Discounts, Markups, and Sales Tax Sale prices are often described as percent discounts. (credit: "Close-up of a discount sign" by Ivan Radic/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Calculate discounts. Solve application problems involving discounts. Calculate markups. Solve application problems involving markups. Compute sales tax. Solve application problems involving sales tax. Many people first encounter percentages during a retail transaction such as a percent discount (SALE! 25% off!!), or through sales tax ("Wait, I thought this was $1.99?"), a report that something has increased by some percentage of the previous value (NOW! 20% more!!). These are examples of percent decreases and percent increases. In this section, we discuss decrease, increase, and then the case of sales tax. Calculating Discounts Retailers frequently hold sales to help move merchandise. The sale price is almost always expressed as some amount off the original price. These are discounts , a reduction in the price of something. The price after the discount is sometimes referred to as the reduced price or the sale price. When a reduction is a percent discount, it is an application of percent, which was introduced in Understanding Percent . The formula used was part = percentage × total . In a discount application, the discount plays the role of the part, the percent discount is the percentage, and the original price plays the role of the total. The formula for a discount based on a percentage is discount = percent discount × original price , with the percent discount expressed as a decimal. The price of the item after the discount is sale price = original price - discount . These are often combined into the following formula sale price = original price - percent discount × original price = original price × ( 1 - percent discount ) When the original price and the percent discount are known, the discount and the sale price can be directly computed. Calculating Discount for a Percent Discount Calculate the discount for the given price and discount percentage. Then calculate the sale price. Original price = $75.80; percent discount is 25% Original price = $168.90; percent discount is 30% Substituting the values into the formula discount = percent discount × original price , we find that the discount is discount = 0 .25 × 75.80 = 18.95 . The discount is $18.95. The sale price of the item is then sale price = original price - discount = 75.80 - 18.95 = 56.85 , or $56.85. Substituting the values into the formula discount = percent discount × original price , we find that the discount is discount = 0 .30 × 168.90 = 50.67 . The discount is $50.67. The sale price of the item is then sale price = original price - discount = 168.90 - 50.67 = 118.23 , or $118.23. Sometimes the original price and the sale price of an item is known. From this, the percent discount can be computed using the formula discount = percent discount × original price , by solving for the percent discount. Calculating the Percent Discount from the Original and Sale Prices Determine the percent discount based on the given original and sale prices. Original price = $1,200.00; sale price = $900.00 Original price = $36.70; sale price = $29.52 Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula sale price = original price - discount . Substituting and calculating, we find the discount to be 900.00 = 1 , 200.00 - discount . Solving for the discount gives $300.00. Step 2. Find the percent discount. Substituting the discount of $300.00 and the original price of $1,200.00, into the formula discount = percent discount × original price , we can find the percent discount. 300 .00 = percent discount × 1,200 .00 300 .00 1 , 200.00 = percent discount 0.25 = percent discount Converting to percent form, the percent discount is 25%. Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula sale price = original price - discount . Substituting and calculating, we find the discount to be 29.52 = 36.70 - discount . Solving for the discount gives $7.38. Step 2. Find the percent discount. Substituting the discount of $7.38 and the original price of $36.70, into the formula discount = percent discount × original price , we can find the percent discount. 29 .52 = percent discount × 36 .70 29 .52 36.70 = percent discount 0.2 = percent discount Converting to percent form, the percent discount is 20%. Sometimes the sale price and the percent discount of an item are known. From this, the original price can be found. To avoid multiple steps, though, the formula that we will use is sale price = original price × (1 - percent discount) . The original price can be found by solving this equation for the original price. Calculating the Original Price from the Percent Discount and Sale Price Determine the original price based on the percent discount and sale price. Percent discount 10%; sale price = $450.00 Percent discount 75%, sale price = $90.00 Using the percent discount and the sale price, we can find the original price with the formula sale price = original price × (1 - percent discount) . Substituting and solving for the original price, we find sale price = original price × (1 - percent discount) 450 .00 = original price × (1 - 0 .10) 450 .00 = original price × (0 .90) 500 .00 = original price The original price of the item was $500.00. Using the percent discount and the sale price, we can find the original price with the formula sale price = original price × (1 - percent discount) . Substituting and solving for the original price, we find sale price = original price × (1 - percent discount) 90 .00 = original price × (1 - 0 .75) 90 .00 = original price × (0 .25) 360 .00 = original price The original price of the item was $360.00. Solve Application Problems Involving Discounts In application problems, identify what is given and what is to be found, using the terms that have been learned, such as discount, original price, percent discount, and sale price. Once you have identified those, use the appropriate formula (or formulas) to find the solution(s). Determine Discount and New Price a Sale Rack Item The sale rack at a clothing store is marked “All Items 30% off.” Ian finds a shirt that had an original price of $80.00. What is the discount on the shirt? What is the sale price of the shirt? We are asked to find the discount, and the sale price. We know the percent discount is 30%, or 0.30 in decimal form. The original price was $80. Substituting into the percent discount formula, we find that the discount is discount = percent discount × original price = 0 .30 × 80 = 24 . The discount is $24 on that shirt. The sale price is the original price minus the discount, so the sale price is $80 – $24 = $56. Determine the Percent Discount of a Bus Pass An annual pass on the city bus is priced at $240. The student price, though, is $168. What is the percent discount for students for the bus pass? We know the original price of the item, $240. We also know the sale price of the item, $168. From this we know the discount is $ 240 - $ 168 = $ 72 . Substituting these values into the formula discount = percent discount × original price , we can find the percent discount. discount = percent discount × original price 72 = percent discount × 240 72 240 = percent discount × 240 240 0.3 = percent discount The student percent discount on the bus pass is 30%. Finding the Original Price of a New Pair of Tires Kendra’s car developed a flat, and the tire store told her that two tires had to be replaced. She got a 10% discount on the pair of tires, and the sale price came to $189.00. What was the original price of the tires? Using the percent discount and the sale price, we can find the original price with the formula sale price = original price × (1 - percent discount) . Substituting and solving for the original price, we find sales price = original price × (1 - percent discount) 189 .00 = original price × (1 - 0 .10) 189 .00 = original price × (0 .90) 210 .00 = original price The original price of the two tires Kendra bought was $210.00. Computing Price Based on a Percent Off Coupon There are cases where retailers allow multiple discounts to be applied. However, it is rare that the discount percentages are added together. For example, if you have a 15% coupon and qualify for a 20% price reduction, the retailer typically does not add those two percentages together to determine the new price. The retailer instead applies one discount, then applies the second discount to the price obtained after the first discount was deducted. Research the original prices of two different laptops offered by one retail outlet. Assume you will receive a student discount of 12% and your outlet of choice is having a 15% off sale on all laptops. For each laptop: List the original price and calculate the price after applying the student discount (12%) only. Then find the price after applying the sale discount (15% off) to the price found in Step 1. Determine the total saved on the laptop and what percent discount the total savings represents. Now, apply the discounts in reverse order (first the sale discount, then the student discount). Note anything interesting about your findings. Calculate Markups When retailers purchase goods to sell, they pay a certain price, called the cost . The retailer then charges more than that amount for the goods. This increase is called the markup . This selling price, or retail price , is what the retailer charges the consumer in order to pay their own costs and make a profit. Markup, then is very similar to discount, except we add the markup, while we subtract the discount. The formula for a markup based on a percentage is markup = percent markup × cost , with the percent markup expressed as a decimal. The price of the item after the markup is retail price = cost + markup . These are often combined into the following formula retail price = cost + percent markup × cost = cost × ( 1 + percent markup) It should be noted that the formulas used for a markup are very similar to those for a discount, with addition replacing the subtraction. Determining the Retail Price Based on the Cost and the Percent Markup Calculate the markup for the given cost and markup percentage. Then calculate the retail price. Cost = $62.00; percent markup is 15% Cost = $750.00; percent markup is 45% Substituting the values into the formula markup = percent markup × cost , we find that the markup is markup = 0.15 × 62.00 = 9.30 . The markup is $9.30. The retail price of the item is then retail price = cost + markup , or $62.00 + $9.30 = $71.30. Substituting the values into the formula markup = percent markup × cost , we find that the markup is markup = 0.45 × 750.00 = 337.50 . The markup is $337.50. The retail price of the item is then retail price = cost + markup , or $750.00 + $337.50 = $1,087.50. Sometimes the cost and the retail price of an item are known. From this, the percent markup can be computed using the formula markup = percent markup × cost , by solving for the percent markup. Calculating the Percent Markup from the Cost and Retail Price Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places. Cost = $90.00; retail price = $103.50 Cost = $5.20; retail price = $9.90 Step 1: Using the cost and the retail price, we can find the markup with the formula retail price = cost + markup . Substituting and calculating, we find the markup to be 103.50 = 90.00 + markup . Solving for the markup gives $13.50. Step 2: After substituting the markup, $13.50, and the original price, $90.00, into the formula markup = percent markup × cost , we can find the percent markup. 13 .50 = percent markup × 90.00 13 .50 90.00 = percent markup 0.15 = percent markup Converting to percent form, the percent markup is 15%. Step 1: Using the cost and the retail price, we can find the markup with the formula retail price = cost + markup . Substituting and calculating, we find the markup to be 9.90 = 5.20 + markup . Solving for the markup gives $4.70. Step 2: After substituting the markup, $4.70, and the original price, $5.20, into the formula markup = percent markup × cost , we can find the percent markup. 4 .70 = percent markup × 5.20 4.70 5.20 = percent markup 0.9038 = percent markup Converting to percent form, the percent markup is 90.38%. Sometimes the retail price and the percent markup of an item are known. From this, the cost can be found. To avoid multiple steps, though, the formula that we will use is retail price = cost × (1 + percent markup) . The cost can be found by solving this equation for the cost. Calculating the Cost from the Percent Markup and Retail Price Determine the cost based on the percent markup and retail price. Percent markup 20%; retail price = $10.62 Percent markup 125%; retail price = $26.55 Using the percent markup and the retail price, we can find the cost with the formula retail price = cost × (1 + percent markup) . Substituting and solving for the cost, we find retail price = cost × (1 + percent markup) 10 .62 = cost × ( 1 + 0.2 ) 10 .62 = cost × (1 .2) 8 .85 = cost The cost of the item was $8.85. Using the percent markup and the retail price, we can find the cost with the formula retail price = cost × (1 + percent markup) . Substituting and solving for the original price, we find retail price = cost × (1 + percent markup) 26 .55 = cost × (1+2 .25) 26 .55 = cost × (3 .25) 11 .80 = cost The cost of the item was $11.80. Solve Application Problems Involving Markups As before when working with application problems, be sure to look for what is given and identify what you are to find. Once you have evaluated the problem, use the appropriate formula to find the solution(s). These application problems address markups. Determine Retail Price of a Power Bar Janice works at a convenience store near campus. It sells protein bars at a 60% markup. If a bar costs the store $1.30, how much is the retail price at the convenience store? We are asked to find the retail price. We know the percent markup is 60%. The cost of the bar was $1.30. Substituting into the percent markup formula, we find that the markup is markup = percent markup × cost = 0.60 × 1.30 = 0.78 . The markup is $0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is retail price = cost + markup = 1.30 + 0.78 = 2.08 . The retail price is $2.08. Determine the Percent Markup of a Phone Javi began working at a phone outlet. In a recent shipment, he noticed that the cost of the phone to the store was $480.00. The phone sells for $840.00 in the store. What is the percent markup on the phone? We know the cost of the phone, $480. We also know the retail price of the phone, $840.00. From this we know the markup is $ 840.00 - $ 480.00 = $ 360.00 . Substituting these values into the formula markup = percent markup × cost , we can find the percent markup. markup = percent markup × cost 360 = percent markup × 480 360 480 = percent markup × 480 480 0.75 = percent markup The markup on the phone is 75%. Finding the Cost of a T-Shirt Bob decided to order a t-shirt for his gaming friend online for $29.50. He knows the markup on such t-shirts is 18%. What was the t-shirt’s cost before the markup? Using the percent markup and the retail price, $29.50, we can find the cost with the formula retail price = original price × (1 + percent markup) . Substituting and solving for cost we find retail price = cost × (1 + percent markup) 29 .50 = cost × (1+0 .18) 29 .50 = cost × (1 .18) 25 .00 = cost The cost of the t-shirt was $25.00. Compute Sales Tax Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. None of these sales tax revenues go to the federal government. For example, North Carolina has a state sales tax of 4.75% while New Mexico has a state sales tax of 5%. Additionally, many counties in North Carolina charge an additional 2% sales tax, bringing the total sales tax for most (72 