to the 12 Raise the result to the 12 10 =120 10 =120thth power power

5.5. Multiply by $4000Multiply by $4000

.0525.0525

0.00437500000.0043750000

1.00437500001.0043750000

1.68852421381.6885242138

$6754.10$6754.10

Continuous CompoundingContinuous Compounding

• What if the compounding period is less than What if the compounding period is less than a day? How about each hour? Each a day? How about each hour? Each minute? Each second? Each nanosecond?minute? Each second? Each nanosecond?

• The ultimate in compounding is called The ultimate in compounding is called continuous compoundingcontinuous compounding..

• When continuous compounding occurs, the When continuous compounding occurs, the formula is different:formula is different:

A = PeA = Pertrt

Continuous CompoundingContinuous Compounding

• When continuous compounding occurs, the When continuous compounding occurs, the formula is different:formula is different:

A = PeA = Pertrt

• ee is a mathematical quantity equal tois a mathematical quantity equal to

(1 + 1/(1 + 1/nn))nn

when when nn becomes really big. becomes really big.

• ee ≈≈ 2.718282.71828

Comparing LoansComparing LoansP P = = $4000$4000, , rr = = .0525.0525, , tt = = 1010• Simple interest:Simple interest:

AA = 4000(1+.0525 = 4000(1+.052510) = 4000(1.525) = $6,10010) = 4000(1.525) = $6,100• Compounded annually: Compounded annually:

AA = 4000(1+.0525) = 4000(1+.0525)1010 = $6,672.38 = $6,672.38• Compounded monthly: Compounded monthly:

AA = 4000(1+.0525/12) = 4000(1+.0525/12)120120 = $6,754.10 = $6,754.10• Compounded daily: Compounded daily:

AA = 4000(1+.0525/360) = 4000(1+.0525/360)36003600 = $6,761.58 = $6,761.58Compounded continuously: Compounded continuously:

AA = 4000 = 4000ee.525.525 = $4000 = $4000 2.71828 2.71828.525.525 = $6,761.84 = $6,761.84

Effective Annual YieldEffective Annual Yield

TheThe effective annual yieldeffective annual yield is the simple is the simple interest rate that produces the same interest rate that produces the same amount of money in an account at the amount of money in an account at the end of one year as there is when the end of one year as there is when the account is subjected to compound account is subjected to compound interest at a stated rate.interest at a stated rate.


The Effective Annual Yield, or Effective The Effective Annual Yield, or Effective Rate Rate YY is computed as follows:is computed as follows:

YY = (1 + ) = (1 + )nn - 1 - 1rrnn


A bank compounds interest daily (360 A bank compounds interest daily (360 times a year) at 5% on money in an times a year) at 5% on money in an account. After one year, you would haveaccount. After one year, you would have

YY = = (1 + .05/360)(1 + .05/360)360 360 - 1- 1

YY = = 1.0513 – 1 = 0.05131.0513 – 1 = 0.0513

So you are earning at an effective annual So you are earning at an effective annual rate of about rate of about 5.13%.5.13%.


Section 4Section 4

, Stocks and Bonds, Stocks and BondsAnnuitiesAnnuities

AnnuitiesAnnuities

• An An annuityannuity is a savings account in which is a savings account in which an equal amount of money is paid each an equal amount of money is paid each year (or each month, or some other year (or each month, or some other period).period).

• An IRA is an example of an annuityAn IRA is an example of an annuity

AnnuitiesAnnuities

• In doing annuity calculations, the In doing annuity calculations, the Principal (Principal (PP) is the amount of each ) is the amount of each payment into the annuity.payment into the annuity.

• Pretty awful, isn’t it? (It gets worse!)Pretty awful, isn’t it? (It gets worse!)

AnnuitiesAnnuities

• For example, invest $1200 each year into an For example, invest $1200 each year into an annuity at 8% yield.annuity at 8% yield.

• Formula:Formula:

• After 10 years, this annuity is worthAfter 10 years, this annuity is worth

• on $12,000 invested.on $12,000 invested.

AnnuitiesAnnuities• If we make If we make nn payments a year (for example, payments a year (for example,

monthly payments of $100 for 10 years at 8%), monthly payments of $100 for 10 years at 8%), the formula becomes:the formula becomes:

• on $12,000 invested.on $12,000 invested.

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Thinking Mathematically Chapter 8 Consumer Math. Thinking Mathematically Section 1 Percent