Your name: Your section: MTH 150 SURVEY OF MATHEMATICS Chapter 11 CONSUMER MATHEMATICS 11.1 Percent 11.2 Personal Loans and Simple Interest 11.3 Personal Loans and Compound Interest 11.4 Installment Buying 11.5 Buying a House with a Mortgage
11.1 Percent What is a Percent or Percentage? Convert a decimal into a percentage 1. Multiply the decimal number by 2. Add a percent sign Examples: Convert 0.49 to a percentage Convert 0.651 to a percentage Convert a percent into a decimal 1. Divide the number by 2. Remove the percent sign Examples: Convert 37% into a decimal Convert 65.3% in to a decimal Convert a fraction into a percent 1. Divide the by the using a calculator. 2. Multiply the quotient by 100 (which has the effect of moving the decimal point places to the 3. Add a percent sign Page 2 of 30
Examples: Convert ¼ into a percent Convert 5/8 into a percent Convert 10/8 into a percent Percent Applications Percent Change Percent Change Change Amount in original period x 100 Amount in Amount in latest period previous period Amount in original ( previous) period x 100 If the amount in the latest period is greater than the amount in the previous period, the answer will be and will indicate a percent If the amount in the latest period is smaller than the amount in the previous period, the answer will be and will indicate a percent Examples: The population in GA in 1990 was 6,478,216. By 2000 the population had grown by 1,708,237. What was the percentage gain (increase) in population for the state of GA? Page 3 of 30
The number of people audited by the IRS fell from 1942 in 1996 to 618 in 2000. What is the percentage decrease from 1996 to 2000? The graph shows the rise of the minimum hourly wage from 1938 to 1997 Determine the percent increase from 1938 to 1997 Problems of type N% of a = b To find a percentage of a total, multiply the percent (in decimal form) by the total. Find 50% of 8 apples To determine a percent, divide the number by the total I m taking 30 of the 90 computers. What percent is that? Page 4 of 30
EXAMPLES: Find 30% of 20 toys Find 80% of 30 students Find 10% of 60 computers Of the 15 people invited to my party, 12 came. What percent is that? Only 17 of my 30 students passed. What percent is that? 3% of what number is 15? 5% of what number is 13? 80% of what number is 320? Page 5 of 30
More Examples The top 5 advertisers spent $8105 million in 2001. General Motors is responsible for a 27% share of this amount spent. How much did GM spend? US companies lost $32.2 billion due to various errors and frauds. Employee theft is responsible for a 46% share. How much was lost due to Employee theft? Consider the following circle graph of the Milton Family s budget. Assume the family has a monthly income of $2500. How much do they spend on rent? How much do they save? Page 6 of 30
Some background terminology: Principal of the loan 11.2 Personal Loans and Simple Interest Credit Security Interest Simple Interest Rate The simple interest formula Interest = Principal * Rate* time I PRT Notes: Always assume 30 days per month, 12 months and 360 days per year. Examples: If we borrowed $2000 for 2 years at a rate of 4.5% how much interest would we pay? Page 7 of 30
If we borrowed $2000 for 9 months at a rate of 4.5% how much interest would we pay? If we borrowed $2000 for 30 days at a rate of 4.5% how much interest would we pay? Find the missing quantity Principal Rate Time Interest Amount repaid a. $ 1,200 8% / year 6 months b. $6,000 2 years $1,080 c. 4 years $2,000 $12,000 d. $7,000 10% / year $8,750 Page 8 of 30
How to Find the Number of Days Between Two Specific Dates? Page 9 of 30
Examples: Find the exact time from May 3 to November 15. Find the exact time from January 4 to April 8 when the loan is due in a leap year Find the exact time from October 8 to December 3 when the loan is due in a leap year. Find the due date of the loan, using the exact time, if the loan is made on April 14 for 100 days. Find the due date of the loan, using exact time, if the loan is made on July 27 for 230 days. What is a Partial Payment? The United States Rule for Making Partial Payments A Supreme Court decision specified the method by which partial payments are credited and it s called the United States rule. The partial payment first pays off the interest for the time that s gone by (find this using I = prt), and then the rest of the partial payment is applied to the principal. The Banker s Rule considers a year to have 360 days, and any fractional part of a year is the exact number of days of the loan. The Banker s rule is used to calculate the simple interest when applying the United States rule. Page 10 of 30
