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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. We fill with zeros as necessary. multiply .12 by 100 1.2 divide 12 by 100 • .1.2 0 12 12. 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 = 0.625 Step 2 Multiply the quotient by 100. 0.625 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. 0.47 47 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 5/8 Decimal .625 4 0 0.% 0 5% . Percent .62 5% 400% 5% Finding Percent Increase • • Use subtraction to find the amount of increase. Find the fraction for the percent increase, using • 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? Use subtraction to find the amount of increase. $5.50 - $5.00 = $0.50 Find the fraction for the percent increase, using amount of increase original amount = $0.50 $5.00 • • • Find the percent increase by expressing the fraction in step 2 as a percent. 0.1 = 10% Finding Percent Decrease • • Use subtraction to find the amount of decrease. Find the fraction for the percent decrease, using • 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? Use subtraction to find the amount of decrease. $5.00 - $4.50 = $0.50 Find the fraction for the percent decrease, using • • • • • Find the percent decrease by expressing the fraction in step 2 as a percent. 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%. • 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. • It costs less now! • Think about it. Increase and Decrease Starting price: $5.00 $5.00 + 10% of $5.00 = $5.50. $5.50 – 10% of $5.50 = $5.50 - 55¢ = $4.95 Final price: $4.95 Thinking Mathematically Section 2 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. • Why? • 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. What is simple interest? • 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 (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 0.06. The time of the deposit (t) is one year. The interest is: I = Prt = ($2000)(0.06)(1) = $120. 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. 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? Solution The amount deposited, or principal (P), is $2000. The rate (r) is 6%, or 0.06. The time of the deposit (t) is 5 years. The interest is: I = Prt = ($2000)(0.06)(5) = $600. At the end of 5 years, the interest is $600. Accumulated Amount: A = P + I = $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) = $2000 + 2000r r = $180/$2000 = 0.09 0.09 = 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 = ($10,000)(.05)(1) = $500 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 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 0.0526 which is 5.26% !!! Sometimes, 5% isn't really 5%!!! Quick Quiz A mother puts $1000 in an account for her newborn 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(1 + .05 20) = $1000 (1 + 1) A = $1000 2 = $2000. Thinking Mathematically Section 3 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 Year 1 2 3 4 5 Simple Compounded Annually $2000 $2000 $2000 $2000 $2000 $120 $120 $120 $120 $120 $2120 $2240 $2360 $2480 $2600 $2000 $2120 2247.20 2382.03 2524.95 $120 127.20 134.83 142.92 151.50 $2120 2247.20 2382.03 2524.95 2676.45 $2000 at 6% 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)2. . . . 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 0.06. 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 = $2382.03 Rounded to the nearest cent, the amount in the savings account after three years is $2382.03 Because the amount in the account is $2382.03 and the original principal is $2000, the interest is $2382.03 - $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: nt r A = P 1+ n n: quarterly: 4 semi-annually: n: daily: 365 12 2 monthly: 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 = $4000, r = .0525, t = 10 and n = 12 On your calculator, 1. Enter .0525 2. Divide by 12 3. Add 1 P .0525 0.0043750000 1.0043750000 4. Raise the result to the 12 10 =120th power 1.6885242138 5. Multiply by $4000 $6754.10 Continuous Compounding • 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. • When continuous compounding occurs, the formula is different: A= rt Pe Continuous Compounding • When continuous compounding occurs, the formula is different: A = Pert • e is a mathematical quantity equal to (1 + 1/n)n when n becomes really big. • e ≈ 2.71828 Comparing Loans P = $4000, r = .0525, t = 10 • Simple interest: A = 4000(1+.052510) = 4000(1.525) = $6,100 • Compounded annually: A = 4000(1+.0525)10 = $6,672.38 • Compounded monthly: A = 4000(1+.0525/12)120 = $6,754.10 • Compounded daily: A = 4000(1+.0525/360)3600 = $6,761.58 Compounded continuously: A = 4000e.525 = $4000 2.71828 = $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: r n Y = (1 + n ) - 1 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)360 - 1 Y = 1.0513 – 1 = 0.0513 So you are earning at an effective annual rate of about 5.13%. Thinking