Thinking Mathematically Chapter 8 Consumer Math

Thinking Mathematically Section 1 Percent

What is a percent? 1.A percent, such as 12%, represents a fraction of 100. So 12% is the same as 12/100 2.Much of what we work with in consumer mathematics is based on percents: interest rates on loans and credit cards, for example.

Expressing a Fraction as a Percent 1.Divide the numerator by the denominator. 2.Multiply the quotient by 100. Equivalently, move the decimal point in the quotient two places to the right. 3.Add a percent sign.

Multiplying and dividing by 100 Whenever we multiply a decimal by 100, we merely move the decimal point two places to the right. When we divide, we move two places to the left.Whenever we multiply a decimal by 100, we merely move the decimal point two places to the right. When we divide, we move two places to the left. We fill with zeros as necessary.We fill with zeros as necessary. multiply.12 by multiply 1.2 by divide 12 by

Example 1: Expressing a Fraction as a Percent Express 5/8 as a percent.

Solution Step 1 Divide the numerator by the denominator. 5  8 = Step 2 Multiply the quotient by x 100 = 62.5 Step 3 Add a percent sign. 62.5%

Expressing a Decimal Number as a Percent Same as expressing fractions, except it's already in decimal format. a)Move the decimal point two places to the right. b)Add a percent sign.

Example 2: Expressing a Decimal Number as a Percent Express 0.47 as a percent.

Solution Step 1 Move the decimal point two places to the right  47 Step 2 Add a percent sign. Step 2 Add a percent sign. 47  47%

Expressing a Percent as a Decimal Just go the other way: 1. Move the decimal point two places to the left. 2. Remove the percent sign

Example 3: Expressing Percents as Decimals Express 180% as a decimal.

Solution Step 1 Move the decimal point two places to the left. 180%  1.80% Step 2 Remove the percent sign 1.80%  1.80 or 1.8

Review Fraction DecimalPercent 5/ % 400% %. 5% 5%.0

Finding 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, usingFind the fraction for the percent increase, using Find the percent increase by expressing the fraction in step 2 as a percent.Find the percent increase by expressing the fraction in step 2 as a percent.

Finding Percent Increase At the convenience store, the 6-pack that was $5.00 last week is now $5.50. What is the percent increase in price?At the convenience store, the 6-pack that was $5.00 last week is now $5.50. What is the percent increase in price? Use subtraction to find the amount of increase.Use subtraction to find the amount of increase. $ $5.00 = $0.50 Find the fraction for the percent increase, usingFind the fraction for the percent increase, using amount of increase $0.50 amount of increase $0.50 original amount $5.00 original amount $5.00 Find the percent increase by expressing the fraction in step 2 as a percent.Find the percent increase by expressing the fraction in step 2 as a percent. 0.1 = 10% 0.1 = 10% =

Finding Percent Decrease Use subtraction to find the amount of decrease.Use subtraction to find the amount of decrease. Find the fraction for the percent decrease, usingFind the fraction for the percent decrease, using Find the percent decrease by expressing the fraction in step 2 as a percent.Find the percent decrease by expressing the fraction in step 2 as a percent.

Finding Percent Decrease At the convenience store, another 6-pack that was $5.00 last week is now $4.50. What is the percent decrease in price?At the convenience store, another 6-pack that was $5.00 last week is now $4.50. What is the percent decrease in price? Use subtraction to find the amount of decrease.Use subtraction to find the amount of decrease. $ $4.50 = $0.50$ $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 step 2 as a percent.Find the percent decrease by expressing the fraction in step 2 as a percent. 0.1 = 10% 0.1 = 10%

Increase and Decrease At the convenience store, one week a 6-pack increases 10% in price. The next week the price decreases by 10%. At the convenience store, one week a 6-pack increases 10% in price. The next week the price 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 price is more money than 10% of the old price. So the price goes down MORE than it has gone up. No. There's a cumulative effect. 10% of the new price is more money than 10% of the old price. So the price goes down MORE than it has gone up. It costs less now! It costs less now! Think about it. Think about it.

Increase and Decrease Starting price: $5.00 $ % of $5.00 = $5.50. $5.50 – 10% of $5.50 = $ ¢ = $4.95 Final price: $4.95

Thinking Mathematically Section 2 Simple Interest Simple Interest

What is simple interest? Whenever you borrow or lend money, a certain percentage of the total amount will be paid in addition to paying back the loan.Whenever you borrow or lend money, a certain percentage of the total amount will be paid in addition to paying back the loan. Why?Why? Because while you have the money, the lender doesn't and can't invest it and make money off it.Because while you have the money, the lender doesn't and can't invest it and make money off it. Simple interest is a fixed percentage of the amount borrowed that will be paid for each of the years the loan is not paid off.Simple interest is a fixed percentage of the amount borrowed that will be paid for each of the years the loan is not paid off.