Examples: Joe borrows $4000 on a note dated March 14 with simple interest of 5% for 90 days. He made a partial payment of $500 on April 1. How much will he owe on the date of maturity? The Cookie Company signed a $2000 note dated August 5 th with simple interest of 7% for 110 days. The company made a partial payment of $300 on September 10. Find the amount due on the maturity date of the loan. Page 11 of 30
The Sweet Tooth Restaurant borrowed $3000 on a note dated May 15 with simple interest of 11%. The maturity date of the loan is September 1. The restaurant made partial payments of $875 on June 15 and $940 on August 1. Find the amount due on the maturity date of the loan. Page 12 of 30
What is compound interest? 11.3 Personal Loans and Compound Interest Comparison of simple interest versus compound interest: $1000 loan for 4 years at 10% a year First, let s look at the way interest rate are expressed. Compounded annually means how many times a year? Compounded semiannually means Compounded quarterly means Compounded monthly means Compounded biweekly means Compounded twice a month Page 13 of 30
Compound Interest formula The final amount (A) includes principal and interest. r A P 1 n nt Examples 1. $2000 for 5 years at 2% compounded semiannually. Use the formula to compute a. The total amount b. The interest earned 2. $2000 for 5 years at 2% compounded monthly.. Use the formula to compute a) The total amount b) The interest earned 3. $2000 for 5 years at 2% compounded daily. a) The total amount b) The interest earned Page 14 of 30
4. Find the compound amount if $5,000 is invested for 10 years at 6% compounded: Annually Semiannually Quarterly Which one is the better investment and why? 5. Michelle deposits $2,500 in her savings account that pays 5% compounded monthly. After 2 years she withdraws $500 and leaves the rest of her money in the bank for another four years. How much is in her account after four years? Page 15 of 30
6. Tom needs to borrow $1500 to expand his farm implement maintenance business. He learns that the local bank will lend him the money for 2 years at a rate of 10% compounded quarterly. After hearing this, Tom s grandfather offers to lend him the money for 2 years with a simple interest rate of 7%. How much money will Tom save by borrowing the money from his grandfather? 7. Ramiro Rodriquez would like to establish a fund of $40,000 to help pay for his son s college education ten years from now. If a bank has a 10 year CD that will pay 9% compounded quarterly, how much must Ramiro deposit in that account today? Page 16 of 30
Effective Annual Yield or Annual Percentage Yield (APY) It s the simple interest rate that gives the same amount of interest as a compound rate over the same period of time. To determine the effective annual yield, calculate the amount using the compound interest formula where P = $1. Then subtract $1 from that amount. This difference, written as a percent, is the effective annual yield. Examples: 1. What is the APY for a $1 invested for 1 year at 12% compounded monthly? 2. What is the APY for a $1 invested for 1 year at 12% compounded weekly? 3. What is the effective annual yield for an investment of $10,000 invested at 5% compounded semiannually for 4 years? Page 17 of 30
What is a Fixed Installment Loan? 11.4 INSTALLMENT BUYING What is an Open-End Installment Loan? What is the Annual Percentage Rate (APR)? What is the Finance Charge? The finance charge can also be found by: Finance Charge = Total repaid Amount borrowed What is the Total Installment Price (or Cost)? Fixed Installment Loan Examples: Mrs. Stewart needs to borrow $1000 for an automobile repair. She finds that State National Bank charges 5% simple interest on the amount borrowed for the duration of the loan and requires the loan to be repaid in 6 equal monthly payments. Consumer s Credit Union offers loans of $1000 to be repaid in 12 monthly payments of $86.30. A. How much interest is charged by the State National Bank? B. How much interest is charged by the Consumer s Credit Union? Page 18 of 30