Mathematically Section 4 Annuities, Stocks and Bonds 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 IRA is an example of an annuity Annuities • In doing annuity calculations, the Principal (P) is the amount of each payment into the annuity. • Pretty awful, isn’t it? (It gets worse!) Annuities • For example, invest $1200 each year into an annuity at 8% yield. • Formula: • After 10 years, this annuity is worth • 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: • on $12,000 invested. Thinking Mathematically Section 5 Installment Buying The Vocabulary of Fixed Installment Loans The amount financed is what the consumer borrows: Amount financed = Cash price - Down payment The Vocabulary of Fixed Installment Loans The total installment price is the sum of all monthly payments plus the down payment: Total installment price = Total of monthly payments + Down payment The Vocabulary of Fixed Installment Loans The finance charge is the interest on the installment loan: Finance charge = Total installment price - cash price Computing interest on Installment Loans • The computation of the amount of interest to be paid on an installment load at a given interest rate is very complicated. • Generally it is done either using a preprogrammed business calculator, or • by using an Installment Loan Table Installment Loan Table Number of Monthly Payments Annual Percentage Rate (APR) 10.0% 10.5% 11.0% 11.5% 12.0% 12.5% 6 12 18 24 36 48 60 21.74 27.48 22.90 28.96 24.06 30.45 At 10%, for a four year installment loan of $100, you would pay a finance charge of $21.74. total paid: $121.74. For a $200 loan you would pay a finance charge of total paid: $243.48. $43.48 For a $1000 loan you would pay a finance charge of $217.40. total paid: $1217.40. Example What would the finance charge be if you bought a car for $18,000 and got a 5 year loan at 10.5% if you pay $3,000 down payment? Installment Loan Table Number of Monthly Payments Annual Percentage Rate (APR) 10.0% 10.5% 11.0% 11.5% 12.0% 12.5% 6 12 18 24 36 48 60 21.74 27.48 22.90 28.96 24.06 30.45 Example What would the finance charge be if you bought a car for $18,000 and got a 5 year loan at 10.5% if you pay $3,000 down payment? $18000 - $3000 = $15,000. Amount financed $15,000 is 150 $100 units. 150 * 28.96 (from table) = $4344 Finance charge So you will pay a total of $22,344 for that car. However you have already paid $3000. Each month you pay part of the finance charge; you must pay back $19,344. Since $19,344 must be paid over 60 months. Divide $19,344 into 60 equal payments. The monthly payment is $322.40. Example 2 You are buying a car for $18,000 and you are paying $5000 down payment. You are going to take out a four year loan. You are told your monthly payment will be $330. What is the approximate interest rate? $330 for 48 months = $15,840 = Amount Financed + Finance Charge = Finance Charge + $13,000. Finance Charge is $15,840 - $13,000 = $2840 You are borrowing $18,000 - $5000 = $13,000. There are 130 $100 units in $13,000 so your finance charge per $100 is $21.85 Look at the table: Installment Loan Table Number of Monthly Payments Annual Percentage Rate (APR) 10.0% 10.5% 11.0% 11.5% 12.0% 12.5% 6 12 closest amount to $21.85 18 24 36 48 60 21.74 27.48 22.90 28.96 24.06 30.45 Example 2 You are buying a car for $18,000 and you are paying $5000 down payment. You are going to take out a four year loan. You are told your monthly payment will be $330. What is the approximate interest rate? Look at the table: Your approximate interest rate is 10%. Prepayment Sometimes you can pay a loan off early. Perhaps you won the lottery and now have enough to pay for the car. Paying off early should reduce the Total Finance Charge because you borrowed the money for less time. Beware, however, of prepayment penalties: an additional amount penalizing you for borrowing for less time. (Uncommon among reputable lenders.) Prepayment How much will you have to pay, for example if you pay off that 4 year loan instead making your 18th monthly payment? How much money do you save? This is hard to calculate. Rule of thumb: “The Rule of 78”. The amount is approximated by this calculation: amount saved (unearned interest) = k(k +1)F n(n +1) What???? Prepayment How much will you have to pay, for example if you pay off that 4 year loan instead of making your 18th monthly payment? amount saved (unearned interest) = k(k +1) n(n +1) F F is the Finance Charge ($2840) k is the remaining number of payments (excluding this one) = 48 – 18 = 30 n is the number of payments according to the original loan = 48. Prepayment How much will you have to pay, for example if you pay off that 4 year loan instead of your 18th monthly payment? amount saved (unearned interest) = k(k +1) F n(n +1) amount saved = (30)(31)($2840) (48)(49) amount saved = $1122.96 Prepayment How much will you have to pay, for example if you pay