Credit Cards Credit cards are both a convenience and a danger. (credit: "Credit Cards" by Sean MacEntee/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Apply for a credit card armed with basic knowledge. Distinguish between three basic types of credit cards. Compare and contrast the benefits and drawbacks of credit cards. Read and understand the basic parts of a credit card statement. Compute interest, balance due, and minimum payment due for a credit card. It can be difficult to get along these days without at least one credit card. Most hotels and rental car agencies require that a credit card is used. There are even a number of retailers and restaurants that no longer accept cash. They make online purchasing easier. And nothing contributes more to a good credit rating than a solid history of making credit card payments on time. Being granted a credits card is a privilege. Used unwisely that privilege can become a curse and the privilege may be withdrawn. In this section, we will talk about the different types of credit cards and their advantages and disadvantages. The more knowledge a cardholder has about the credit card industry, the better able credit accounts can be managed, and that knowledge may cause major adjustments to a cardholder’s lifestyle. All credit cards are not equal, but they all represent consumers borrowing money, usually from a bank, to pay for needs and “wants.” As such, they are a type of loan, and your repayment may include interest. (You might want to review Section 6.8, which discusses loans and repayment plans.) There are many institutions and credit cards to choose from. Use caution as you shop around for a credit card that suits you. Your top concern is likely the interest rates on purchases and cash advances. But be careful to also read the small print regarding charges for late payments, and other fees such as an annual fee, where the credit card charges you (the cardholder) a fee each year for the privilege of using the cards. Many cards charge no such fee, but there are many that charge modest to heavy fees. Make sure to understand rules for reward programs , where the credit card issuer grants benefits based on one’s spending. Finally, once one applies for and is granted a credit card, pay attention to the credit limit the bank offers. Once a company is owed that much money, use of the card for purchases should be curtailed until some of the debt is paid off. The interest rate will not matter if the balance is paid every month. When the balance is paid every month, there is NO INTEREST charged. Types of Credit Cards There are basically three types of credit cards: bank-issued credit cards, store-issued credit cards, and travel/entertainment credit cards. We will look at all three and explain the good and the bad qualities of each. Bank-Issued Credit Cards Perhaps the most widely used credit card type is the bank-issued credit card , like Visa or MasterCard (and even American Express and Discover cards). These types of cards are an example of revolving credit, meaning that additional credit is extended before the previous balance is paid—but only up to the assigned credit limit. Bank-issued cards are considered the most convenient, as they can be used to purchase anything, including apparel, furniture, groceries, fuel for automobiles, meals, hotel bills, and so on, just as if paying with cash. The interest rates on bank-issued credit cards are usually lower than those for other credit cards we’ll discuss, and the credit limits are generally higher. Currently, bank-issued cards have an average 20.09% APR. Store-Issued Credit Cards Store-issued credit cards are issued by retailers. One can hardly walk into a store these days without being offered a discount on purchases if one applies for the store credit card. These cards can only be used in that store or family of stores that issues the card. However, if a store credit card is associated with Visa, MasterCard, or American Express, then the card might be used the same way that the bank-issued cards are used. This is called cobranding. The logo of the bank-issued card will be present on the store card. Many stores offer both types. Like other credit cards, they may come with an annual fee. Store credit cards usually charge higher interest rates than bank-issued cards. Currently, store credit cards have an APR (annual percentage rate) of 24.15%. Any rewards offered by store credit cards are usually limited to purchases made in their own store, and it typically takes longer to accumulate enough rewards or points to redeem them, whereas cobranded credit cards offer opportunities to earn rewards on all purchases, regardless of whether purchases are made in the issuing store or not. Store credit cards usually offer lower credit limits, at least in the beginning. After being proved to be a responsible credit card owner, credit limits can be raised. Nevertheless, store credit cards are a good choice for those new to the credit card industry. If on-time payments are consistently made, it is an excellent way to get started building a credit history. Travel/Entertainment Cards, or Charge Cards This is the third type of credit card. The travel and entertainment cards, also known as charge cards, first and foremost offer very high limits or unlimited credit, but they must be paid in full every month. They generally charge high annual fees and impose expensive penalties should a payment be late. On the other hand, they typically have longer grace periods and offer many and various kinds of rewards. Check out this nerdwallet article about the differences between a charge card and a credit card . An interest rate will greatly depend on credit score. Responsible use of credit cards will increase a credit score. See the WHO KNEW? from The Basics of Loans . Choosing a Credit Card Top Travel and Entertainment Cards At one time, Diner’s Club was the premier entertainment card. To be accepted into the Diner’s Club and be rewarded with a charge card meant one was special. Today, an American Express card is held with the same reverence as Diner’s Club was in the past. It was a privilege to own one of these cards. There is a lot of competition going on to supplant Diner’s Club, as shown by the Chase Sapphire Preferred Card’s travel rewards and benefits. Another thing worth mentioning here and something that appears to be an unusual offering is part of the American Express Gold and higher-level cards. Cardholders can actually open a savings account, buy a CD, or apply for a personal or business loan from American Express. The company boasts a higher interest rate than what is paid on traditional savings accounts and CDs. Comparing Credit Cards Which type of credit card is paid off every month so has no interest to be paid, but comes with high fees? Which type of credit cards are the most widely accepted? Which type of credit cards are the most limited? Charge cards are to be paid off completely each month. Bank-issued credit cards are the most flexible to use, because they are not limited to which retailers or service providers accept them. Store issues credit cards are the most limited, since they only work in that family of stores. Credit Card Statements Cardholders usually receive monthly statements and have 21 days to pay the minimum amount due. The statements itemize and summarize activity on the credit card for that statement’s billing period . The billing period for a credit card is generally a month long, but typically does not start and end on the first and last days of the month. The statement will include the current balance, interest rate, the minimum payment due, and the due date. Be aware, different companies produce statements that are laid out differently. The information will be clearly labeled though. The due date is a top concern. Missing a due date is one of the worst things a cardholder can do financially, and this is by far the biggest downfall of owning a credit card. Not only is the cardholder subject to late fees, but when a payment is late more than once there is a high probability that the cardholder will be negatively reported to the credit bureaus, which can quickly erode a credit score. shows an excerpt from an actual statement from a Chase Bank Visa card, based on the current $668.25 balance. Credit card statement Specifically pay attention to the late payment penalty and minimum payment warning statements. stating that if no other purchases are made and you continue making only the minimum payment, it will take 19 months to pay off the balance and you will pay $754.00. You can’t say you were not warned. It is critical that you examine your statement every month because it is always a possibility that your account may have been compromised. If you should notice fraudulent charges on your statement, notifying the credit card company is often enough to have those charges researched by the company and removed. The card with the fraudulent charges will be canceled and a new card with a new account number will be sent to you. Reading a Credit Card Statement On the credit card statement , identify The balance due The minimum required payment The length of time it takes to pay off the balance by paying the minimum payments and without charging more to the card The interest rate for purchases Credit card account statement The balance due is under the payment information heading and is $3,663.23. The minimum payment due is also under the payment information heading, and is $36.63. The time to pay off the balance using only minimum payments is below the payment information, and says it takes 2 years and 4 months to pay off the balance. The interest rate for purchases is toward the bottom of the statement. It is 19.99%. Reading Credit Card Statements Compute Interest, Balance Due, and Minimum Payment Due for a Credit Card Computing all of these values depends on understanding and computing the average daily balance on a credit card. Once that is known, the interest, balance due, and minimum payment can be found. Above all else, if you pay off the entire balance each month, interest is not charged. Average Daily Balance Most credit card companies compute interest using the average daily balance method. To find the average daily balance on your credit card, determine the balance on the card each day of the billing period (often that month), and take the average. One process to find that average daily balance follows these steps: Start with a list of transactions with their dates and amounts. For each day that had transactions, find the total of the transactions for the day. Expenditures are treated as positive values, payments are treated as negative values. Create a table containing each day with a different balance. The balance is the previous balance plus the day’s total transactions. Add a column for the number of days those balances until the balance changed. Add a column that contains the balances multiplied by the number of days until the balance changed. Find the sum of that last column. Divide the sum by the number of days in the billing period (often the number of days in the month). This is the average daily balance. Computing Average Daily Balance The billing cycle goes from May 1 to May 31. The balance at the start of the billing cycle is $450.21. The list of transactions on the card is below. Date Activity Amount 1-May Billing Date Balance $450.21 10-May Payment $120.00 15-May Groceries $83.43 26-May Auto Parts $45.12 26-May Restaurant $85.34 30-May Shoes $98.23 Find the average daily balance for the credit card during the month of May. To find the average daily balance, we use the following steps. Start with a list of transactions with their dates and amounts. This list is provided. For each day that had transactions, find the total of the transactions for the day. The only day with more than one transaction was May 26. The sum of those transactions is $130.46. Treating the payment as a negative value, the daily transaction amounts are Date Amount 1-May $450.21 10-May -$120.00 15-May $83.43 26-May $130.46 30-May $98.23 Create a table containing each day with a different balance. The new table with dates that had different balances is below. Date