How to compute the Finance Charge? Table 11.2 Annual Percentage Rate Table for Monthly Payment Plans Number Of Payments Annual Percentage Rate 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7/0% 7.5% 8.0% 8.5% 9.0% 9.5% 10.0% Finance charge per $100 of amount financed 6 1.17 1.32 1.46 1.61 1.76 1.90 2.05 2.20 2.35 2.49 2.64 2.79 2.93 12 2.18 2.45 2.73 3.00 3.28 3.56 3.83 4.11 4.39 4.66 4.94 5.22 5.50 18 3.20 3.60 4.00 4.41 4.82 5.22 5.63 6.04 6.45 6.86 7.28 7.69 8.10 24 4.22 4.75 5.29 5.83 6.37 6.91 7.45 8.00 8.54 9.09 9.64 10.19 10.75 30 5.25 5.92 6.59 7.26 7.94 8.61 9.30 9.98 10.66 11.35 12.04 12.74 13.43 36 6.29 7.09 7.90 8.71 9.52 10.34 11.16 11.98 12.81 13.64 14.48 15.32 16.16 48 8.38 9.46 10.54 11.63 12.73 13.83 14.94 16.06 17.18 18.31 19.45 20.59 21.74 60 10.50 11.86 13.23 14.61 16.00 17.40 18.81 20.23 21.66 23.10 24.55 26.01 27.48 Examples: Sam Bazzi purchased a 2006 PT Cruiser GT for $25,000,. Sam obtained a 60-month, nomoney-down installment loan with an APR of 8%. A. Determine Sam s finance charge. B. Determine Sam s monthly payment. Juan Avalos paid $7000 for a new central air-conditioning unit for his house. He paid 20% as a down payment and financed the balance with a 36-month fixed installment loan with an APR of 5% A. Determine Juan s finance charge. B. Determine Juan s monthly payment. Page 19 of 30
How to compute the APR? Examples: An entertainment system that sells for $1,000 can be purchased on an installment plan for $200 down and 12 monthly payments of $70 each. Find the finance charge and true annual rate of interest (APR). Several suits, with a total cash price of $1320, were purchased on an installment plan for $20 down and $221.76 per month for 6 months. Find the finance charge and true annual rate of interest (APR). Page 20 of 30
Open-End Installment Loan What is it? Examples: Harry s credit card company determines his minimum monthly payment by adding all new interest to 2% of the outstanding principal. The credit card company charges an interest rate of 0.0576241% per day. On May 8, Harry uses his credit card to purchase a piece of equipment for his garage for $2600. He makes no other purchases during May. o Assuming Harry had no new interest, determine Harry s minimum payment due on June 1 st, his billing date. o On June 1, instead of making the minimum payment, Harry makes a payment of $100. Assuming there are no additional charges of cash advances, determines Harry s minimum payment due on July 1. NOTE: When finding the minimum payment.always round up to the nearest whole dollar. Page 21 of 30
Jane s credit card company determines her minimum monthly payment by adding all new interest to 1% of the outstanding principal. The credit card company charges an interest rate of 0.0479452% per day. On October 23, Jane uses her credit card to make purchases of $219.75, $82.13, and $24.70. She makes no other purchases during October. o Assuming Jane had no new interest, determine Jane s minimum payment due on November 1 st, her billing date. o On November 1, instead of making the minimum payment, Jane makes a payment of $200. Assuming there are no additional charges of cash advances, determines Jane s minimum payment due on December 1. Page 22 of 30
How to Compute the Finance Charge on an Open-end Installment Loans or Credit Cards? 1. Unpaid Balance Method : Or 2. Average Daily Balance Method: Example for the Unpaid Balance Method (#26): On September 5, the billing date, Verna Brown had a balance due of $567.20 on her credit card. The transactions during the following month were: September 8 Payment $275.00 September 21 Charge: Airline ticket $330.00 September 27 Charge: Hotel bill $190.80 October 2 Charge: Clothing $84.75 a) Find the finance charge on October 5, using the unpaid balance method. Assume that the interest rate is 1.1% per month. b) Find the new balance on October 5. Page 23 of 30
Example for the Average Daily Balance Method (#32): Instead of paying the unpaid balance method, suppose that Verna s bank uses the average daily balance method. a) Find Verna s average daily balance for the billing period from September 5 to October 5. Date Balance Due Number of Days Balance * # Days September 5 September 8 September 21 September 27 October 2 Total Average Daily Balance is the sum of the third column divided by the @ of days: b) Find the finance charge to be paid on October 5. c) Find the balance due on October 5. d) Compare those answers with the previous example Page 24 of 30