off that 4 year loan instead of your 18th monthly payment? unearned interest = amount saved = $1122.96 Payoff amount = Payment #18 + Payments #19 through #48 - unearned interest Payoff amount = 31 * $330 - $1122.96 = $9107.04 Prepayment How much will you have to pay, for example if you pay off that 4 year loan instead of your 18th monthly payment? Payoff amount = 31 $330 - $1122.96 = $9107.04 Would have paid a total of 48 $330 = $15,840 $9107.04 + 17 $330 = $14,717.04 instead of $15,840 on a $13,000 loan. Comparing Loans Problem You've found the car of your dreams. It's a great deal: $16,000. You figure you can sell your current car for $2000 and use that as a down payment. You don't have the $14,000 you need, so you go looking for a loan. Problem You want to find a 4-year loan at 6% using simple interest. How much interest will you wind up paying? I = Prt = ($14,000)(.06)(4) = $3,360 Too bad you can't find such a loan. Problem Well, maybe there's a 4-year loan at 6% compounded annually. How much interest will you wind up paying? A = P(1+r)t = ($14,000)(1.06)4 = $17,674.67 A=P+I I = A – P = $17,674.67 - $14,000 = $3,674.67 Too bad you can't find such a loan. Problem How about a 4-year loan at 6% compounded monthly. How much interest will you wind up paying? A = P(1+r/12)12t = ($14,000)(1.005)48 = $17,786.85 A=P+I I = A – P = $17,786.85 - $14,000 = $3,786.85 But maybe an installment loan would be better than that. Would it? APR Table 4.00% 12 18 24 36 48 60 $2.18 $3.20 $4.22 $6.29 $8.38 $10.50 4.50% $2.45 $3.60 $4.75 $7.09 $9.46 $11.86 5.00% $2.73 $4.00 $5.29 $7.90 $10.54 $13.23 5.50% $3.00 $4.41 $5.83 $8.71 $11.63 $14.61 6.00% $3.28 $4.82 $6.37 $9.52 $12.73 $16.00 6.50% $3.56 $5.22 $6.91 $10.34 $13.83 $17.40 7.00% $3.83 $5.63 $7.45 $11.16 $14.94 $18.81 7.50% $4.11 $6.04 $8.00 $11.98 $16.06 $20.23 8.00% $4.39 $6.45 $8.55 $12.81 $17.18 $21.66 8.50% $4.66 $6.86 $9.09 $13.64 $18.31 $23.10 9.00% $4.94 $7.28 $9.64 $14.48 $19.45 $24.55 Problem The table says the Finance Charge (Interest) per $100 borrowed would be $12.73: I = (140)($12.73) = $1,782.20 But you're lucky. After 2 years, you can pay off the loan. Interest saved = (23)(24)($1782.20) = $418.22 (48)(49) Interest = $1,782.20 - $418.22 = $1,363.98 Summary • Interest on a 6% loan on $14,000 for four years. • $3,360.00 simple interest • $3,674.67 compound annually • $3,786.85 compound monthly • $1,782.20 installment loan • $1,363.98 installment loan paid off after 2 years. Problem • Just for the hell of it, how about a 12% loan? Summary • Interest on a 12% loan on $14,000 for four years. • $6,720.00 simple interest • $8,029.27 compounded annually • $8,571.17 compounded monthly • $3,696.00 installment loan • $2,751.50 installment loan paid off after 2 years. Credit Cards and “Revolving Credit” Credit cards are essentially loans that are compounded monthly. But the monthly amount changes on the basis of how much you owed at the start of the month, and sometimes how much you spent during the month. Different cards may work in different ways and are calculated in a number of ways. Good card: low interest rate only pay interest on what you carried over from last month (grace period for purchases this month) 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. Credit Card Payment (Unpaid balance) Monthly interest rate = Billing cycle: 1.3% per month at start of month May 1 Unpaid Balance : $1350 Payment on May 5: Purchases this month: $250 $497 Payment due date: June 9th Minimum payment only if there is an unpaid balance at least $10, or 1/36 of balance if it is higher than $360. Credit Card Payment (Unpaid balance) Monthly interest rate = Billing cycle: 1.3% per month at start of month May 1 Unpaid Balance : $1350 Payment on May 5: $250 Purchases this month: $497 Payment due date: June 9th Interest due = ($1350 – $250) * .013 = $14.30 Balance owed: $1350 - $250 + $14.30 + $497 = 1611.30 Minimum payment: 1611.30/36 = about $45. Credit Card Payment (Unpaid balance) What happens if you keep paying the minimum payment and charge about $500 a month? The balance owed was $1611.30 but you paid $45.30. New balance is $1566. Last month it was $1100 ($1350 – $250) Do it again paying $60 minimum and it becomes about $1566 + $20 + $500 - $60 = $2070. Do it again paying $minimum and it becomes about $2070 + $27 + $500 - $72 = $2525 Eventually the monthly interest becomes bigger than the minimum payment!!! Do it again and it becomes bigger and BIGGER and BIGGER and BIGGER Credit Card Payment (Unpaid balance) What happens if you keep paying the minimum payment and charge about $500 a month? Do it again and it becomes bigger and BIGGER and BIGGER and BIGGER Eventually, your interest payment becomes bigger than your minimum payment and you don’t even have to charge anything to make your debt grow. Credit Cards Advice: shop for the best rate look for a card that uses the unpaid balance