What is simple interest? When you borrow or lend money, the amount borrowed or loaned is called the principal (P).When you borrow or lend money, the amount borrowed or loaned is called the principal (P). The interest rate (r) is a percentage of the principal that will be paid back in addition to the principal.The interest rate (r) is a percentage of the principal that will be paid back in addition to the principal. The interest (I) is the amount paid back in addition to the principal.The interest (I) is the amount paid back in addition to the principal.

Calculating Simple Interest Interest =(Principal)(Interest Rate)(Time) I = Prt The accumulated amount (A) is the total value including the principal. Accumulated Amount: A = P + I A = P + Prt = P(1+rt)

Calculating Simple Interest for a Year 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 The amount deposited, or principal ( P ), is $2000. The rate ( r ) is 6%, or The time of the deposit ( t) is one year. The interest is: I = Prt = At the end of the first year, the interest is $120. You can withdraw the $120 in interest, and you still have $2000 in the savings account. ($2000)(0.06)(1) = $120 = $120.

Calculating Simple Interest You deposit $2000 in a savings account at Hometown Bank, which has a rate of 6%. Find the interest after 5 years. How much will be accumulated in the bank after 5 years if you never take out any money? How much will be accumulated in the bank after 5 years if you never take out any money?

Solution The amount deposited, or principal (P), is $2000. The rate (r) is 6%, or The time of the deposit (t) is 5 years. The interest is: I = Prt = At the end of 5 years, the interest is $600. Accumulated Amount: A = P + I Accumulated Amount: A = P + I ($2000)(0.06)(5) = $600 = $600. = $2600

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

Solution The amount deposited, or principal (P), is $2000. Since you now have $2180, and the time is 1 year (t = 1) A = P(1+rt) = P(1+r) $2180 = ($2000)(1+r) = $ r r = $180/$2000 = = 9%, which is the interest rate.

Discounted Loans  Similar to the Simple Interest Loan is something called a Discounted Loan.  If the Simple Interest Loan is unrealistic and hardly ever seen, the Discounted Loan is even rarer.  I have no idea why the book even brings it up.

Discounted Loans  In a Discounted Loan, the borrower must pay the interest "up front" at the time of the loan.  Therefore the borrower only gets the desired amount minus the interest charged.  However, since he has already paid the interest, he only has to pay the loan amount when he pays back.

Discounted Loans  For example, you want to borrow $10,000 and the lender offers you a Discounted Loan for one year at the rate of 5%.  I = Prt  The lender gives you $10,000 and you immediately pay the lender $500. Or, simply put,  the lender gives you $9,500 and, after a year, you have to pay back $10,000 = ($10,000)(.05)(1) = $500

Discounted Loans The lender gives you $9,500 and, after a year, you have to pay back $10,000. The Effective Interest Rate is computed by A = P(1+rt) $10,000 = $9,500(1 + r) since t is 1 year $500 = $9,500r r = about which is 5.26% !!! Sometimes, 5% isn't really 5%!!!

Quick Quiz  A mother puts $1000 in an account for her new- born daughter. The account offers 5% simple interest. If the account is left alone for the next 20 years, how much will be in the account when the daughter turns 20?  A = P(1 + rt), with P = $1000, r =.05 (or 1/20), and t = 20.  A = $1000(  20) = $1000 (1 + 1)  A = $1000  2 = $2000.

Thinking Mathematically Section 3 Compound Interest Compound Interest

What is Compound Interest? Since the lender could have earned money each year of the loan at some interest rate, the amount of principal is recomputed every year (if the loan is "compounded annually"). After one year, the lender could have made money on the principal. The lender should compute the amount he or she could make the next year on the accumulated amount.

Simple Interest vs. Compound Interest $2000$120$2120 $ $2000$120$2120$2000$120$2240 $2000$120$2360 $2000$120$2480 $2000$120$2600 $2000 at 6% Simple Simple Compounded Annually Year Year

Calculating the Amount in an Account for Compound Interest Paid Once a Year If P dollars are deposited at a rate r, in decimal form, subject to compound interest, then the amount, A, of money in the account after 1 year is given by: A = P(1+r). after 2 years A = P(1+r) (1+r) = P(1+r) after t years A = P(1+r)(1+r)(1+r)(1+r) = P(1+r) t. t times

Calculating the Amount in an Account for Compound Interest Paid Once a Year If P dollars are deposited at a rate r, in decimal form, subject to compound interest, then the amount, A, of money in the account after t years is given by: A = P(1+r) t.