Balance 1-May $450.21 10-May $330.21 15-May $413.64 26-May $544.10 30-May $642.33 Now, add a column for the number of days those balances until the balance changed. The days until the balance changes is found by finding the difference in the dates. For instance, from May 15 to May 26 was 11. Adding that column to the table we have Date Balance Days Until Balance Changes 1-May $450.21 9 10-May $330.21 5 15-May $413.64 11 26-May $544.10 4 30-May $642.33 2 The last entry was 2 since there are 31 days in May. Add a column that contains the balances multiplied by the number of days until the balance changed. We create the column and multiply the values. Date Balance Days Until Balance Changes Balance Times Days 1-May $450.21 9 $4,051.89 10-May $330.21 5 $1,651.05 15-May $413.64 11 $4,550.04 26-May $544.10 4 $2,176.40 30-May $642.33 2 $1,284.66 Find the sum of that last column. Adding that last column we have a sum of $13,714.04. There are 31 days in May, so divide the sum by 31, which gives an average of $442.39, which is the average daily balance. Calculating the Interest for a Credit Card The interest charged for a credit card is based on the daily interest rate of the card, the number of days in the billing cycle, and the average daily balance on the card. The interest charge, I , for a credit card during a billing cycle is I = ADB × r × d 365 , where ADB is the average daily balance, r is the annual percentage rate, and d is the number of days in the billing cycle. As before, interest is rounded up to the next penny. Calculating Interest for a Credit Card Billing Cycle Compute the interest charged for the credit card based on the given average daily balance (ABD), annual interest rate, and number of days in the billing cycle. ADB = $2,765.00, annual interest rate 13.99%, billing cycle of 30 days ADB = $789.30, annual interest rate 17.99%, billing cycle of 31 days ADB = $1,037.85, annual interest rate 11.99%, billing cycle of 28 days Substituting $2,765.00 for ADB, 0.1399 for r and 30 for d and calculating, we find the interest charge to be I = ADB × r × d 365 = $ 2,765.00 × 0.1399 × 30 365 = $ 31.80 . Substituting $789.30 for ADB, 0.1799 for r and 31 for d and calculating, we find the interest charge to be I = ADB × r × d 365 = $ 789.30 × 0.1799 × 31 365 = $ 12.06 . Substituting $1,037.85 for ADB, 0.1199 for r and 28 for d and calculating, we find the interest charge to be I = ADB × r × d 365 = $ 1,037.85 × 0.1199 × 28 365 = $ 9.55 . Credit Cards Charge Stores Fees The interest you pay is not the only way a credit card company generates revenue. It also charges fees to the retailers, online stores, and service providers that allow the consumer, you, to use your credit card to pay them. These are called processing fees. Currently they typically range from 2.87% to 4.35% of each transaction. That means if you use your credit card at a store and spend $100.00, the store will have to pay the credit card company somewhere between $2.87 and $4.35. One type of processing fee is the interchange fee. Mastercard charges the vendor 1.35% of the sale, plus an additional percentage up to 3.25%, and a fixed $001 fee for each transaction. Added to that is an assessment fee. This fee is 0.14% for Visa cards. Calculating the Balance of a Credit Card The balance, or sometimes balance due, on a credit card is the previous balance, plus all expenses, minus all payments and credits, plus the interest on the card. As stated before, if the card was paid off, there is no interest to be paid. Calculating the Balance of a Credit Card Find the balance on the credit card with the given interest charge and balance before interest was charged. The cards were not paid off previously. Balance before interest is $708.50, interest charge is $8.15 Balance before interest is $1,395.10, interest charge is $21.32 Adding the balance before interest to the interest charge, we find the balance to be $716.65. Adding the balance before interest to the interest charge, we find the balance to be $1,416.42. The next example puts all those steps together. Find Balance Due from Transactions and Interest Rate Kaylen’s credit card charges 16.9% annual interest. His current billing period is from November 1 to November 30. The balance on November 1 was $1,845.23. Use Kaylen’s following transactions to determine his balance due at the end of the billing cycle. Date Activity Amount 1-Nov Billing Date Balance $1,845.23 3-Nov Groceries $78.50 4-Nov Tablet $159.00 4-Nov Online Game Purchase $39.99 4-Nov Restaurant $47.10 10-Nov Payment $300.00 13-Nov Gasoline $58.75 13-Nov Clothing $135.00 18-Nov Gift $30.00 18-Nov Restaurant $21.75 28-Nov Gasoline $43.79 The first step is to find Kaylen’s average daily balance. To find the average daily balance, we use the following steps. Start with a list of transactions with their dates and amounts. This list is provided. For each day that had transactions, find the total of the transactions for the day. The days with more than one transaction were Nov. 4, Nov. 13, and Nov. 18. Treating the payment on November 10 as a negative value, the daily transaction amounts are Date Amount 1-Nov $1,845.23 3-Nov $78.50 4-Nov $246.09 10-Nov -$300.00 13-Nov $193.75 18-Nov $51.75 28-Nov $43.79 Create a table containing each day with a different balance. The new table with dates that had different balances is below. Date Balance 1-Nov $1,845.23 3-Nov $1,923.73 4-Nov $2,169.82 10-Nov $1,869.82 13-Nov $2,063.57 18-Nov $2,115.32 28-Nov $2,159.11 Now, add a column for the number of days those balances until the balance changed. The days until the balance changes is found by finding the difference in the dates. For instance, from May 15 to May 26 was 11. Adding that column to the table we have The last entry was 3 since there are 30 days in November (the 28th, 29th, and 30th). Date Balance Days Until Balance Changes 1-Nov $1,845.23 2 3-Nov $1,923.73 1 4-Nov $2,169.82 6 10-Nov $1,869.82 3 13-Nov $2,063.57 5 18-Nov $2,115.32 10 28-Nov $2,159.11 3 Add a column that contains the balances multiplied by the number of days until the balance changed. We create the column and multiply the values. Date Balance Days Until Balance Changes Balance Times Days 1-Nov $1,845.23 2 $3,690.46 3-Nov $1,923.73 1 $1,923.73 4-Nov $2,169.82 6 $13,018.92 10-Nov $1,869.82 3 $5,609.46 13-Nov $2,063.57 5 $10,317.85 18-Nov $2,115.32 10 $21,153.20 28-Nov $2,159.11 3 $6,477.33 Find the sum of that last column. Adding that last column we have a sum of $62,190.95. There are 30 days in November, so divide the sum by 30, which gives an average of $2,073.03, which is the average daily balance. With the average daily balance, we can determine the interest that is charged for November. Substituting ADB = $2,073.03, r = 0.169, and d = 30 into the formula I = ADB × r × d 365 and calculating, we find the interest to be I = ADB × r × d 365 = $ 2,073.03 × 0.169 × 30 365 = $ 28.80 . This interest is added to the final balance from the table in step 3, $2,159.11, which yields a balance due of $2,101.83. Minimum Payment Due The minimum payment due is the smallest required amount to be paid on a credit card to avoid late fees and penalties, such as an increased interest rate. The calculations for this may differ from card to card. They also depend in the balance of the credit card. General guidelines for minimum payment due are: For larger balances (usually over $1,000), the minimum payment will be some percentage of the balance due. For moderate balances (between $25 and $1,000), the minimum would be a specified dollar amount. $25 seems to be a common value. If the balance is small (under $25 for instance), then the minimum payment is the balance. Those are just guidelines. Individual cards may vary in these values. Minimum payments should only be paid if money is short in a given month. The length of time to pay off a credit card using minimum payments is quite long, and results in paying a lot of interest. It is strongly discouraged. Calculate the Minimum Payment Due The FYA credit card company has the following minimum payment policy. For balances over $1,000, the minimum payment is 2.5% of the balance due plus fees, but not interest. For balances between $500.00 and $999.99, the minimum payment is $50.00. For balances $499.99 and under, the minimum payment is $25.00 or the balance due, whichever is smaller. In the following, calculate the minimum payment due given the credit card minimum payment policy, the balance due and fees charged. Balance due is $1,309.00, no fees Balance due is $265.50, $35 in fees Balance due is $784.90, no fees The balance is over $1,000, so the minimum payment is 2.5% of the balance due plus fees. 2.5% of the balance due is 0.025 × $ 1,309.00 = $ 32.73 . Since there are no fees, the minimum payment due is $32.73. The balance is under $499.99, so the minimum payment due is $25.00. The balance is between $500.00 and $999.00, so the minimum payment due is $50.00. Check out this nerdwallet article about minimum payments for more! Check Your Understanding Key Terms Reward program Annual fee Credit limit Bank-issued credit card Store-issued credit card Travel and entertainment cards Charge cards Billing period Balance Minimum payment Average daily balance Key Concepts Credit cards can be a flexible way to pay for almost anything, but can become a financial hazard if used unwisely. When deciding which credit card to apply for, evaluate the interest rate, fees (annual and late), reward programs and credit limit. Be sure they meet your criteria. Paying off the balance of your credit card every month will control your spending and will never result in paying interest. Credit card statements hold all important information about your credit card, including payment, balances, charges and billing cycle dates. Although the minimum payment is attractive precisely because it is so small, paying only the minimum results is a long payoff term and higher interest costs. Video Choosing a Credit Card Reading Credit Card Statements Formula I = ADB × r × d 365

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Buying or Leasing a Car The choice to lease or buy is based on cost and other concerns. (credit: “Car Insurance” by Pictures of Money/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Evaluate basics of car purchasing. Compute purchase payments and identify the related interest cost. Evaluate the basics of leasing a car. Identify and contrast the pros and cons of purchasing versus leasing a car. Investigate the types of car insurance. Solve application problems involving owning and maintaining a car. There are people who don’t need a car and won’t purchase one. But for many people, whether or not to have a car is not a question. Having a car is a basic necessity for these people. Obtaining a car can be daunting. The models, the features, the additional costs, and finding funding are all steps that need to be taken. One of the big decisions is whether to buy the car or to lease the car. This section will address some of the issues associated with each option. The Basics of Car Purchasing The biggest questions you will answer before purchasing a car are, what do you want and what do you need? Does it have to be new? Does it have to be a make and model you are familiar with? Does it have to have assisted driving? What other details are important to you? For a new vehicle, every feature beyond standard features comes with additional cost, which leads to the question that constrains all of your decisions about a car. How much can you afford to spend on a car? What you can afford must include insurance costs (discussed later in this section) and maintenance and upkeep. Once you have this in mind, you can search for a car that matches, as closely as possible, what you want and can afford. Most, if not all, dealers have websites that you can search through to identify the car you want. If new cars are not affordable, used cars cost less but come with the wear and tear of use. The sticker price of the car, called the manufacturer’s suggested retail price (MSRP), or the negotiated price you arrive at, isn’t the end of the cost to buying a car. There are many fees that accompany the purchase of the car, and perhaps