It s your turn to practice! On February 3, the billing date, Carol Ann Bluesky had a balance due of $124.78 on her credit card. Her bank charges a interest rate of 1.25% per month. She made the following transactions during the month: February 8 Charge: Art supplies $25.64 February 12 Payment $100.00 February 14 Charge: Flowers delivered $67.23 February 25 Charge: Music CD $13.90 Part I: a) Find the balance charge on March 3, using the unpaid balance method. b) Find the new balance on March 3. Part II: a) Find Carol Ann s average daily balance for the billing period from February 3 to March 3. Assume it is not a leap year. Date Balance Due Number of Days Balance * # Days Total b) Find the finance charge to be paid on March 3 c) Find the balance due on March 3. Page 25 of 30
A balance of $2,000 with 19% interest and a 2% minimum payment will take you over to repay if you only pay the minimum payment every month!!!...it will cost you nearly in interest only! Tips to lower your credit costs? Do not go over your credit limit Avoid using the cash advance feature. Switch to a lower card rate but be careful! Take advantage of the grace period Ask for a lower rate Send your payment on time to avoid late fees and more interest. If you can t pay the total balance, always pay more than the minimum payment If you are paying interest, send your payment as soon as possible to lower your average daily balance Use savings to pay for credit card bills Page 26 of 30
11.5 Buying a House with a Mortgage What is a Mortgage? Two most popular types of Mortgages: 1) 2) What are Points? Examples of Conventional Loans 1) Lynne Myers is buying a condominium for $289,000. Her credit union is requiring a down payment of 10%. To obtain a 15-year mortgage at 8% interest, she must pay 2 points at the time of closing. a. What is the required down payment? b. With the 10% down payment, what is the amount of the mortgage? c. What is the cost of the 2 points Page 27 of 30
How to Find the Monthly Principal and Interest Payment per $1000 of Mortgage? Number of Years Rate (%) 10 15 20 25 30 4.0 $10.12 $7.40 $6.06 $5.28 $4.77 4.5 10.36 7.65 6.33 5.56 5.07 5.0 10.61 7.91 6.60 5.85 5.37 5.5 10.85 8.17 6.88 6.14 5.68 6.0 11.10 8.44 7.16 6.44 6.00 6.5 11.35 8.71 7.46 6.75 6.32 7.0 11.61 8.99 7.75 7.07 6.65 7.5 11.87 9.27 8.06 7.39 6.99 8.0 12.13 9.56 8.36 7.72 7.34 8.5 12.40 9.85 8.68 8.05 7.69 9.0 12.67 10.14 9.00 8.40 8.05 9.5 12.94 10.44 9.32 8.74 8.41 10.0 13.22 10.75 9.65 9.09 8.78 10.5 13.49 11.05 9.98 9.44 9.15 11.0 13.78 11.37 10.32 9.80 9.52 11.5 14.06 11.68 10.66 10.16 9.90 12.0 14.35 12.00 11.01 10.53 10.29 2) Julia Reynolds is buying a house selling for $280,000 her bank requires her to make a 15% down payment. The current mortgage rate is 5%. a. Determine the amount of the required down payment b. Determine the monthly principal and interest payment for a 30-year mortgage with a 15% down payment. Page 28 of 30
3) Lori Edwards is buying a house selling for $200,000. Her bank requires her to make a 10% down payment. The current mortgage rate is 4%. a. Find the amount of the required down payment. b. Find the monthly principal and interest payment for a 15-year mortgage with a 10% down payment. 4) The Robinsons obtain a 30-year, $225,000 conventional mortgage at 6% on a house selling for $250,000. Also, they must pay 2 points at closing. a. What is their monthly mortgage payment, including principal and interest? b. Determine the total amount the Robinsons will pay for their house. c. How much of the cost will be interest (including the 2 points) d. How much of the first payment on the mortgage is applied to the principal? Page 29 of 30
5) The Keelers obtain a 20-year, $300,000 conventional mortgage at 5% on a house selling for $350,000. Also, they must pay 3 points at closing. a. What is their monthly mortgage payment, including principal and interest? b. Determine the total amount the Keelers will pay for their house. c. How much of the cost will be interest (including the 2 points) d. How much of the first payment on the mortgage is applied to the principal? 6) The Smiths are negotiating with two banks for a mortgage to buy a house selling for $175,000. The terms at bank A are a 15% down payment, an interest rate of 7%, a 30- year conventional mortgage, and 1.5 points to be paid at the time of closing. The terms at bank B are a 20% down payment, an interest rate of 8%, a 25-year conventional mortgage, and no points. Which loan should the Smiths select for the total cost of the house to be less? Page 30 of 30