of the 100) counties in North Carolinians to 6.75%. However, in Durham, the county sales tax is 2.25% plus an additional 0.5% tax used to fund public transportation, bringing Durham County’s sales tax to 7%. To find the sales tax in a particular place, then, add other locality sales taxes to the base state sales tax rate. How much we pay in sales tax depends on where we are, and what we are buying. To determine the amount of sales tax on taxable purchase, we need to find the product of the purchase price, or marked price, and the sales tax rate for that locality. To calculate the amount of sales tax paid on the purchase price in a locality with sales tax given in decimal form, calculate sales tax = purchase price × tax rate The total price is then Total price = purchase price + purchase price × tax rate = purchase price × ( 1 + tax rate) When the sales tax calculation results in a fraction of a penny, then normal rounding rules apply, round up for half a penny or more, but round down for less than half a penny. You should notice that this the same as markup, except using a different term. Sales tax plays the role of markup, the purchase price plays the role of cost, and the tax rate plays the role of percent markup. This means all the strategies developed for markups apply to this situation, with the changes indicated. Sales Tax in Kankakee Illinois The sales tax in Kankakee, Illinois, is 8.25%. Find the sales tax and total price of items based on the purchase price listed. Purchase price = $428.99 Purchase price = $34.88 The sales tax is found using sales tax = purchase price × tax rate . The purchase price is $428.99 and the tax rate is 8.25%. Substituting and calculating, the sales tax is sales tax = $ 428.99 × 0.0825 = $ 35.391675 . The sales tax needs to be rounded off. Since the third decimal place (fraction of a penny) is 1, we round down and the sales tax is $35.39. The total price is the sales tax plus the purchase price, so is $ 428.99 + $ 35.88 = $ 464.87 . The sales tax on the item is found using sales tax = purchase price × tax rate . The purchase price is $34.88 and the tax rate is 8.25%. Substituting and calculating, the sales tax is sales tax = $ 34.88 × 0.0825 = $ 2.8776 . The sales tax needs to be rounded off. Since the third decimal place (fraction of a penny) is 7, we round up and the sales tax is $2.88. The total price of the item is the sales tax plus the purchase price, so is $ 34.88 + $ 2.88 = $ 37.76 . As before, the information available might be different than only the purchase price and the sales tax rate. In these cases, use either sales tax = purchase price × tax rate or Total price = purchase price × (1 + tax rate) and solve for the indicated tax, price, or rate. These problems mirror those for percent markup. Be aware, almost all sales tax rates are structured as full percentages, or half percent, or one-quarter percent, or three-quarter percent. This means the decimal value of the sales tax rate, written as a percent, will be either 0, as in 5.0%, 5 as in 7.5%, 25 as in 3.25%, or 75 as in 4.75%. When rounding for the sales tax percentage, be sure to use this guideline. Calculating the Sales Tax from the Purchase Price and the Total Price Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax percentages. Purchase price = $329.50; total price = $354.21 Purchase Price = $13.77; total price = $14.39 Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is $24.71. Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula sales tax = purchase price × tax rate , the sales tax rate can be found. Substituting and solving yields Sales Tax = purchase price × tax rate $ 24.71 = $ 329.50 × tax rate $ 24.71 $ 329.50 = tax rate 0 .07499 = tax rate Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%. Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is $0.62. Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula sales tax = purchase price × tax rate , the sales tax rate can be found. Substituting and solving yields sales tax = purchase price × tax rate $ 0.62 = $ 13.77 × tax rate $ 0.62 $ 13.77 = tax rate 0 .04503 = tax rate Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%. Calculating the Purchase Price from the Sales Tax and Total Price Find the purchase price for the indicated sales tax rate and total price. Sales tax rate = 5.75%; total price = $36.56 Sales tax rate = 4.25%; total price = $97.17 When the sales tax rate and the total price are known, the formula total price = purchase price × (1 + tax rate) can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find Total price = purchase price × (1 + tax rate) $ 36.56 = purchase price × (1 + 0 .0575) $ 36.56 1.0575 = purchase price $ 34.57 = purchase price The purchase price, the price before tax, was $34.57. When the sales tax rate and the total price are known, the formula total price = purchase price × (1 + tax rate) can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find Total price = purchase price × (1 + tax rate) $ 97.17 = purchase price × (1 + 0 .0425) $ 97.17 1.0425 = purchase price $ 93.21 = purchase price The purchase price, the price before tax, was $93.21. Solve Application Problems Involving Sales Tax Solving problems involving sales tax follows the same ideas and steps as solving problems for markups. But here we will use the following formula: total price = purchase price + sales tax We can also use the formula: total price = purchase price × (1 + sales tax rate) . This can be seen in the following examples. Compute Sales Tax for Denver, Colorado The sales tax rate in Denver Colorado is 8.81%. Keven buys a TV in Denver, and the purchase price (before taxes) is $499.00. How much will Keven pay in sales tax and what will be the total amount he spends when he buys the TV? The sales tax rate in Denver is 8.81%. To find the sales tax Keven will pay, find 8.81% of the purchase price. In decimal form, that sales tax rate is 0.0881. Using the formula and substituting 499.00 for purchase price, we find that Keven will pay purchase price × tax rate = $499 × 0.0881 = $ 43.96 in sales tax for the TV. The total price that Keven will pay is the purchase price plus the sales tax, or $ 499.00 + $ 43.96 = $ 542.96 . Compute Sales Tax for Austin, Texas Jillian visits Austin, Texas, and purchases a new set of weights for her home. She spends, including sales tax, $467.64. The sales tax rate in Austin Texas is 8.25%. How much of the total price is sales tax? The sales tax paid for this purchase is the difference in the total price and the purchase price. We know the total price is $467.64. We also know the sales tax rate, which is 8.25%. In decimal form, this is 0.0825. Using these values and the formula total price = purchase price × (1 + tax rate) to find the purchase price. total price = purchase price × (1 + tax rate) $ 467.64 = purchase price × ( 1 + 0.0825 ) $ 467.64 = purchase price × ( 1.0825 ) $ 432 = purchase price Knowing both the total price and the now the purchase price, we can find the difference, which is the sales tax. The total price was $467.64. The purchase price was $432. The difference of the total price and the purchase price, or the sales tax, is then $467.64 − $432.00, which is $35.64. Jillian pays $35.64 in sales tax. Finding Sales Tax Percentage West Virginia was the first state to impose a sales tax. This happened on May 3, 1921. Look up your locality on this website that lists standard state-level sales tax rates and compare the sales tax structure in your state to two nearby states (for the lower 48) and for any two states (Alaska and Hawaii). Check Your Understanding Key Terms Discount Cost Markup Retail price Key Concepts Discounts are markdowns from an original price. Mark-ups are increases to the price paid by a retailer to cover their costs. be able to calculate the markup based on a percentage of the cost Sales taxes vary from state to state and often county to county. Retail prices, sales prices and percent discounts can be calculated if the other two values are known. Original costs, retail prices, and percent markup can be calculated if the other two values are known. In calculations, sales tax acts like a markup. Videos Computing Price Based on a Percent Off Coupon Finding Sales Tax Percentage Formulas discount = percent discount × original price sale price = original price - discount sale price = original price - percent discount × original price = original price × (1 - percent discount) markup = percent markup × cost retail price = cost + markup retail price = cost + percent markup × cost = cost × (1 + percent markup) sales tax = purchase price × tax rate Total price = purchase price + purchase price × tax rate = purchase price × (1 + tax rate)