even sales tax. These include but aren’t necessarily limited to the following: the title and registration fee , which includes registering your car with the state, getting the license plate, and assigning the title of the car to the lender. This cannot be avoided. a destination fee , which covers the cost of delivering the vehicle to the dealer a documentation fee , sometimes referred to a processing fee of handling fee, is the cost of all the paperwork the dealer did to get you the car a dealer preparation fee , which is for washing the car and other preparation of that sort. You should try to negotiate that out of the cost of the dealer tries to charge for that extended warranties and maintenance plans, which help cover some of the costs of caring for the car. Sales tax. You could pay for these immediately, but they are often added to the financing of the car, meaning they become part of the principal of your loan. Total Cost to Purchase a Car Nichole negotiates with her car dealership so that the price is $21,800. She needs to pay the 6.75% sales tax on the car. Other fees are $31.00 for title and registration, $1,000 in destination fees, and a documentation fee of $175. What is the total cost of Nichole’s car? We add the car’s sales cost, sales tax and all other fees to arrive at this value. The sales tax is 6.75% on the price she negotiated, so is $ 21,800 × 0.0675 = $ 1,471.50 . Adding these up, we have a total cost of $ 21,800 + $ 1,471.50 + $ 31.00 + $ 1,000 + $ 175 = $ 24,477.50 . One way to bring down payments on a car is to provide a down payment or a trade in. This is money applied to the purchase price before financing happens. Be warned, the sales tax applies to the full purchase price! If you reduce the amount financed, the payments respond by going down. This often becomes part of the negotiating process. Total Cost to Purchase a Car with Down Payment Sophia negotiates a $19,800 price for her new car. The sales tax is 9.5% in her area, and the dealership charges her $300 in documentation fees. Her title, plates, and registration come to $321.50. The dealership adds to this a destination fee of $1,100. If she places a down payment of $5,000 on the car, what is the total she will finance for the car? The price was $19,800. The sales tax of 9.5% is based on this number. The sales tax comes to $ 19,800 × 0.095 = $ 1,881 . Adding all the fees to the price and the sales tax brings the total cost of the car to $ 19,800 + $ 1,881 + $ 300 + $ 321.50 + $ 1,100 = $ 23,402.50 . Her down payment is applied to this number, so the $5,000 is subtracted from $23,402.50. The subtraction yields the amount to be financed, which is $18,402.50. When purchasing a car, the total cost to obtain the car is not the only factor in your monthly price. You will also pay an interest rate for the loan you obtain. The interest rate you will get is dependent on your credit score (see The Basics of Loans ). But you can choose from different lenders. The dealership will likely offer to finance your car loan. Frequently, dealerships offer special financing with very low rates. This is to help move inventory, and may indicate their desire to make sales. This might make negotiating easier. Even if the dealership offers financing, check with your bank or credit union to determine the interest rates they are offering. To reduce your payments, choose the lowest rate you can find. Purchase Payments and Interest Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. The interest rates available for used cards are frequently higher than those for new cars. These loan payments work exactly the same way as other loans do as far as payments are concerned. The payment function comes from The Basics of Loans . The difference between financing a new car or a used car is that financing a new car typically comes with a lower interest rate and a longer term that financing a used car. The payment, pmt , per period to pay off a loan with beginning principal P is p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 , where r is the annual interest rate in decimal form, t is the term in years, and n is the number of payments per year (typically, loans are paid monthly making n = 12). Note, payment to lenders is always rounded up to the next penny. Often, the formula takes the form p m t = P × ( r ) × ( 1 + r ) n ( 1 + r ) n − 1 , where r is the interest rate per period (annual rate divided by the number of periods per year), and n is the total number of payments to be made. New Car Payments In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term of the car loan. Total to be financed is $31,885, interest rate is 2.9%, for 5 years. Total to be financed is $22,778, interest rate is 4.5%, for 6 years. The amount to be financed is the principal, P , which is $31,885. The rate r is 0.029, and the term is t = 5 years. These are monthly payments, so n = 12. Substituting and calculating, we find the monthly payment to be p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 31,885 × ( 0.029 / 12 ) × ( 1 + 0.029 / 12 ) 12 × 5 ( 1 + 0.029 / 12 ) 12 × 5 − 1 = $ 31,885 × ( 0.00241 6 ¯ ) × ( 1.00241 6 ¯ ) 60 ( 1.00241 6 ¯ ) 60 − 1 = $ 31,885 × ( 0.00241 6 ¯ ) × ( 1.15583736592 ) 1.15583736592 − 1 = $ 89.0635298302 0.15583736592 = $ 571.52 The amount to be financed is the principal, P , which is $22,778. The rate r is 0.045, and the term is t = 6 years. These are monthly payments, so n = 12. Substituting and calculating, we find the monthly payment to be p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 22,778 × ( 0.045 / 12 ) × ( 1 + 0.045 / 12 ) 12 × 6 ( 1 + 0.045 / 12 ) 12 × 6 − 1 = $ 22,778 × ( 0.00375 ) × ( 1.00375 ) 72 ( 1.00375 ) 72 − 1 = $ 22,778 × ( 0.00375 ) × ( 1.3093031051 ) 1.3093031051 − 1 = $ 111.837397673 0.3093031051 = $ 361.58 Used Car Payments Calculate the monthly payment for the used car if the total to be financed is $16,990, the interest rate is 7.5%, and the loan term is 3 years. The amount to be financed is the principal, P , which is $16,990. The rate r is 0.075, and the term is t = 3 years. These are monthly payments, so n = 12. Substituting and calculating, we find the monthly payment to be p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 = $ 16,990 × ( 0.075 / 12 ) × ( 1 + 0.075 / 12 ) 12 × 3 ( 1 + 0.04775 / 12 ) 12 × 3 − 1 = $ 16,990 × ( 0.00625 ) × ( 1.00625 ) 36 ( 1.00625 ) 36 − 1 = $ 16990 × ( 0.00375 ) × ( 1.25144613551 ) 1.25144613551 − 1 = $ 132.887936515 0.25144613551 = $ 528.50 The Basics of Leasing a Car Leasing a car is an alternative to purchasing a car. It is still a loan, and acts like one in many respects. They typically last either 24 months or 36 months, though other terms are available. Leases also come with mileage limits, frequently 10,000, 12,000, or 15,000 miles per year. When the lease is over, the car is returned to the dealer. At that time there may be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit. There are two components to lease costs. One is the monthly payment for the lease. The other is the fees for leasing, These often are paid before the lease is complete. These include: a down payment, which is your initial payment that is applied to the price of the car. It reduces the amount you finance, much the same as when you purchase a car. It is recommended that this be negotiated away. the acquisition fee , sometimes called the bank fee. This is the money charge for the company to set up the lease. It is essentially a paperwork fee. It is not likely that this can be negotiated. a security deposit , which might be required. It is about the same as 1 month’s payment for the lease. The deposit is returned to you if the car is in good shape at the end. This can be negotiated away. disposition fees , which cover the cost the company will incur when they take your car back and are typically between $200 and $450. the title, registration, and license fees, just as with the purchase of a car. sales tax, which will likely be applied. The sales tax only covers the depreciated portion of the car (more on depreciation later) in many states. Since this depends on the state in which the car is leased, you should determine the sales tax rules for where you lease the car. As you can imagine, this can come to a fairly high dollar amount. Cost to Obtain a Lease Donna wants to lease a Subaru Outback in Eden, New York. Find the total cost of obtaining her lease if there is no down payment, $175.00 in acquisition fees, a security deposit of $300.00, $350.00 in disposition fees, $102.50 in title and registration fees, and sales tax of $3,536.05. Adding these values together we find the total cost is $4,463.55. Zollie Frank Zollie Frank and Armund Shoen founded one of the original leasing companies, Four Wheels, in 1939. Their company leased automobiles to corporations. They began by leasing 5 cars to the Petrolager pharmaceutical company in year one. This saved Petrolager money and provided a steady cash flow to the Four Wheels business. In year two, the number of cars leased to Petrolager was 75. Their new idea was to lease cars directly to companies for one year. Previously, such companies might pay for mileage, gas, and a partial down payment. Sadly, the salesmen who were being so helped often left the company before the car was paid for, and so the company lost the down payment money. The lease was for $45 per month per car for one year. You have some obligations when you lease a car. You must keep the car in good condition, cleaned, maintained, and free of anything more than minor damage. If the car is in poor condition when the car is returned, you will be responsible for the cost to bring the car to an acceptable condition. You are also expected to keep the mileage under its limit. If you go over, you will pay 10 to 25 cents per mile over. Lease Payments Lease payments are similar to regular loan payments, but have some other details. Calculating a lease payment involves knowing the following values: The price of the car. This is the cost you would pay for the car after applying all discounts, incentives, and negotiations. Residual Value. This is the manufacturer's estimate of the car's value after a set period of time. The residual value is expressed as a percentage of the manufacturer’s suggested retail price (MSRP). Months. This is the length of the lease. Most leases are either 24- or 36-month leases, but other terms are available. Monthly Depreciation . The monthly depreciation is the difference between the price of the car and the residual value, divided by the number of months of the lease, and represents the monthly loss of value of the car while it’s being used. Money Factor (MF). This is the interest rate, but expressed in a different way for a lease. Converting from the money factor to the annual percentage rate (APR) is done by multiplying the MF by 2400. Naturally, converting an APR to a MF is done by dividing the APR by 2400. The monthly depreciation for a car, MD, is MD = P − R n , P is the price paid for the car, R is the residual value of the car, and n is the number of months of the lease. The annual percentage rate for a lease is APR = 2400 × MF , where MF is the money factor of the lease. The MF for a lease is MF = APR / 2,400 . Monthly Depreciation of a Car The purchase price of a car is $25,000. Its residual price is $14,500. What is its monthly depreciation for a 36-month lease? The purchase price of a car is $30,000. Its residual price is $18,600. What is its monthly depreciation for a 24-month lease? The monthly depreciation formula is MD = P − R n , Substituting $25,000 for P , $14,500 for R , and 36 for n , we find MD to be MD = P − R n = $ 25,000 − $ 14,500 36 = $ 10,500 36 = $ 291.67 . The monthly depreciation formula is MD = P − R n , Substituting $30,000 for P , $18,600 for R , and 24 for n , we find MD to be MD = P − R n = $ 30,000 − $ 18,600 24 = $ 11,400 24 = $ 475.00 . Converting Between APR and MF Find the annual percentage rate if the money factor is 0.00001875. Find the money factor if the APR is 6.25%. The APR is the money factor times 2400, so APR = 2400 × MF = 2400 × 0.00001875 = 0.045 . Expressed as a percentage, the APR is 4.5%. The MF is the APR divided by 2400, so MF = APR / 2400 = 