method; that gives you a "grace period" of no interest on new purchases. always pay off your bill in full; if you can't then pay as much as you can afford/ Thinking Mathematically Section 6 The Cost of Home Ownership Terminology • Mortgage broker- offers to find you a mortgage lender willing to make you a loan. • Fixed rate mortgage - have the same monthly payment during the entire time of the loan • Variable rate mortgage - have payment amounts that change from time to time depending on changes in the interest rate. Mortgages • A fixed rate mortgage is exactly the same as any other installment loan,but – Lots of up-front costs, including points • A point is 1% of the amount borrowed to be paid to the lender. – Much longer payment schedule. – Different kind of table is used. Example Computing the Total Cost of Interest over the Life of a Fixed-Rate Mortgage 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. Find the required down payment. b. Find the amount of the mortgage. c. How much must be paid for the two points at closing? d. Find the monthly payment (excluding escrowed taxes and insurance). e. Find the total cost of interest over 30 years. Solution a. The required down payment is 10% of $195,000 or 0.10 $195,000 = $19,500 b. The amount of the mortgage is the difference between the price of the home and the down payment. = $195,000 - $19,500 = $175,500 Solution c. To find the cost of two points on a mortgage of $175,500, find 2% of $175,500. 0.02 $175,500 = $3510 The down payment ($19,500) is paid to the seller and the cost of two points ($3510) is paid to the lending institution. Solution d. We need to find the monthly mortgage payment. To do so we will use a monthly payment table. For a mortgage of 30 years at 7.5% the monthly payment per $1000 is found in the mortgage table. Mortgage Table 4.00% 15 4.50% 5.00% 5.50% 6.00% 6.50% 7.00% 7.50% 8.00% $7.40 $6.06 $5.28 $4.77 $4.43 $4.18 $7.65 $6.33 $5.56 $5.07 $4.73 $4.50 $7.91 $6.60 $5.85 $5.37 $5.05 $4.82 $8.17 $6.88 $6.14 $5.68 $5.37 $5.16 $8.44 $8.71 $7.16 $7.46 $6.44 $6.75 $6.00 $6.32 $5.70 $6.04 $5.50 $5.85 $8.99 $7.75 $7.07 $6.65 $6.39 $6.21 $9.27 $8.06 $7.39 $6.99 $6.74 $6.58 $9.56 $8.36 $7.72 $7.34 $7.10 $6.95 years of loan 20 25 30 35 40 monthly payment per $1000 of loan monthly payment of $6.99 per $1000 of loan over 30 years Solution d. We need to find the monthly mortgage payment. For a mortgage of 30 years at 7.5% the monthly payment per $1000 is $6.99. We can multiply this amount, $6.99, by the number of thousands of dollars in the mortgage amount to find the monthly payment for the entire mortgage amount. We divide the mortgage amount $175,500 by $1000 to find the number of thousands of dollars in the mortgage amount. Solution part d cont. $175,5000/$1000 = 175.5 The monthly mortgage payment is found by multiplying the number of thousands of dollars in the mortgage, 175.5, by the amount in the table, $6.99. 175.5 $6.99 = $1226.75 The monthly mortgage payment for principal and interest is $1226.75. Solution 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 the monthly payment in (d): $1226.75. 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 30 = 360. Thus the total of all monthly payments = $1226.75 360. Solution part e cont. Now we can calculate the interest over 30 years. = $1226.75 360 - $175,500 = $441,630 - $175,500 = $266,130 The total cost of interest over 30 years is $266,130. Review • Percents – how many out of 100? • Multiplication and Division by 100 – Multiply by 100, shift decimal point left by 2 places – Divide by 100, shift decimal point right by 2 places • Simple Interest: – I = Prt, A = P(1 + rt) • Compound Interest: – A = P(1 + r/n)nt • Effective Annual Yield: – Y = (1 + r/n)n - 1 Review • Installment Loan – – – – – Find entry in Installment Loan Table Multiply by the Principal Finance Charge: I = above value / 100 A=P+I Monthly payment = A / (number of months) (k(k+1)/n(n+1)) F • Rule of 78 – Unearned interest = Review • Credit Cards – Unpaid Balance Method: • Monthly Interest Rate • Paid on last month's balance – Amount Paid last month. – example: 1.5% per month, last month's bill: $500, paid: $200 Interest this month .015 $300 = $4.50 • Bill this month: last month's bill – last month's payment + interest + new purchases – example: $500 - $200 + $4.50 + $237 (new purchases) = $541.50 • Minimum Payment: varies, but described in contract – example: total amount owed / 36, or $10 (whichever is larger) – $541.50/36 = about $15 Review • Home Mortgages – Price of house – Pay up front: • down payment (to seller): % of Price • points (to lender): 1 point = 1% of Amount Borrowed – Monthly Payment: • from table • multiply figure from table by amount borrowed and divide by 1000. (e.g. $7.00 $360,000 / 1000 = $2520) – Total Payment • Monthly payment number of months (e.g. - $2520 360) – Interest = Total Payment – Amount Borrowed