Math


Simple Interest Interest is how savings earns money. (credit: “Interest Rates” by Mike Mozart/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Compute simple interest. Understand and compute future value. Compute simple interest loans with partial payments. Understand and compute present value. There is truth in the phrase “You need to have money to make money.” In essence, if you have money to lend, you can lend it at a cost to a borrower and make money on that transaction. When money is borrowed, the person borrowing the money (borrower) typically has to pay the person or entity that lent the money (the lender) more than the amount of money that was borrowed. This extra money is the interest that is to be paid. Interest is sometimes referred to as the cost to borrow, the cost of the loan, or the finance cost. This idea also applies when someone deposits money in a bank account or some other form of investment. That person is essentially lending the money to the bank or company. The money earned by the depositor is also called interest. The interest is typically based on the amount borrowed, or the principal . The pairing of borrower and lender can take various forms. The borrower may be a consumer using a credit card or taking out a loan from a bank, the lender. Companies also borrow from lending banks. Someone who invests in a company’s stock is the lender in this case; the company is essentially the borrower. In this section, we examine the basic building block of interest paid on loans and borrowed credit and also the returns on investments like bank accounts, simple interest. Compute Simple Interest Let’s get some terminology understood. Interest to be paid by a borrower is often expressed as an annual percentage rate , which is the percent of the principal that is paid as interest for each year the money is borrowed. This means that the more that is borrowed, the more that must be paid back. Sometimes, the interest to be paid back is simple interest , which means that the interest is calculated on the amount borrowed only. The length of time until the loan must be paid off is the term of the loan. The date when the loan must be paid off is when the loan is due . The day that the loan is issued is the origination date . We’ll put this terminology to use in the following examples. Note that in this section we will use letters, called variables, to represent the different parts of the formulas we’ll be using. This will help keep our formulas and calculations manageable. Simple Interest Loans with Integer Year Terms Calculating simple interest is similar to the percent calculations we made in Understanding Percent and Discounts, Markups, and Sales Tax , but must be multiplied by the term of the loan (in years, if dealing with an annual percentage rate). The simple interest, I , to be paid on a loan with annual interest rate r for a number of years (term of the loan) t , with principal P , is found using I = P × r × t , where the decimal form of the interest rate, r , is used. The total repaid, then is T = P + I or, more directly, T = P + P × r × t . This total is often referred to as the loan payoff amount , or more simply just the payoff. When the annual interest rate, the principal, and the number of years that the money is borrowed is known, the interest to be paid can be found and from there the total to be repaid can be calculated. Be aware, interest paid to a lender is almost uniformly rounded up to the next cent. Simple Interest on Loans with Integer Year Terms Calculate the simple interest to be paid on a loan with the given principal, annual percentage rate, and number of years. Then, calculate the loan payoff amount. Principal P = $4,000, annual interest rate r = 5.5%, and number of years t = 4 Principal P = $14,800, annual interest rate r = 7.9%, and number of years t = 7 Substitute the principal P = $4,000, the decimal form of the annual interest rate r = 0.055, and number of years t = 4 into the formula for simple interest, and calculate. I = P × r × t = 4,000 × 0.055 × 4 = 880 . The simple interest, or cost of the loan, to be paid on the loan is $880. The loan payoff amount, or the total to be repaid, is T = P + I = 4,000 + 880 = 4,880 , or $4,880.00. Substitute the principal P = $14,800, the decimal form of the annual interest rate r = 0.079, and number of years t = 7 into the formula for simple interest, and calculate. I = P × r × t = 14,800 × 0.079 × 7 = 8,184.4 . The simple interest, or cost to borrow, to be paid on the loan is $8,184.40. The loan payoff amount, or the total to be repaid, is T = P + I = 14,800 + 8,184.4 = 22,984.4 , or $22,984.40. Simple Interest Equipment Loan Riley runs an auto repair shop, and needs to purchase a new brake lathe, which costs $11,995. She takes out a two-year, simple interest loan at an annual interest rate of 14.9%. How much interest will she pay and how much total will she repay on the loan? Step 1. Determine the variables, or parts of the formula. The principal P is the cost of the brake lathe, so P = $11,995. The interest rate Riley pays is 14.9%, or r = 0.149 in decimal form. The length of the loan is two years, so t = 2. We are first asked to find I , the interest Riley will pay. Step 2. Substitute the known variables into the formula for simple interest I = P × r × t and solve for I . From Step 1 we have I = P × r × t = 11995 × 0.149 × 2 = 3,574.51 . This tells us that the simple interest, or cost to borrow, to be paid on the loan is $3,574.51. Step 3. Use the formula T = P + I to determine the total amount Riley will repay, T . The total to be repaid is T = P + I = 11,995 + 3,574.51 = 15,569.51 , or $15,569.51. Simple Interest Loans with Other Lengths of Terms In the previous example and Your Turn exercise, the loans were paid back in one payment after an integer number of years. However, there are also loans lasting a length of time not equal to an integer number of years (like 1, 2, or 3 years or more), but in a number of months (like 4 months, 18 months, and so on). What model would apply to these situations? When the loan is paid back after a term that is not an integer number of years but is instead a number of months, the term of the loan, or time, t , is expressed as a fraction of the year. So for a 2-month loan, the time, in years, is 2/12 = 1/6. For a 5-month loan, the time in years is 5/12. For an 18-month term, the term in years is 18/12 = 1.5. Loan to Purchase Equipment Abeje needs a loan to purchase equipment for the gym she is going to open. She visits the bank and secures a 4-month loan of $20,000. Her annual percentage rate is 6.75%. How much interest will Abeje pay and what is her loan payoff amount? Abeje’s loan is for $20,000, so her principal is P = 20,000. The interest rate Abeje will pay is 6.75%, or r = 0.0675 in decimal form. The length of the loan is 4 months, so t = 4 12 . Substituting these in the formula for simple interest, we find her interest to be I = P × r × t = 20,000 × 0.0675 × 4 / 12 = 450 The simple interest, or cost to borrow, to be paid on the loan is $450.00. The payoff is T = P + I = 20,000 + 450 = 20,450 , or $20,450.00. Those examples dealt in months. However, some loans are for days only (45 days, 60 days, 120 days). In such cases, we find the daily interest rate. The fraction we will use for the daily interest rate is the interest rate (as a decimal) divided by 365. This may be referred to as Actual/365. In order to find the term of the loan, divide the number of days in the term of the loan by 365. To determine the interest, I , on a loan with term t expressed in days, with principal of P , and interest rate in decimal form of r , calculate I = P × r 365 × t . Here, r 365 represents the daily interest rate. Alternately, the above formula is equivalent to I = P × r × t 365 , where the interest rate remains an annual rate, but the time is expressed as a fraction of the year. It seems reasonable to use 365 as the number of days in the year, since there are 365 days in most years. However, sometimes, banks have used (and continue to use) 360 as the number of days in a year. They may also treat all months as if they have 30 days. These differences lead to (sometimes small) differences in how much interest is paid. Since the number of days is in the denominator, a smaller denominator (360) will result in larger numbers (interest) that is 365 is used for the denominator. See this page from ACRE for a comparison. Loan for Moving Costs David plans to move his family from Raleigh, North Carolina to Tempe, Arizona. His company will reimburse (pay after the move) David for the move. David does research and determines that movers will cost $5,600 to move his family’s belongings to Tempe. He takes out a simple interest, 45-day loan at 11.75% interest to pay this cost. How much interest will be paid on this 45-day loan, and what is David’s loan payoff amount? This loan is in terms of days, so we will use the formula I = P × r 365 × t , where t is the number of days and r is the annual interest rate. The principal for the loan is the moving cost, or P = 5,600. The annual interest rate that David will pay is 11.75%, which in decimal is 0.1175. The length of time for the loan is 45 days, so t = 45. Substituting these values into the formula and calculating, we find that the interest to be paid is I = 5,600 × 0.1175 365 × 45 = 81.13 , or $81.13 (remember, interest is almost always rounded up to the next cent). The payoff for the loan is $5,681.13. Understand and Compute Future Value Money can be invested for a specific amount of time and earn simple interest while invested. The terminology and calculations are the same as we’ve already seen. However, instead of the total to be paid back, the investor is interested in the total value of the investment after the interest is added. This is called the future value of the investment. The future value, FV , of an investment that yields simple interest is F V = P + I = P + P × r × t , where P is the principal (amount invested at the start), r is the annual interest rate in decimal form, and t is the length of time the money is invested. The time t will be an integer if the term of the deposit is an integer number of years, will be number of months/12 if the term is in months, will be actual/365 if the deposit is for a number of days. Simple Interest on a Deposit In the following, determine how much interest was earned on the investment and the future value of the investment, if the investment yields simple interest. Principal is $1,000, annual interest rate is 2.01%, and time is 5 years Principal is $5,000, annual interest rate is 1.85%, and time is 30 years Principal is $10,000, annual interest rate is 1.25%, and time is 18 months Principal is $7,000, annual interest rate is 3.26%, and time is 100 days The principal is P = $1,000, the annual interest rate, in decimal form, is 0.0201, and the term is 5 years, or t = 5. Since the term is an integer number of years, the interest earned on the investment is I = P × r × t = 1,000 × 0.0201 × 5 = 100.5 , or the interest earned was $100.50. To find the future value, we use the formula F V = P + I . Substituting the values and calculating, we find the future value of the investment to be F V = P + I = 1,000 + 100.5 = 1100.5 . The future value of the investment at the end of 5 years is $1,100.50. Notice that the future value could have been calculated directly with F V = P + P × r × t The principal is P = $5,000, the annual interest rate, in decimal form, is 0.0185, and the term is 30 years, or t = 30. Since the term is an integer number of years, the interest earned on the investment is I = P × r × t = 5,000 × 0.0185 × 30 = 2,775 , or the interest earned was $2,775.00. To find the future value, we use the formula F V = P + I . Substituting the values and calculating, we find the future value of the investment to be F V = P + I = 5,000 + 2,775 = 7,775 . The future value of the investment at the end of 30 years is $7,775.00. The principal is P = $10,000, the annual interest rate, in decimal form, is 0.0125, and the term is 18 months. Since the term is in months, we have to write the months in terms of years. For 18 months, we use 18/12 as t . The interest earned on the investment is I = P × r × t = 10,000 × 0.0125 × 18 12 = 187.5 , or the interest earned was $187.50. To find the future value, we use the formula F V = P + I . Substituting the values and calculating, we find the future value of the investment to be F V = P + I = 10,000 + 187.5 = 10,187.5 . The future value of the investment at the end of 18 months is $10,187.50. Principal is $7,000, annual interest rate is 3.26%, and time is 100 days. The principal is P = $7,000, the annual interest rate, in decimal form, is 0.0326, and the term is 100 days. Since the term is in days, we have to write the time using actual/365, or t = 100/365. The interest earned on the investment is I = P × r × t = 7,000 × 0.0326 × 100 365 = 62.52 , or the interest earned was $62.52. To find the future value, we use the formula F V = P + I . Substituting the values and calculating, we find the future value of the investment to be F V = P + I = 7,000 + 62.52 = 7,062.52 . The future value of the investment at the end of 100 days is $7,062.52. You may have noticed that for these problems, the future value was rounded down. When the future value is paid, the amount is typically rounded down. A certificate of deposit (CD) is a savings account that holds a single deposit (the principal) for a fixed term at a fixed interest rate. Once the term of the CD is over, the CD may be redeemed (cashed in or withdrawn) and the owner of the CD receives the original principal plus the interest earned. The deposit often cannot be withdrawn until the term is up; if it can be withdrawn early, there is often a penalty imposed to do so. Certificate of Deposit Jonas deposits $2,500 in a CD bearing 3.25% simple interest for a term of 3 years. When he redeems his CD at the end of the 3 years, how much will he receive? This is a future value example. We know that P = $2,500 is the amount deposited. The annual simple interest rate in decimal form is r = 0.0325. The term of the investment is t = 3 years. Substituting those values into the future value formula, we have F V = P + P × r × t = 2,500 + 2,500 × 0.0325 × 3 = 2,500 + 243.75 = 2,743.75 . When the CD is redeemed, Jonas will receive $2,743.75. The reason CD (certificate of deposit) rates look so small is because they are extremely safe investments. Though overall interest rates for CDs change over time and individual returns vary with the terms of the CD, investors are offered predictable interest income for their investments. To investigate this yourself, search online to determine the strengths and weaknesses of CDs (investopedia.com offers good, basic information on investing). Then, online, identify five national banks and two local banks who offer CDs. Track the interest rates for the CDs at various terms (1 year, 3 years, 5 years) for each of the banks you found that offer