0.0625 / 2400 = 0.000026041 6 ¯ . Once the values above are found, the payment for the lease can be calculated. The payment, PMT , for a lease is P M T = ( P − R ) n + ( P + R ) × MF , where P is the price paid for the car, R is the residual value of the car, n is the number of months of the lease, and MF is the money factor for the lease. Calculating Car Lease Payments Calculate the lease payments for car with the following price, residual price, length of lease, and money factor or APR. Price is $28,344, residual price is $18,140.16, 24-month lease, money factor is 0.000025. Price is $22,500, residual price is $13,050, 36-month lease, APR is 7.5%. Substituting the values P = $28,344, R = $18,140.16, n = 24 and MF = 0.000025 into the formula and calculating, the monthly lease payment is P M T = ( P − R ) n + ( P + R ) × MF = ( $ 28,344 − $ 18,140.16 ) 24 + ( $ 28,344 + $ 18,140.16 ) × 0.000025 = $ 10,203.84 24 + ( $ 46,484.16 ) × 0.000025 = $ 426.33 Given the APR, we find the MF which is MF = APR / 2400 = 0.075 / 2400 = 0.00003125 . Substituting the values P = $28,344, R = $18,140.16, n = 24 and the MF into the formula and calculating, the monthly lease payment is P M T = ( P − R ) n + ( P + R ) × MF = ( $ 22,500 − $ 13,050 ) 36 + ( $ 22,500 + $ 13,050 ) × 0.00003125 = $ 9,450 24 + ( $ 35,550 ) × 0.000025 = $ 263.62 Comparing Purchasing and Leasing When deciding to buy or lease a car, the differences between the two options should be carefully evaluated. The following is a list of points of comparison between the two. The payments for a lease are likely less than the payment for purchasing. When leasing, you get a new car after the lease term is over, typically 24 or 36 months. Buying the car means the same car is driven until it is re-sold and a new one bought. Essentially, leasing a car is equivalent to renting a car. The leased car is new, so all warrantees are in force and you drive the car during its best years. When the car is purchased, it may be kept past its warrantees and may be driven until it is quite old. Each time you lease a new car, all the fees and taxes must be paid again. When buying a car, these fees are only paid once. Leasing contracts carry restrictions on the mileage you can drive per year, and going over incurs more cost at the end of the lease. Buying the car means no such mileage limits. When leasing, you are obligated to keep the vehicle in good condition and maintained according to the dealer’s schedule. Some dealerships will even pay for oil changes over the life of the lease. When the car is purchased, the upkeep schedule is the choice of the owner. When a car is purchased and kept for long enough, the warranty expires and the owner is responsible for all maintenance items and repairs. The warrantee for a car won’t expire during the lease term. When a new car is purchased and the loan is paid off the car is still owned by the buyer and may be traded in when a new car is to be purchased. When leasing, the car is returned to the dealer when the lease term is over. When deciding between the two, you are choosing between these features. If you aren’t willing to drive an older car or deal with the upkeep that accompanies an older car, you may want to lease. This means you will need to pay those beginning costs each time the lease is up. If you want to own the car after the payments are over, then you may want to buy a car. This means you are paying for all the upkeep after the warrantees expire, but you have no limits on mileage and own the car at the end. It really depends on your preferences. Lease or Buy In the following, determine if a lease or purchase of a car is better. Joyce is concerned with large repairs and does not want to deal with them. Maurice prefers to drive newer cars. Since Joyce does not want to deal with repairs, so leasing would be a better choice. This way, the warranty covers most of the big repairs that could need to be done. Since Maurice likes to drive new cars, leasing is a better option, since he will lease a new car every 2 to 3 years. Car Insurance Car insurance is meant to cover costs associated with accidents involving cars. Most states (all except New Hampshire and Virginia) require some insurance. Without insurance, the state may not let you get a license for your car or register your car. Your state’s requirements can be hard to follow. Fortunately, insurance companies and brokers will make sure your insurance is sufficient for your state and will warn you if you try to not meet the requirements. Of course, they may offer more than what is sufficient, so it is your responsibility to determine how much coverage you want, as long as the minimum insurance requirements are met. The cost of insurance should be accounted for when evaluating the affordability of buying or leasing a car. Whether your car is leased or owned, you do need insurance. This contributes to the cost of having the vehicle. Leasing or owning makes no difference to the insurance company you choose, because they are insuring you based on what you are driving, your driving record, and other information about you including where you live and your age. These insurance policies have many components that address different costs that can come from auto accidents. This may make details confusing, and you may not realize what you are paying for until you must use it. Here is a brief outline of the different components of auto insurance, many of which are required by the state that issues your driver’s license. Liability insurance is mandatory coverage in most states. Liability insurance covers property damage and injuries to others should you be found legally responsible for an accident. You are required to have the minimum amount of coverage, as determined by your state, in both areas. Collision insurance is insurance covering the damage caused to your car if involved in an accident with another vehicle. Comprehensive insurance is an extra level of coverage if involved in an accident with another vehicle and covers other things like theft, vandalism, fire, or weather events as outlined in your policy. There is a deductible assigned to each type of insurance, an amount that you pay out of pocket before your comprehensive coverage takes effect. Comprehensive insurance is often required if you lease or finance the purchase of a vehicle. Uninsured or underinsured motorist insurance: If you are hit by an uninsured or underinsured motorist, this insurance will help pay medical bills and damage to your car. Medical payments insurance is mandatory in some states and helps pay for medical costs associated with an accident, regardless of who is at fault. Personal injury protection insurance is coverage for certain medical bills and other expenses due to a car accident. Other covered expenses may include loss of income or childcare, depending on your policy. Gap insurance is designed to cover the gap between what is owed on the car and what the car is worth in the event your car is a total loss. Rental reimbursement insurance is coverage for a rental car while your car is under repair resulting from an accident. You can also purchase other special insurance policies, such as classic car insurance, new car replacement insurance, and sound system replacement insurance, to name a few. It is important that you determine exactly what you need, as insurance policies can be expensive and vary according to your age, driving history, and where you live. Types of Insurance Which component of insurance pays if you are in an accident with a motorist without insurance? Which component of insurance pays for the remaining principal owed on your car in the case of a total loss? Uninsured motorist insurance covers accidents with those who have no insurance. Gap insurance will cover the gap between what is owed on the car and what it is worth if an accident results in a total loss. Monthly Cost of Owning a Car If your car payment is $287.50 per month and your car insurance is $930 every 6 months, what is the cost of the car per month when accounting for the insurance? The cost of the car including insurance is the monthly payment, $287.50, plus the monthly cost of the insurance. The insurance cost per month is $ 930 / 6 = $ 155 since the insurance cost is for every 6 months. Adding those the cost with insurance is $442.50. Maintaining a Car Cars are not a buy it and forget it item. They require upkeep, which adds to the cost of owning the car. Tires, brakes, and wipers need replacing. Oli changes, inspections, so many things other than gasoline. Below is a list of some maintenance requirements for cars, along with cost and roughly how often they should happen. Maintenance Frequency Cost Range New Tires Every 5 years $25–$300 per tire Oil Change Every 3,000–6,000 miles $35–$75 Wipers Every 6–12 months $20–$40 Inspection Annual $10–$50 Brake pads 10,000–20,000 miles $200–$300 Air Filter 15,000–30,000 miles $35–$80 When designing a budget, these expected costs should be accounted for. Extra money per month should be saved in addition to this budget category, to handle unanticipated, and perhaps very costly, repairs. Estella needs to budget for her car maintenance. She expects to buy new tires each 4 years, which will cost her $480 to replace them all. Oil changes near her cost $49.99, and she believes she will get one every 4 months. Her inspection costs $15 per year. Wipers for her car cost $95 for all three and she anticipates changing them every year. She drives less than 30,000 miles per year, so she plans to replace the air filter once per year. The air filter for her car costs $57.50. How much should she budget per month to cover these costs? Her yearly costs are the wipers, inspection, and tires, which total $167.50. Tires will be bought every 5 years, so per year she should budget $96. Her oil changes, which will happen three times per year, cost $49.99 each, so she’ll spend $149.97 for the year on oil changes. Adding these up, her yearly budget should include $413.47 for maintenance. Dividing by 12 gives the monthly budget for maintenance, which is $34.46 (rounded up to the next penny). Check Your Understanding Key Terms Title and registration fees Destination fee Documentation fee Dealer preparation fee Extended warranty Down payment Acquisition fee Security deposit Disposition fees Liability insurance Collision insurance Comprehensive insurance Uninsured or underinsured motorist insurance Medical payment insurance Personal injury insurance Gap insurance Rental reimbursement insurance Key Concepts There are many factors to consider when choosing to buy or lease a car. The cost of the car is increased by a number of fees and sales tax. There are advantages to buying a car and advantages to leasing a car. The decision between the two depends on the preference of the buyer. Insurance covers costs associate with accidents. It is made up of various components. The costs of owning a car, including insurance and maintenance, should be a part of the budgeting process. Budgeting for unexpected repairs can ease the stress of encountering large repair bill. Formula p m t = P × ( r / n ) × ( 1 + r / n ) n × t ( 1 + r / n ) n × t − 1 MD = P − R n APR = 2400 × MF MF = APR / 2,400 P M T = ( P − R ) n + ( P + R ) × MF