Example: Using the Compound Interest Formula You deposit P = $2000 in a savings account at Hometown Bank, which has a rate of 6% (r =.06). a.Find the amount, A, of money in the account after 3 years subject to compound interest. b.Find the interest.

Solution The amount deposited, or principal (P), is $2000. The rate (r) is 6%, or The time of the deposit (t) is three years. The amount in the account after three years is: A = P(1+r) t = $2000(1+0.06) 3 = $2000(1.06) 3 = $ Rounded to the nearest cent, the amount in the savings account after three years is $ Because the amount in the account is $ and the original principal is $2000, the interest is $ $2000 = $382.03

Other compound interest loans Loans are not usually compounded annually. More typical are loans compounded quarterly (4 times a year), monthly [such as car loans and mortgages] (12 times a year) or even daily [such as credit cards] (360 times a year). This causes us to modify our formula.

Calculating the Amount in an Account for Compound Interest Paid n Times a Year If P dollars are deposited at rate r, in decimal form, subject to compound interest paid n times per year, then the amount, A, of money in the account after t years is given by: AP1 r n   nt n: semi-annually: 2 n: quarterly: 4n: daily: 365 n: monthly: 12

Other compound interest loans You take out a loan for $4,000 at an annual interest rate of 5.25% compounded monthly. If you pay the loan back 10 years from now, how much will you owe? We use the formula from the previous slide, with P = $4000, r =.0525, t = 10 and n = 12

Other compound interest loans We use the formula from the previous slide, with P = $4000, r =.0525, t = 10 and n = 12 On your calculator, 1.Enter Divide by 12 3.Add 1 4.Raise the result to the 12  10 =120 th power 5.Multiply by $ $

Continuous Compounding What if the compounding period is less than a day? How about each hour? Each minute? Each second? Each nanosecond?What if the compounding period is less than a day? How about each hour? Each minute? Each second? Each nanosecond? The ultimate in compounding is called continuous compounding.The ultimate in compounding is called continuous compounding. When continuous compounding occurs, the formula is different:When continuous compounding occurs, the formula is different: A = Pe rt

Continuous Compounding When continuous compounding occurs, the formula is different:When continuous compounding occurs, the formula is different: A = Pe rt e is a mathematical quantity equal to (1 + 1/n) ne is a mathematical quantity equal to (1 + 1/n) n when n becomes really big. when n becomes really big. e ≈ e ≈

Comparing Loans P = $4000, r =.0525, t = 10 Simple interest:Simple interest: A = 4000(  10) = 4000(1.525) = $6,100 A = 4000(  10) = 4000(1.525) = $6,100 Compounded annually:Compounded annually: A = 4000( ) 10 = $6, Compounded monthly:Compounded monthly: A = 4000( /12) 120 = $6, Compounded daily:Compounded daily: A = 4000( /360) 3600 = $6, Compounded continuously: A = 4000e.525 = $4000  = $6,761.84

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

Effective Annual Yield The Effective Annual Yield, or Effective Rate Y is computed as follows: Y = (1 + ) n - 1 Y = (1 + ) n - 1 rnrnrnrn

Effective Annual Yield A bank compounds interest daily (360 times a year) at 5% on money in an account. After one year, you would have Y = (1 +.05/360) Y = – 1 = So you are earning at an effective annual rate of about 5.13%.

Thinking Mathematically Section 4, Stocks and Bonds Annuities

Annuities An annuity is a savings account in which an equal amount of money is paid each year (or each month, or some other period).An annuity is a savings account in which an equal amount of money is paid each year (or each month, or some other period). An IRA is an example of an annuityAn IRA is an example of an annuity

Annuities In doing annuity calculations, the Principal (P) is the amount of each payment into the annuity.In doing annuity calculations, the Principal (P) is the amount of each payment into the annuity. Pretty awful, isn’t it? (It gets worse!)Pretty awful, isn’t it? (It gets worse!)

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

Annuities If we make n payments a year (for example, monthly payments of $100 for 10 years at 8%), the formula becomes:If we make n payments a year (for example, monthly payments of $100 for 10 years at 8%), the formula becomes: on $12,000 invested.on $12,000 invested.