CDs. Calculate the amount of interest earned for a $10,000 deposit for each CD at each of the terms. Compare the results from the various banks, CDs, and terms and decide which is the best investment. You may want to consider both the length of time that the money is locked up, and the return. Paying Simple Interest Loans with Partial Payments In every example above, there was one payment for the loan, or one withdrawal for the investment. However, for many loans (house, car, in-ground swimming pool), the loan will be paid back in two or more payments. Such a payment is called a partial payment , because they only pay off part of the loan. When a partial payment is made, some of the payment pays for the principal, but the rest of the payment pays for interest on the principal. When making the first partial payment, the interest is calculated on the principal for the time between the origination date of the loan and the date of the payment. If another partial payment is made, the interest is calculated based on the remaining principal and the time between the previous partial payment and the current partial payment date. Interest Paid in a Partial Payment on a Loan A simple interest loan for $6,500 is taken out at 12.6% annual percentage rate. A partial payment is made 45 days into the loan period. How much of the partial payment will be for interest? A simple interest loan for $13,700 is taken out at 6.55% annual interest rate. A partial payment is to be made after 60 days. How much of the partial payment will be for interest? To find the interest paid in this partial payment, we calculate the interest on the principal for the time between the origination of the loan and the payment day, or 45 days. The principal is $6,500. The annual interest rate, in decimal form, is 0.126. The interest paid for 45 days is found by substituting the values for principal P , rate r , and time t into the formula I = P × r 365 × t . Calculating, we have I = P × r 365 × t = 6,500 × 0.126 365 × 45 = 100.9726 . Rounding up, the portion of the partial payment that will be paid for interest is $100.98. To find the interest paid in this partial payment, we calculate the interest on the principal for the time between the origination of the loan and the payment day, or 60 days. The principal is $13,700. The annual interest rate, in decimal form, is 0.0655. The interest paid for those 60 days is found by substituting those values into the formula I = P × r 365 × t . Calculating, we have I = P × r 365 × t = 13,700 × 0.0655 365 × 60 = 147.5096 . Rounding up, the portion of the partial payment that will be paid for interest is $147.51. Remaining Balance The previous examples demonstrated how to determine the interest paid in a partial payment. Using this, we can determine the remaining balance after a partial payment. Step 1: determine the amount of the payment, P , that is applied to interest, I . Step 2: subtract the amount paid in interest from the payment, ( P − I ) . This is the amount applied to the balance. Step 3: subtract the amount applied to the balance (the value obtained in Step 2) from the balance of the loan, B − ( P − I ) . This is the remaining balance after the partial payment. Determining the Remaining Balance on a Loan After a Partial Payment A simple interest loan for $45,500 is taken out at 11.8% annual percentage rate. A partial payment of $20,000 is made 50 days into the loan period. After this payment, what will the remaining balance of the loan be? A simple interest loan for $150,000 is taken out at 5.85% annual percentage rate. A partial payment of $50,000 is made 70 days into the loan period. After this payment, what will the remaining balance of the loan be? The principal is $45,500, which will be treated as the balance, B , of the loan. The annual simple interest rate, in decimal form, is 0.118. The time is t = 50 days. Step 1: Determine the amount of the partial payment that is applied to interest. To find this, substitute the values above into the formula I = P × r 365 × t and calculate. Calculating, the amount of the payment that is applied to interest is I = 45,500 × 0.118 365 × 50 = 735.4795 . Rounding up, we have I = $735.48. Step 2: The amount of the payment that is to be applied to the balance of the loan is partial payment minus the amount of the partial payment that is applied to the interest. The payment is $2,000. The amount that is applied to the balance is P − I = $ 20,000 − $ 735.48 = $ 19,264.52 . Step 3: The remaining balance is found by subtracting the amount applied to the balance from the previous balance, or B − ( P − I ) = $ 45,500 − $ 19,264.52 = $ 26,235.48 . The remining balance after the partial payment is $26,235.48. The principal is $150,000, which will be treated as the balance, B , of the loan. The annual simple interest rate, in decimal form, is 0.0585. The time is t = 70 days. Step 1: Determine the amount of the partial payment that is applied to interest. To find this, substitute the values above into the formula I = P × r 365 × t and calculate. Calculating, the amount of the payment that is applied to interest is I = 150,000 × 0.0585 365 × 70 = 1,682.8767 . Rounding up, we have I = $1,682.88. Step 2: The amount of the payment that is to be applied to the balance of the loan is partial payment minus the amount of the partial payment that is applied to the interest. The payment is $50,000. The amount that is applied to the balance is P − I = $ 50,000 − $ 1,682.88 = $ 48,317.12 . Step 3: The remaining balance is found by subtracting the amount applied to the balance from the previous balance, or B − ( P − I ) = $ 150,000 − $ 48,317.12 = $ 101,682.88 . The remining balance after the partial payment is $101,682.88. Loan Payoff Finally, we will determine the amount to be paid at the end of the loan. To do so, we apply the formula for the loan payoff to the remaining balance. However, the length of time for that remaining balance is the time between the partial payment and the day the loan is paid off. Step 1: Determine the remaining balance after the partial payment. Step 2: Calculate the number of days between the partial payment and the date the loan is paid off. This will be the time t in the payment formula. Step 3: Calculate the amount to be paid at the end of the loan, or the payoff amount, using Payoff = P + P × r 365 × t , where P is the remaining balance and t is the time found in Step 2. Finding Loan Pay Off After a Partial Payment Laura takes out an $18,400 loan for 120 days at 17.9% simple interest. She makes a partial payment of $7,500 after 45 days. What is her payoff amount at the end of the loan? The initial balance, or principal, of her loan is $18,400. The interest rate in decimal form is 0.179. Her partial payment of $7,500 is made after 45 days. Using these values, we can determine how much of the partial payment is applied to the balance. From there, we can determine her final loan payoff after 120 days. Step 1: Determine the remaining balance after the partial payment. Using the partial payment process outlined in the previous example, we first find that the amount of the partial payment that is applied to the balance. Their interest paid in the partial payment is I = P × r 365 × t = 18,400 × 0.179 365 × 45 = 406.0603 , or $406.07 (remember to round up!). Using this and that the loan amount was for $18,400, the remaining balance on the loan after the partial payment is B - ( P - I ) = $ 18,400 - ( $ 7,500 - $ 406.07 ) = $ 11,306.07 . Step 2: The number of days between the partial payment and the date that the loan is to be paid off is 120 – 45 = 75. This means that the time between the partial payment and the final payment is 75 days. Step 3: To calculate the payoff amount, use payoff = P + P × r 365 × t , with P = $11,306.07 (the remaining balance), t = 75 (from Step 2) and r = 0.179. The payoff amount, then, is payoff = 11,306.07 + 11,306.07 × 0.179 365 × 75 = 11,721.9165 . Rounding up, the payoff amount is $11,721.92. Repeated Partial Payments Car loans and mortgages (loans for homes) are paid off through repeated partial payments, most often monthly payments. Since car loans are often 3 to 6 years, and mortgages 15 to 30 years, calculating each individual monthly payment one at a time is time consuming and tedious. Even a 3-year loan would involve applying the above steps 36 times! Fortunately, there is a formula for determining the amount of each partial payment for monthly payments on a simple interest loan. The amount of monthly payments, A , for a loan with principal P , monthly simple interest rate r (in decimal form), for t number of months is found using the formula A = P × r × ( 1 + r ) t ( 1 + r ) t − 1 . The monthly interest rate is the annual rate divided by 12. The number of months is the number of years times 12. Calculating Car Payments Desiree buys a new car, by taking a loan out from her credit union. The balance of her loan is $27,845.00. The annual interest rate that Desiree will pay is 7.3%. She plans to pay this off over 4 years. How much will Desiree’s monthly payment be? To use the formula for monthly payments, we need the principal, the interest rate, and the number of years. The principal is $27,845. The annual rate, in decimal form, is 0.073. Dividing 0.073 by 12 gives the monthly interest rate 0.073 / 12 = 0.00608 3 ¯ . She takes the loan out for 4 years, which is t = 12 × 4 = 48 months. Substituting these values into the formula, A = P × r × ( 1 + r ) t ( 1 + r ) t − 1 , we calculate: A = 27,845 × 0.00608 3 ¯ × ( 1 + 0.00608 3 ¯ ) 48 ( 1 + 0.00608 3 ¯ ) 48 - 1 = 27,845 × 0.00608 3 ¯ × ( 1.00608 3 ¯ ) 48 ( 1.00608 3 ¯ ) 48 - 1 = 27,845 × 0.00608 3 ¯ × 1.337918996 1.337918996 - 1 = 27,845 × 0.008139007 0.337918996 = 27,845 × 0.024085675 = 670.6656 Using the formula and rounding up to the next cent, we see that Desiree’s monthly payment will be $670.67. The calculation of payments is long, and involves many steps. However, most spreadsheet programs, including Google Sheets, have a payment function. In Google Sheets, that function is PMT. To find the payment for an installment loan (like for a car), you need to enter the interest rate per period, the number of payments, and the loan amount. From Your Turn 6.36 , the rate was 0.0476/12, the number of payments was 60, and the loan amount was $23,660. In Google Sheets, select any cell and enter the following: =PMT(0.0476/12,60,23660) And click the enter key. Immediately, in the cell you selected, the payment of $443.90 appears, though with a negative sign. The negative sign indicates it is a payment out of an account. Since we want to know the payment amount, we ignore the negative sign. The result, with the formula in the formula bar, is shown in . Google Sheets payment function In general, to use the PMT function in Google Sheets, enter =PMT( r /12, t *12, P ) where r is the annual interest rate, t is the number of years, and P is the principal of the loan. Understand and Compute Present Value for Simple Interest Investments When finding the future value of an investment, we know how much is deposited, but we have no idea how much that money will be worth in the future. If we set a goal for the future, it would be useful to know how much to deposit now so an account reaches the goal. The amount that needs to be deposited now to hit a goal in the future is called the present value . The present value, P V , of money deposited at an annual, simple interest rate of r (in decimal form) for time t (in years) with a specified future value of F V , is calculated with the formula P V = F V ( 1 + r t ) . Note: Present value, in this calculation, is always rounded up. Otherwise, future value may fall short of the target future value. Understanding what this tells you is important. When you find the present value, that is how much you need to invest now to reach the goal F V , under the conditions (time and rate) at which the money will be invested. Compute the present value of the investment described. Interpret the result. F V = $10,000, t = 15 years, annual simple interest rate of 5.5% F V = $150,000, t = 20 years, annual simple interest rate of 6.25% F V = $250,000, t = 486 months, annual simple interest rate of 4.75% The future value is F V = $10,000. The time of the investment is in years, so t = 15. The annual, simple interest rate is 5.5%, which in decimal form is 0.055. We substitute those values into the formula and calculate. P V = F V ( 1 + r t ) = 10,000 ( 1 + ( 0.055 ) × ( 15 ) ) = 10,000 ( 1 + 0.825 ) = 10,000 ( 1.825 ) = 5,479.452 . Rounding up, we see that the present value of $10,000 invested at a simple annual interest rate of 5.5% for 15 years is $5,479.46. This means that $5,479.46 needs to be invested so that, after 15 years at 5.5% interest, the investment will be worth $10,000. The future value is F V = $150,000. The time of the investment is in years, so t = 20. The annual, simple interest rate is 6.25%, which in decimal form is 0.0625. We substitute those values into the formula and calculate. P V = F V ( 1 + r t ) = 150,000 ( 1 + ( 0.0625 ) × ( 20 ) ) = 150,000 ( 1 + 1.25 ) = 150,000 ( 2.25 ) = 66,666. 6 ¯ . Rounding up, we see that the present value of $150,000 invested at a simple annual interest rate of 6.25% for 20 years is $66,666.67. This means that $66,666.67 needs to be invested so that, after 20 years at 6.25% interest, the investment will be worth $150,000. The future value is FV = $250,000. The time of the investment is 486 months. This needs to be converted to years. To do so, divide the number of months by 12, giving y e a r s = 486 12 = 40.5 , so t = 40.5 years. The annual, simple interest rate is 4.75%, which in decimal form is 0.0475. We substitute those values into the formula and calculate. P V = F V ( 1 + r t ) = 250,000 ( 1 + ( 0.0475 ) × ( 40.5 ) ) = 150,000 ( 1 + 1.92375 ) = 150,000 ( 2.92375 ) = 129,954.5159 . Rounding up, we see that the present value of $250,000 invested at a simple annual interest rate of 4.75% for 486 months is $129,954.52. This means that $129,954.52 needs to be invested so that, after 486 months at 4.75% interest, the investment will be worth $150,000. Present Value of a CD Beatriz will invest some money in a CD that yields 3.99% simple interest when invested for 30 years. How much must Beatriz invest so that after those 30 years, her CD is worth $300,000? Beatriz needs to know how much to deposit now so that her CD is worth $300,000 after 30 years. This means she needs to know the present value of that $300,000. The time is 30 years and the annual simple interest rate, in decimal form, is 0.0399. Using that information and the formula for present value, we calculate the present value of that $300,000. P V = F V ( 1 + r t ) = 300,000 ( 1 + ( 0.0399 ) × ( 30 ) ) = 300,000 ( 1 + 1.197 ) = 300,000 ( 2.197 ) = 136,549.8407 . Rounding up, Beatriz needs to invest $136,549.85 so that she has $300,000 in 30 years. Check Your Understanding Key Terms Interest Principal Annual percentage rate Simple interest Term Due Origination date Payoff amount Future value Partial payment Present value Key Concepts Interest is money that is paid by a borrower for the privilege of borrowing the money. Simple interest is computed by substituting the principal, interest rate, and number of years into the formula I = P × r × t The payoff for a loan is the amount of principal remaining on a loan plus the interest that accumulated on the loan since the last payment. The future value of an investment yielding simple interest is the original principal plus the interest earned on the investment. When making a partial payment, some of the payment pays off all the accumulated interest, while the remainder of the payment is applied to the principal of the loan. Finding the present value of an investment is used to determine how much should be invested now in order to achieve a specific goal. Formulas I = P × r × t T = P + I T = P + P × r × t I = P × r 365 × t F V = P + I = P + P × r × t A = P × r × ( 1 + r ) t ( 1 + r ) t − 1 P V = F V ( 1 + r t )