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Renting and Homeownership Purchasing a home is a big investment, while renting is a lower cost alternative. (credit: "New construction, new development house for sale" by jongorey/houseandhammer.com, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Evaluate advantages and disadvantages of renting. Evaluate advantages and disadvantages of homeownership. Calculate the monthly payment for a mortgage and related interest cost. Read and interpret an amortization schedule. Solve application problems involving affordability of a mortgage. After renting an apartment for 10 years, you realize that it may be time to purchase a home. Your job is stable, and you could use more space. It is time to investigate becoming a homeowner. What are the things that you must consider, and what is the financial benefit of owning as opposed to renting? This section is about the advantages, disadvantages, and costs of homeownership as opposed to renting. Advantages and Disadvantages of Renting When renting, you will likely sign a lease , which is a contract between a renter and a landlord. A landlord is the person or company that owns property that is rented. The lease will detail your responsibilities, restrictions on activities, deposits, fees, maintenance, repairs, and rent during the term of the lease. It also defines what your landlord can, and cannot, do with the property while you occupy the property. Like leasing a car, there are advantages to renting but also some disadvantages. Some advantages are: Lower cost. Short-term commitment. Little to no maintenance cost. The landlord pays for or performs most maintenance. You need not stay at end of lease. Once the lease term is over (the lease is up), you are not obligated to stay. If renting in an apartment complex, there may be a pool, gym, or community room for renters to use. Of course, there are disadvantage too: No tax incentives. Housing cost is not fixed. When the lease is up, the rent can change. No equity. When you are done living in a rental, you have built no value. Restrictions on occupants. There may be a limit on how many can live in the apartment. Restrictions on decorating. The property is not yours, so any decorating or improvements need landlord permission. Limits on pets. Permission for pets, and their number and type, will be set forth in the lease. May not be able to remain when lease term is over. The landlord can, at the end of your lease, invite you to leave. The building may be sold, and the new landlord may institute changes to the lease when the previous lease expires. Renting has fees to be paid at the start of the lease. Typically, when you rent, you will pay first and last months’ rent and a security deposit. A security deposit is a sum of money that the landlord holds until the renter leaves the rental property. The deposit will cover repairs for damage to the apartment during the renter’s stay but may be returned if the apartment is in good condition. If your landlord runs a credit check on you, the landlord may charge you for that. Advantages of Buying a Home The advantages to buying a home mirror the disadvantages of renting, and the disadvantages of home ownership mirror the advantages of renting. Some advantages to buying a home are: There are tax incentives. The interest you pay for your mortgage (more on that later) is deductible on your federal income tax. There are no restrictions on pets or occupants, unless laws in your area specify limits for homes. You can redecorate any way you wish, limited only by the laws in your area. Once your mortgage is set with a fixed-interest rate, your housing cost is fixed. Your home grows equity, that is, the difference between what you owe and what the house is worth grows. You can use the equity to secure loans, and you recover the equity (and more if you’re fortunate) when you sell the house. As long as you pay your mortgage and maintain the home to the standards of your community, you can stay as long as you wish. Some disadvantages to home ownership are: The cost is higher than renting. Mortgages and associated costs are typically higher than rent for a similar living space. The owner is responsible for upkeep, maintenance, and repairs. These can be extremely costly. The owner cannot walk away from the property. It can be sold, but simply leaving the property, especially if not paid off yet, has serious consequences. The big question of affordability looms large over the decision to rent or buy. Renting, strictly from an affordability viewpoint, comes with much less initial outlay and smaller commitment. If you do not have sufficient income to regularly save for possibly expensive repairs, or your credit isn’t quite as good as it needs to be, then renting may be the best choice. Of course, even if you can afford to buy a home, you may choose to rent based on the comparative advantages. Rent or Buy Buying a home really involves two buyers. You and the mortgage company. The mortgage company has interest in the home, as they are providing the funds for the home. They want to protect their investment, and many fees are about the bank as much as the buyer. They fund a mortgage based on the value they assign the property. Not you. This means they will want some certainty that the home is sound, and you are a good investment. Closing Costs When a home is bought, there are many costs that need to be paid at the time of purchase, which are lumped under the term closing costs. At the start of 2022, the average closing costs for a single-family home exceeded $6,800. These costs include: The appraisal fee, which is what is paid to someone to establish the home’s worth. The value of the home to the bank may differ from what the home is listed for, or what an app tells you the home is worth. It may run approximately $350. The home inspection fee. The inspection should reveal any problems with the house that will need to be fixed either before or after you obtain the home. The title search. The is a records search to insure there are no issues with who actually owns the property. It can cost about 0.5% to 1% of the amount you are financing. Prepaid taxes. You will need to pay about 6 months of taxes at the time of purchase. The credit report fee. This is a fee for checking your credit. You might pay $25 or more for this. The origination fee. This is the price the mortgage company charges you to cover the costs of creating the mortgage. This could be 0.5% to 1% (or more) of the amount you are borrowing. The application fee. This is just a processing fee and could come to several hundred dollars. The underwriting fee. This covers the cost of verifying your financial qualifications. It could be a flat fee, or some small percentage of the amount financed. Such as 0.5% or 1%. Attorney fees. If you use an attorney, you will have to pay the attorney. State of local fees. This may include a filing fee charged by the county or municipality in which you reside. That’s a long list, and it is not even complete. When buying, be prepared to see these costs. It can be surprising. But in the end, you will have equity in the home, which means when you sell your home, you will get some of your money back. Closing Costs In the end, you must weigh your options and carefully consider your priorities in choosing to rent or buy a home. Mortgages Some people will purchase a home or condo with cash, but the majority of people will apply for a mortgage. A mortgage is a long-term loan and the property itself is the security. The bank decides the minimum down payment (with your input), the payment schedule, the duration of the loan, whether the loan can be assumed by another party, and the penalty for late payments. The title of the home belongs to the bank. Since a mortgage is a loan, everything about loans from The Basics of Loans holds true, including the formula for the payments. Monthly Mortgage Payments The formula to calculate your monthly payments of principal and interest uses APR as the annual interest rate. The payment, pmt , per month to pay down a mortgage with beginning principal P is p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 , where r is the annual interest rate in decimal form and t is the number of years of the payment. Note, payment to lenders is always rounded up to the next penny. To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments. 30-Year Mortgage at 4.8% Interest Evan buys a house. His 30-year mortgage comes to $132,650 with 4.8% interest. Find Evan’s monthly payments. Using the information above, P = $132,650, r = 0.048 and t = 30. Substituting those values into the formula p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 and calculating, we find the payment is p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 = $ 132,650 × ( 0.048 / 12 ) × ( 1 + 0.048 / 12 ) 12 × 30 ( 1 + 0.048 / 12 ) 12 × 30 − 1 = $ 132,650 × ( 0.004 ) × ( 1.004 ) 360 ( 1.004 ) 360 − 1 = $ 2,233.07781448 3.20858992551 = $ 695.97 His mortgage payment is $695.97. To find the total amount of your payments over the life of the loan, multiply your monthly payments by the number of payments. This can be useful information, but not too many people reach the end of their mortgage. They tend to move before the mortgage is paid off. The total paid, T , on an t year mortgage with monthly payments pmt is T = p m t × 12 × t . 30-Year Mortgage at 5.35% Interest Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. If Cassandra pays off the mortgage over those 30 years, how much will she have paid in total? To find the total paid over the life of the mortgage, use the formula T = p m t × 12 × t . To calculate this, the payment must be found. Using the information above, P = $99,596, r = 0.0535 and t = 30. Substituting those values into the formula p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 and calculating, we find the payment is p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 = $ 99,596 × ( 0.0535 / 12 ) × ( 1 + 0.0535 / 12 ) 12 × 30 ( 1 + 0.048 / 12 ) 12 × 30 − 1 = $ 99.596 × ( 0.004458 3 ¯ ) × ( 1.004458 3 ¯ ) 360 ( 1.004458 3 ¯ ) 360 − 1 = $ 2,202.45911222 3.96013414693 = $ 556.16 Using the mortgage payment of $556.16 and t = 30 years in the formula T = p m t × 12 × t , the total that Cassandra will pay for the mortgage is $200,217.60. With the principal of the mortgage and how much total is paid over the life of the mortgage, the cost of financing can be found by subtracting the principal of the mortgage from the total paid over the life of the mortgage. The cost of financing a mortgage, CoF, is CoF = T − P where P is the mortgage’s starting principal and T is the total paid over the life of the mortgage. 30-Year Mortgage at 5.35% Interest Cassandra buys a house. Her 30-year mortgage comes to $99,596 with 5.35% interest. What was Cassandra’s cost of financing? In , we found that the total Cassandra will pay for the $99,569 mortgage is $200,217.60. Subtracting those we find the cost of financing CoF = T − P = $ 200,217.60 − $ 99,596 = $ 100,621.60 . Private Mortgage Insurance (PMI) When you purchase a home, you will have to pay a down payment. This means you have money tied to the property, which lenders believe makes you less likely to walk away from a property. The amount of the down payment will be decided between you and the mortgage company. However, if your down payment is less than 20% of the property value, you will be required to pay private mortgage insurance (PMI). This is insurance you pay for so that the mortgage company is protected if you default on the loan. It often comes to between 0.5% and 2.25% of the original loan amount. It increases your monthly payment. Once you reach 20% of the loan value, you can request that the PMI be dropped. Even if you do not request cancelling the PMI, it will eventually and automatically be dropped. For more, see this article on ways to get rid of PMI . Reading and Interpreting Amortization Tables Amortization tables were addressed in The Basics of Loans . They are most frequently encountered when analyzing mortgages. The amortization table for a 30-year mortgage is quite long, containing 360 rows. A full table will not be reproduced here. We can, though, read information from a portion of an amortization table. Amortization Table for a 30-Year, $165,900 Mortgage shows a portion of an amortization table for a 30-year, $165,900 mortgage. Use that table to answer the following questions. What is the interest rate? How much are the payments? How much of payment 175 goes to principal? How much of payment 180 goes to interest? What’s the remaining balance on the mortgage after payment 170? Amortization table Reading at the top of the table, we see the interest rate is 5.61%. Reading from the top of the table or from the column labeled Payment, we see the payments are $953.44 per month. In the row for payment 175, we see that the amount that goes to principal is $400.44. In the row for payment 180, we see that the amount that goes to interest is $543.56. In the row for payment 170, we see the remaining balance is $119,873.35. Escrow Payments The last few examples have looked at mortgage payments, which cover the principal and interest. However, when you take out a mortgage, the payment is sometimes much higher than that. This is because your mortgage company also has you pay into an escrow account , which is a savings account maintained by the mortgage company. Your insurance payments will be set by your insurer and the mortgage company will pay them on time for you from your escrow account. Your property taxes are set by where you live and are typically a percentage of your property’s assessed value . The assessed value is the estimation of the value of your home and does not necessary reflect the purchase or resale value of the home. Your property taxes will also be paid on time by the mortgage company from your escrow account. For example, in Kalamazoo, Michigan, the effective tax rate for property is 1.69% of the assessed value of the home. These escrow payments, which cover bills for the home, can increase the monthly payments for your home well beyond the basic principal and interest payment. Adding Escrow Payments to Mortgage Payments Jenna decides to purchase a home, with mortgage of $108,450 at 6% interest for 30 years. The assessed value of her home is $75,600. Her property taxes come to 5.7% of her assessed value. Jenna also has to pay her home insurance every 6 months, which is $744 per six months. How much, including escrow, will Jenna pay per month? Using the payment function to find her mortgage payments, p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 , with P = $108,405, r = 0.06, and t = 30, her payments are p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 = $ 108,405 × ( 0.06 / 12 ) × ( 1 + 0.06 / 12 ) 12 × 30 ( 1 + 0.06 / 12 ) 12 × 30 − 1 = $ 649.95 Jenna also pays into escrow 1/12 of her property taxes per month. Her property taxes are 5.7% of the assessed value of $75,600, which comes to 0.057 × $ 75,600 = $ 4309.20 . This is an annual tax, so she pays 1/12 of that each month, or $359.10. Jenna’s home insurance is $744 per 6 months, so each month she pays $124.00 for insurance. Adding these together, her monthly payment is $ 649.95 + $ 359.10 + $ 124.00 = $ 1,133.05 . This is quite a bit more than the $649.95 for the principal and interest. Check Your Understanding Key Terms Lease Landlord Mortgage Escrow account Assessed value Key Concepts There are many points of comparison between renting and buying a house. Before deciding to buy a house, you should carefully consider all the responsibilities that come with home ownership. Renting comes with more restrictions on the renter, but with fewer costs and is easier to move from. Owning a house has more costs but has more freedom, plus the owner creates equity. Mortgages are loans, and payments are calculated in the same way as any other loan. Amortization tables help a homeowner understand the mortgage and how the payments are applied to the principal and interest. In addition to paying the amount financed for a mortgage, the monthly payment will include an escrow payment, which covers insurance and taxes. Video Rent or Buy Closing Costs Formula p m t = P × ( r / 12 ) × ( 1 + r / 12 ) 12 × t ( 1 + r / 12 ) 12 × t − 1 T = p m t × 12 × t CoF = T − P

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Income Tax Federal income tax is a concern for most US citizens. (credit: "1040 US tax form" by Marco Verch Professional Photographer/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Determine gross, adjusted gross, and taxable income. Apply exemptions, deductions, and credits to basic income tax calculations. Compute FICA tax. Solve tax application problems for working students. Before the start of the American Civil War in 1861, most of the country’s revenue came from tariffs on trade and excise taxes. However, this fell far short of the high cost of the war. Because of this, the federal government enacted the nation’s first income tax with the Revenue Act of 1861, which created the Internal Revenue Service as we know it today. No one likes paying income tax, but it is a reality of life. In this section, we will learn about Form 1040, the U.S. Individual Income Tax Return, and ways to prepare for tax time. The U.S. tax code may change from year to year. Because of this, this section includes examples of how taxes, deductions, and exemptions might be computed. The types of income, deductions, and exemptions that are used in the examples are used in the current tax code. Gross, Adjusted Gross, and Taxable Income Your income drives how much you pay in taxes. The more you earn, the more you are likely to pay. But your income alone is not the full story. When you add all the money you earned from your job, freelance work, interest from savings, and other sources, you have your gross income . If you are an employee, your income from your job will be reported on a W-2, which is sent to you by your employer. Income from freelance work will be reported on a 1099-MISC form, and is sent by the company that paid you. Income from interest is reported on a 1099-INT form and comes from the entity that paid the interest. Before you determine how much you owe in taxes, you will make certain adjustments to that gross income. You will deduct, or subtract, some of income from the gross income. That’s your adjusted gross income , or AGI. That is still not what you are taxed on. Next, you need to apply exemptions to your income. These are pieces of income that the government does not tax. After that is done, you reach your taxable income . We will look at each of these parts of the taxable income. Gifts and Winning Money given as a gift my be taxed if the gift amount is high enough. If you win $50,000 in the lottery, that money is taxed as income. If you give a family member a large cash gift, that gift will be subject to a tax provided that the gift exceeds the federally set limits. You will notice that your paycheck already has taxes taken out of it. Your employer will withhold some of your income, sending it directly to the federal, state, and local governments. It is an estimate of how much you will owe in income tax. In the end, it reduces how much you will pay when your taxes are due. If they withhold too much income, you will receive the extra they withheld in the form of a refund. Computing Gross Income Roger is preparing to do his taxes. He worked two jobs, with reported income of $27,500 and $13,200. His two CDs yielded $327 together. He also won a split club raffle for $2,000. What was Roger’s gross earnings? This is the sum of his wages, winnings and interest earned. Add these Roger’s gross income was $43,027. Your adjusted gross income (AGI) is computed before your taxes are determined. It begins with the gross income, and then subtracts from that income any deduction. Deductions are expenditures on your part that the government won’t tax. These deductions include money deposited into tax-deferred investments, and mortgage interest that you paid, charitable contributions if you made any, medical bills over a threshold, medical insurance under certain circumstances, and property taxes. If you add all these up, and they are all legal deductions, the sum is subtracted from your gross income, leaving the AGI. Compute Adjusted Gross Income Chandra’s gross income is $58,400. She deposited $3,000 into her tax-deferred retirement account for the year. Her mortgage interest paid was $1,250 for the year, and her property taxes came to $4,200. What is her AGI? To get Chandra ’s AGI, total her deductions and subtract that total from her gross income. The total of her deductions is $ 3,000 + $ 1,250 + $ 4,200 = $ 8,450 . She subtracts that total from her gross income, making her AGI $ 58,400 − $ 8,450 = $ 57,950 . Remember that your AGI is not your taxable income . Exemptions need to be subtracted from the AGI to reach your taxable income. Exemptions are income that the government does not tax. Some examples of exempt income are disbursements from health savings accounts for qualified medical expenses, bond interest, some IRA distributions, and gifts given that are under $16,000. Note that exemptions are different from deductions: exemptions are excused incomes, whereas deductions are excused expenditures. Taxable Income Yelizaveta’s AGI is $75,490. However, she gave a $2,000 gift to her mother. What is her taxable income? Taxable income is AGI minus any exemptions. Gifts below $16,000 are exempted, so her taxable income is $ 75,490 − $ 2,000 = $ 75,490 . Tax Credits Another piece of the tax puzzle is tax credits . This is money subtracted from the tax you owe. Tax credits are very different from deductions or exemptions. Deductions and exemptions are taken away from your gross income before the tax you owe is calculated. A tax credit, is subtracted, dollar for dollar, from your tax bill. Once the tax you owe is calculated, subtract the any tax credits from that calculated tax. Some of the tax credits are refundable. This means that if subtracting them from your tax results in a negative number, you receive a tax refund. For more details, see this article about tax credits . The federal government has placed income limits and restrictions and on those eligible to receive tax credits because their value is so high. Here is a partial list of tax credits that you might qualify for: Earned income credit is a refundable tax credit for low- to moderate-income workers and ranges from $560 to $6,935 depending on dependents and income. This is refundable American opportunity credit is a credit taken by parents who have children enrolled in college at least half time and pursuing a degree. This credit is worth $2,500 per student for the first 4 years of undergraduate school, subject to income limits. This is a refundable tax credit. Lifetime learning credit is a credit is equivalent to 20% of educational expenses, up to $2,000 per year, subject to income limits. There is no cap to how many years you can apply for this credit. Child tax credit is worth $2,000 per child under the age of 17 if that child lives at home at least half the year, subject to income limits. This is a refundable tax credit. Child and dependent care tax credit was designed to help pay for child care while the parent works. The amount of the credit is dependent on your income. However, the maximum amount that can be received is, in 2022, $4,000 for one eligible person, or $8,000 for two or more qualifying people. A dependent qualifies if they are a child under 13 years old, a spouse who is unable to care for themselves, or some other qualifying person. This is a refundable tax credit. Premium tax credit was created by the Affordable Care Act, and it is one that is received by many people throughout the year. In essence it is a health insurance premium subsidy. The amount of the credit is based on your income and the price of health insurance in your area. This is a refundable tax credit. Apply a Tax Credit Kaitlyn has calculated the tax she owes to be $5,200. However, she receives an earned income tax credit of $1,715. How much does Kaitlyn owe after applying the earned income tax credit? The amount of taxes Katilyn owes is her calculated tax of $5,200 minus the credit she receives. The amount she owes in taxes is $ 5,200 − $ 1,715 = $ 3,485 . Tax Credits and a Refund Chanajah calculated their tax owed, which came to $4,300. They have an earned income tax credit of $2,190, a child tax credit of $2,000, and a child and dependent care tax credit of $4,000. How much tax does Chanajah owe, or how much will they get in a refund? Adding Chanajah’s tax credits together, we find their total to be $8,190. That is more than the tax they owe, which was $4,300. Each of those credits is refundable, which means they will receive a refund. Subtracting the credits from the tax owed yields $ 4,300 − $ 8,190 = − $ 3,890 . This is negative, so represents a refund of $3,890. Deductions Versus Credits Computing FICA Taxes FICA stands for the Federal Insurance Contributions Act of 1935. FICA taxes are used solely to fund Social Security and Medicare and are separate from federal income tax. It amounts to 7.65% of your gross pay, which is withheld from your paycheck automatically. Your employer is required to match the 7.65% amount. Of the 7.65%, 6.2% goes to Social Security (SSI), and 1.45% goes to Medicare. As of 2022, SSI tax only applies to the first $147,000 of earnings. Any gross income above that is not taxed for social security. This limit changes every year. Medicare tax, on the other hand, applies to the entirety of your gross income. Computed FICA Taxes McKenzi earned $2,700 in gross income, before taxes, in a given 2-week period. How much does she owe in FICA taxes, and how much of that is for SSI? The FICA tax is 7.65% of her gross earnings. 7.65% of her $2,700 is $206.55. Also, the SSI is 6.2% of her income, or $167.40 Social Security Tax with Higher Income Renard earned $195,000 in wages for the year. How much in SSI taxes does Renard owe for the year? Since Renard earned more than the taxable limit of $147,000 dollars, he only pays the 6.2% SSI tax on $147,000. This comes to $9,114. Calculating Your Income Tax Your income tax bill and your income tax rate are based on your taxable income. The tax system in the United States is progressive, meaning that the tax rates are marginal so the higher your taxable income the higher the tax rate you will pay. Taxable income is broken into brackets, or ranges of income. Each bracket has a different tax rate. The tax brackets and rates for single filers as of 2022 are given below: Bracket Lower Income Limit Upper Income Limit Tax Rate 1 0 $10,275 10% 2 $10,276 $41,775 12% 3 $41,776 $89,075 22% 4 $89,076 $170,050 24% 5 $170,051 $215,950 32% 6 $215,951 $539,900 35% 7 $539,901 37% Federal Income Tax Brackets for Single Filers, 2021–2022 (data source: https://www.irs.gov/newsroom/irs-provides-tax-inflation-adjustments-for-tax-year-2022) So if your taxable income is $76,500 and you are filing as a single filer, your tax bill will be 22% of that $76,500, right? Wrong. Your income is split among those brackets and the money in each bracket is taxed at that bracket’s tax rate. Seems confusing. Here is a list of steps to follow to find the tax owed. Step 1: Find the bracket for the income. Step 2: For each bracket below the income bracket, the tax from that bracket is: Step 2a: Find the difference between the upper limit of that bracket and upper limit of the next lower bracket. If this is bracket 1, use 0 as the upper limit of the previous bracket. Step 2b: The tax from that bracket is the bracket tax rate applied to the difference from Step 2a. Step 3: For the bracket that the income belongs to, find the income minus the lower limit for the bracket. Step 4: The tax for the bracket of the income is tax rate for that bracket applied to the difference found in Step 3. Step 5: Add these various tax values to get the total income tax. There are various tax brackets, and the rates may change in any given year. The income limits may also change. For all examples going forward, we will use the single filer tax brackets, even if those brackets are not appropriate (e.g., married or head of household filers). Income Tax on Taxable Income Faith has a taxable income of $103,650. How much income tax does Faith owe? Step 1: Faith’s income belongs to the 4th tax bracket, $89,076 to $170,050. So the taxes from the first three brackets follow steps 2a and 2b. For bracket 1: Step 2a: The upper limit is $10,275, the upper limit of the previous bracket was 0, so the difference is $10,275. Step 2b: For the first bracket, she owes 10% of $10,275, or $1,027.50. For bracket 2: Step 2a: The upper limit is $41,775, the upper limit from the previous bracket was $10,275, so the difference is $31,500 Step 2b: For the second bracket, she owes 12% of $31,500, or $3,780. For bracket 3: Step 2a: The upper limit is $89,075, the upper limit from the previous bracket is $41,175, so the difference was $47,300 Step 2b: For the third bracket, she owes 22% of $47,300, or $10,406. Step 3: For bracket 4, her income is $103,650, the upper limit of the previous bracket is $89,075. The difference of those is $ 103,650 − $ 89,075 = $ 14,575 . Step 4: The tax he owes for that bracket is 24% of the difference, which is $3,498. Step 5: The total in taxes that Emmanuel owes is the sum of the taxes found above, or $ 1,027.5 + $ 3,780 + $ 10,406 + $ 3,498 = $ 18,711.50 Remember that your employer will estimate how much tax you will owe and withholds it from paychecks. This means you may have already paid some, if not all and more, of the income tax you owe. Tax Brackets Finding Income Tax Owed Emmanuel is preparing his taxes. His W-2 from work shows gross income for the year of $95,250. He also has a 1099-MISC for some freelance art work he did, amounting to $7,500. Emmanuel also deposited $4,500 into a tax-deferred retirement plan. He paid $7,920 in mortgage interest for the year and $3,740 in property taxes. He also qualified for $4,000 in child tax credits. Based on this information, how much tax does Emmanuel owe or how much does he get in a refund? We need to know Emmanuel’s taxable income, based on gross income, deductions, and exemptions. His gross income is the amount from his W-2 and his 1099-MISC. Adding these gives a gross income of $ 95,250 + $ 7,500 = $ 102,750 . Subtracting his deductions will yield his AGI. His deductions are $7,920 for mortgage interest, $3,740 for property taxes, and $4,500 deposited into his retirement account. Adding these, his total deductions are $16,160. Subtracting the deductions from the gross income, we find his AGI to be $86,590. Emmanuel seems to have no exemptions, so his AGI and his taxable income are the same. We find the taxes Emmanuel owes using the process outlined above. Step 1: His income belongs to the third tax bracket, $41,776 to $89,0875. So the taxes from the first two brackets follow steps 2a and 2b. For bracket 1: Step 2a: The upper limit is $10,275, the upper limit of the previous bracket was 0, so the difference is $10,275. Step 2b: For the first bracket, he owes 10% of $10,275, or $1,027.50. For bracket 2: Step 2a: The upper limit is $41,775, the upper limit from the previous bracket was $10,275, so the difference is $31,500 Step 2b: For the second bracket, he owes 12% of $31,500, or $3,780. Step 3: For bracket 3, his income is $86,590, the upper limit of the previous bracket is $41,775. The difference of those is $ 86,590 − $ 41,775 = $ 44,815 . Step 4: The tax he owes for that bracket is 22% of the difference, which is $9,859.30. Step 5: The total in taxes that Emmanuel owes is the sum of the taxes found above, or $ 1027.5 + $ 3780 + $ 9,859.30 = $ 14,666.80 . Once his taxes are computed, he subtracts his tax credits. His only tax credit is $4,000. The total he owes in taxes is $ 14,666.80 − $ 4,000 = $ 10,666.80 . Check Your Understanding Key Terms Gross income Adjusted gross income Exemption Taxable income Deduction Tax credit Earned income credit American opportunity credit Lifetime learning credit Child tax credit Child and dependent care tax credit Premium tax credit Key Concepts Federal income tax is based on income after certain adjustments. Gross income is income from all sources, including gifts and winnings. Before taxes are calculated, the taxable income is found by subtracting deductions and exemptions from gross income. Income tax is progressive, increasing in rate as income increases. Being in the 32% tax bracket means some of your income is taxed at 10%, some at 12%, some at 22%, some at 24%, and the rest at 32%. Income in each tax bracket is taxed at that bracket’s rate, which means in 2022 the first $10,275 earned is taxed at 10% only. Tax credits are subtracted from the taxes that are owed. Some tax credits are refundable, which means they can make the amount you owe negative, which results in a refund. Video Deductions Versus Credits Tax Brackets Project Creating Your Future Budget In this project, you will create a budget based on a job you are likely to have after you graduate. Go online and research the average starting salary for the profession you are studying for. Use at least two sources. Be sure to record the web address from your search. Approximate your monthly take-home pay. You may use the SmartAsset website to estimate this. Use the 50-30-20 budget philosophy to determine how much you should budget for needs, wants and savings, or extra debt reduction. Create a list of likely expenses. This list must include rent/mortgage, utilities, food, and school loan repayment. You may also want to include car payments, gasoline, and other items. Categorize each expense as need, want, or savings. Using the amounts found in step 3, decide how much to allocate to each of your expenses. It may help to quickly research how much rent is where you want to live. Discuss the choices you had to make, and why you prioritized some expenses over others. Interest Rate and Time: What Is the Relationship? The interplay between interest rate and time for an annuity is not easily seen. How the amount that must be deposited per compounding period, pmt , changes based on the time and interest rate would be useful to understand. In this project, you will explore this relationship. We will use a fixed future value of FV = $1,000,000 and a fixed number of periods per year, 12 (monthly compounding). With those, we’ll find various annuity payments that must be made to reach the goal. The annual interest rate for the investment is in the top row. The number of years for the investment is in the left column. In each cell (or box), find the monthly payment necessary to reach the goal of $1,000,000. Annual Interest Rate 1.5% 2.0% 3.0% 5.0% 7.5% 10.0% Number of Years 10 15 20 30 40 45 Describe how the interest rates and number of years impact the payment necessary to reach the goal of $1,000,000. Finding a Home In this project you will identify a home you like, and then estimate the costs associated with that home. Find a home in your region that you would like to buy using an online search of listings in your area. Zillow is a good place to begin. Find the asking price for this home. Assume you would pay that price. Find an estimation for closing costs in your area. Assume you finance those costs also. Estimate the taxes to be paid on the home per year. It is likely that the online listing of the home has an estimate for the taxes for the house. Use Google to determine the average homeowner’s insurance cost in your region. Use the internet to determine the average interest rate for a 30-year mortgage. Find how much you would pay per month, based on the answers to the previous questions, including the escrow payments for taxes and insurance. Assume you will pay $50 per $100,000 borrowed in PMI. Add this to the monthly payment. Chapter Review Understanding Percent Discounts, Markups, and Sales Tax Simple Interest Compound Interest Making a Personal Budget Methods of Savings Investments The Basics of Loans Understanding Student Loans Credit Cards Buying or Leasing a Car Renting and Homeownership Income Tax Chapter Test

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Introduction Roulette is a game whose outcomes are based entirely on the concept of probability. (credit: modification of work "Roulette wheel" by Håkan Dahlström/Flickr, CC BY 2.0) Casinos are big business; according to the American Gaming Association, commercial casinos in the United States brought in over $43 billion in revenue in 2019. Casinos must walk a fine line in order to be profitable. Their customers must lose more money than they win, on average, in order to stay in business. But if the chances of a single customer winning more money than they lose is too small, people will stop coming in the door to play the games. In this chapter, we'll study the techniques a casino must use to determine how likely it is that a customer will win a particular game, and then how the casino decides how much money a winner will rake in so that the customers are happy, but the casino also turns a profit in the long run. In order to figure out those likelihoods, we have to be able to somehow consider every possible outcome of these games. For example, in a game that involves players receiving 5 cards from a deck of 52, there are 2,598,960 possibilities for each player. We'll start off this chapter by learning how to count those possible outcomes.

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The Multiplication Rule for Counting The Multiplication Rule for Counting allows us to compute more complicated probabilities, like drawing two aces from a deck. (credit: “Pair of Aces – Poker” by Poker Photos/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Apply the Multiplication Rule for Counting to solve problems. One of the first bits of mathematical knowledge children learn is how to count objects by pointing to them in turn and saying: “one, two, three, …” That’s a useful skill, but when the number of things that we need to count grows large, that method becomes onerous (or, for very large numbers, impossible for humans to accomplish in a typical human lifespan). So, mathematicians have developed short cuts to counting big numbers. These techniques fall under the mathematical discipline of combinatorics , which is devoted to counting. Multiplication as a Combinatorial Short Cut One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects ( ): Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them ( ). If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since multiplication is repeated addition (i.e., 6 × 4 = 4 + 4 + 4 + 4 + 4 + 4 ), we can use this grouping to quickly see that there are 24 balls. Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns ( Figure 7.5 ). Since there are 3 rows (cakes) and 4 columns (frostings), we have 3 × 4 = 12 possible combinations. This is the reasoning behind the Multiplication Rule for Counting , which is also known as the Fundamental Counting Principle. This rule says that if there are n ways to accomplish one task and m ways to accomplish a second task, then there are n × m ways to accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to our previous product. Using the Multiplication Rule for Counting Every card in a standard deck of cards has two identifying characteristics: a suit (clubs, diamonds, hearts, or spades; these are indicated by these symbols, respectively: ♣ , ♢ , ♡ , ♠ ) and a rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king; the letters A, J, Q, and K are used to represent the words). Each possible pair of suit and rank appears exactly once in the deck. How many cards are in the standard deck? Since there are 4 suits and 13 ranks, the number of cards must be 4 × 13 = 52 ( ). Standard Deck of Cards, Sorted by Rank and Suit (credit: "Playing Cards, USS Arkansas" by Naval History & Heritage Command/Flickr, CC BY 2.0) Using the Multiplication Rule for Counting for 4 Groups The University Combinatorics Club has 31 members: 8 seniors, 7 juniors, 5 sophomores, and 11 first-years. How many possible 4-person committees can be formed by selecting 1 member from each class? Since we have 8 choices for the senior, 7 choices for the junior, 5 for the sophomore, and 11 for the first-year, there are 8 × 7 × 5 × 11 = 3,080 different ways to fill out the committee. Using the Multiplication Rule for Counting for More Groups The standard license plates for vehicles in a certain state consist of 6 characters: 3 letters followed by 3 digits. There are 26 letters in the alphabet and 10 digits (0 through 9) to choose from. How many license plates can be made using this format? Since there are 26 different letters and 10 different digits, the total number of possible license plates is 26 × 26 × 26 × 10 × 10 × 10 = 17 , 576 , 000 . Check Your Understanding Key Terms combinatorics Multiplication Rule for Counting (Fundamental Counting Principle) Key Concepts The Multiplication Rule for Counting is used to count large sets.

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Permutations We can use permutations to calculate the number of different orders of finish in an Olympic swimming heat. (credit: “London 2012 Olympics Park Stratford London” by Gary Bembridge/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Use the Multiplication Rule for Counting to determine the number of permutations. Compute expressions containing factorials. Compute permutations. Apply permutations to solve problems. Swimming events are some of the most popular events at the summer Olympic Games. In the finals of each event, 8 swimmers compete at the same time, making for some exciting finishes. How many different orders of finish are possible in these events? In this section, we’ll extend the Multiplication Rule for Counting to help answer questions like this one, which relate to permutations . A permutation is an ordered list of objects taken from a given population. The length of the list is given, and the list cannot contain any repeated items. Applying the Multiplication Rule for Counting to Permutations In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are. Using the Multiplication Rule for Counting to Find the Number of Permutations The final heat of Olympic swimming events features 8 swimmers (or teams of swimmers). How many different podium placements (first place, second place, and third place) are possible? How many different complete orders of finish (first place through eighth place) are possible? Let’s start with the first place finisher. How many options are there? Since 8 swimmers are competing, there are 8 possibilities. Once that first swimmer completes the race, there are 7 swimmers left competing for second place. After the second finisher is decided, there are 6 swimmers remaining who could possibly finish in third place. Thus, there are 8 possibilities for first place, 7 for second place, and 6 for third place. The Multiplication Rule for Counting then tells us there are 8 × 7 × 6 = 336 different ways the winners’ podium can be filled out. To look at the complete order of finish, we can continue the pattern we can see in part 1 of this example: There are 5 possibilities for fourth place, 4 for fifth place, 3 for sixth place, 2 for seventh place, and then just 1 swimmer is left to finish in eighth place. Using the Multiplication Rule for Counting, we see that there are 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40 , 320 possible orders of finish. Factorials The pattern we see in occurs commonly enough that we have a name for it: factorial . For any positive whole number n , we define the factorial of n (denoted n ! and read " n factorial") to be the product of every whole number less than or equal to n . We also define 0! to be equal to one. We will use factorials in a couple of different contexts, so let's get some practice doing computations with them. Computing Factorials Compute the following: 4 ! 8 ! 6 ! 9 ! 3 ! 4 ! 4 ! = 4 × 3 × 2 × 1 = 24 There are two ways to approach this calculation. The first way is to compute the factorials first, then divide: 8 ! 6 ! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 6 × 5 × 4 × 3 × 2 × 1 = 40,320 720 = 56 However, there is an easier way! You may notice in the second step that there are several terms that can be canceled; that’s always the case whenever we divide factorials. In this case, notice that we can rewrite the numerator like this: 8 ! = 8 × 7 × ( 6 × 5 × 4 × 3 × 2 × 1 ) = 8 × 7 × 6 ! With that in mind, we can proceed this way by canceling out the 6!: 8 ! 6 ! = 8 × 7 × 6 ! 6 ! = 8 × 7 = 56 That’s much easier! Let’s approach this one using our canceling technique. When we see two factorials in either the numerator or denominator, we should focus on the larger one first. So: 9 ! 3 ! 4 ! = 9 × 8 × 7 × 6 × 5 × 4 ! 3 ! 4 ! = 9 × 8 × 7 × 6 × 5 3 × 2 × 1 = 9 × 8 × 7 × 5 = 2,520 Permutations As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to make that connection explicit. Let’s define some notation first. If we have a collection of n objects and we wish to create an ordered list of r of the objects (where 1 ≤ r ≤ n ), we’ll call the number of those permutations n P r (read “the number of permutations of n objects taken r at a time”). We formalize the formula we'll use to compute permutations below. n P r = n ! ( n − r ) ! If you wondered why we defined 0 ! = 1 earlier, it was to make formulas like this one work; if we have n objects and want to order all of them (so, we want the number of permutations of n objects taken n at a time), we get n P n = n ! ( n − n ) ! = n ! 0 ! = n ! 1 = n ! . Next, we’ll get some practice computing these permutations. Computing Permutations Find the following numbers: The number of permutations of 12 objects taken 3 at a time The number of permutations of 8 objects taken 5 at a time The number of permutations of 32 objects taken 2 at a time 12 P 3 = 12 ! ( 12 − 3 ) ! = 12 ! 9 ! = 12 × 11 × 10 × 9 ! 9 ! = 12 × 11 × 10 = 1,320 8 P 5 = 8 ! ( 8 − 5 ) ! = 8 × 7 × 6 × 5 × 4 × 3 ! 3 ! = 8 × 7 × 6 × 5 × 4 = 6,720 32 P 2 = 32 ! ( 32 − 2 ) ! = 32 × 31 × 30 ! 30 ! = 32 × 31 = 992 Applying Permutations A high school graduating class has 312 students. The top student is declared valedictorian, and the second-best is named salutatorian. How many possible outcomes are there for the valedictorian and salutatorian? In the card game blackjack, the dealer’s hand of 2 cards is dealt with 1 card faceup and 1 card facedown. If the game is being played with a single deck of (52) cards, how many possible hands could the dealer get? The University Combinatorics Club has 3 officers: president, vice president, and treasurer. If there are 18 members of the club, how many ways are there to fill the officer positions? This is the number of permutations of 312 students taken 2 at a time, and 312 P 2 = 97,032 . We want the number of permutations of 52 cards taken 2 at a time, and 52 P 2 = 2,652 . Here we’re looking for the number of permutations of 18 members taken 3 at a time, and 18 P 3 = 4,896 . Very Big Permutations Permutations involving relatively small sets of objects can get very big, very quickly. A standard deck contains 52 cards. So, the number of different ways to shuffle the cards—in other words, the number of permutations of 52 objects taken 52 at a time—is 52 ! ≈ 8 × 10 67 (written out, that’s an 8 followed by 67 zeroes). The estimated age of the universe is only about 4 × 10 17 seconds. So, if a very bored all-powerful being started shuffling cards at the instant the universe began, it would have to have averaged at least 8 × 10 67 4 × 10 17 ≈ 2 × 10 50 shuffles per second since the beginning of time to have covered every possible arrangement of a deck of cards. That means the next time you pick up a deck of cards and give it a good shuffle, it’s almost certain that the particular arrangement you created has never been created before and likely never will be created again. Check Your Understanding Key Terms permutation factorial Key Concepts Using the Multiplication Rule for Counting to enumerate permutations. Simplifying and computing expressions involving factorials. Using factorials to count permutations. Formulas P n r = n ! ( n − r ) !

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