Math


Compound Interest The impact of compound interest (credit: "English Money" by Images Money/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Compute compound interest. Determine the difference in interest between simple and compound calculations. Understand and compute future value. Compute present value. Compute and interpret effective annual yield. For a very long time in certain parts of the world, interest was not charged due to religious dictates. Once this restriction was relaxed, loans that earned interest became possible. Initially, such loans had short terms, so only simple interest was applied to the loan. However, when loans began to stretch out for years, it was natural to add the interest at the end of each year, and add the interest to the principal of the loan. After another year, the interest was calculated on the initial principal plus the interest from year 1, or, the interest earned interest. Each year, more interest was added to the money owed, and that interest continued to earn interest. Since the amount in the account grows each year, more money earns interest, increasing the account faster. This growth follows a geometric series ( Geometric Sequences ). It is this feature that gives compound interest its power. This module covers the mathematics of compound interest. Understand and Compute Compound Interest As we saw in Simple Interest , an account that pays simple interest only pays based on the original principal and the term of the loan. Accounts offering compound interest pay interest at regular intervals. After each interval, the interest is added to the original principal. Later, interest is calculated on the original principal plus the interest that has been added previously. After each period, the interest on the account is computed, then added to the account. Then, after the next period, when interest is computed, it is computed based on the original principal AND the interest that was added in the previous periods. The following example illustrates how compounded interest works. Interest Compounded Annually Abena invests $1,000 in a CD (certificate of deposit) earning 4% compounded annually. How much will Abena’s CD be worth after 3 years? Since the interest is compounded annually, the interest will be computed at the end of each year and added to the CD’s value. The interest at the end of the following year will be based on the value found form the previous year. Step 1: After the first year, the interest in Abena’s CD is computed using the interest formula I = P × r × t . The principal is P = 1,000, the rate, as a decimal, is 0.04, and the time is one year, so t = 1. Using that, the interest earned in the first year is I = P × r × t = 1,000 × 0.04 × 1 = 40 , so the interest earned in the first year was $40.00. This is added to the value of the CD, making the CD worth $ 1,000 + $ 40 = $ 1,040 . Step 2: At the end of the second year, interest is again computed, but is computed based on the CD’s new value, $1,040. Using this new value and the interest formula ( r and t are still 0.04 and 1, respectively), we see that the CD earned I = P × r × t = 1,040 × 0.04 × 1 = 41.6 , or $41.60. This is added to the value of the CD, making the CD now worth $ 1,040.00 + $ 41.60 = $ 1,081.60 . Step 3: At the end of the third year, interest is again computed, but is computed based on Abena’s CD’s new value, $1,081.60. Using this value and the interest formula ( r and t are still 0.04 and 1, respectively), we see that the CD earned I = P × r × t = 1,081.60 × 0.04 × 1 = 43.264 , or $43.26 (remember to round down). This is added to the value of the CD, making the CD now worth $ 1,081.60 + $ 43.26 = $ 1,124.86 . After 3 years, Abena’s CD is worth $1,124.86. Determine the Difference in Interest Between Simple and Compound Calculations It is natural to ask, does compound interest make much of a difference? To find out, we revisit Abena’s CD. Comparing Simple to Compound Interest on a 3-Year CD Abena invested $1,000 in a CD that earned 4% compounded annually, and the CD was worth $1,124.86 after 3 years. Had Abena invested in a CD with simple interest, how much would the CD have been worth after 3 years? How much more did Abena earn using compound interest? Had Abena invested $1,000 in a 4% simple interest CD for 3 years, her CD would have been worth P + P × r × t = 1,000 + 1,000 × 0.04 × 3 = 1,120 , or $1,120.00. With interest compounded annually, Abena’s CD was worth $1,124.86. The difference between compound and simple interest is $ 1,124.86 - $ 1,120.00 = $ 4.86 . So compound interest earned Abena $4.86 more than the simple interest did. Compound Interest Understand and Compute Future Value Imagine investing for 30 years and compounding the interest every month. Using the method above, there would be 360 periods to calculate interest for. This is not a reasonable approach. Fortunately, there is a formula for finding the future value of an investment that earns compound interest. The future value of an investment, A , when the principal P is invested at an annual interest rate of r (in decimal form), compounded n times per year, for t years, is found using the formula A = P ( 1 + r n ) n t . This is also referred to as the future value of the investment. Note, sometimes the formula is presented with the total number of periods, n , and the interest rate per period, r . In that case the formula becomes A = P ( 1 + r ) n . Computing Future Value for Compound Interest In the following, compute the future value of the investment with the given conditions. Principal is $5,000, annual interest rate is 3.8%, compounded monthly, for 5 years. Principal is $18,500, annual interest rate is 6.25%, compounded quarterly, for 17 years. The principal is P = $5,000, interest rate, in decimal form, r = 0.038, compounded monthly so n = 12, and for t = 5 years. Substituting these values into the formula, we find A = P ( 1 + r n ) n t = 5 , 000 ( 1 + 0.038 12 ) 12 × 5 = 5 , 000 ( 1 + 0.0031 6 ¯ ) 60 = 5 , 000 ( 1.0031 6 ¯ ) 40 = 5 , 000 × 1.20888663572 = 6 , 044.4332 The future value of the investment is $6,044.43. The principal is P = $18,500, interest rate, in decimal form, r = 0.0625, compounded quarterly so n = 4, and for t = 17 years. Substituting these values into the formula, we A = P ( 1 + r n ) n t = 18 , 500 ( 1 + 0.0625 4 ) 4 × 17 = 18 , 500 ( 1 + 0.015625 ) 68 = 18 , 500 ( 1.015625 ) 68 = 18 , 500 × 2.86992151999 = 53 , 093.5481 The future value of the investment is $53,093.54. Interest Compounded Quarterly Cody invests $7,500 in an account that earns 4.5% interest compounded quarterly (4 times per year). Determine the value of Cody’s investment after 10 years. Cody’s initial investment is $7,500, so P = $7,500. The annual interest rate is 4.5%, which is 0.045 in decimal form. Compounding quarterly means there are four periods in a year, so n = 4. He invests the money for 10 years. Substituting those values into the formula, we calculate A = P ( 1 + r n ) n t = 7 , 500 ( 1 + 0.045 4 ) 4 × 10 = 7 , 500 ( 1 + 0.01125 ) 40 = 7 , 500 ( 1.01125 ) 40 = 7 , 500 × 1.564376865391 = 11 , 732.8265 After 10 years, Cody’s initial investment of $7,500 is worth $11,732.82. Interest Compounded Daily Kathy invests $10,000 in an account that yields 5.6% compounded daily. How much money will be in her account after 20 years? Kathy’s initial investment is $10,000, so P = $10,000. The annual interest rate is 5.6%, which is 0.056 in decimal form. Compounding daily means there are 364 periods in a year, so n = 365. She invests the money for 20 years, so t = 20. Substituting those values into the formula, we calculate A = P ( 1 + r n ) n t = 10,000 ( 1 + 0.056 365 ) 365 × 20 = 10,000 ( 1 + 0.000153424657534 ) 7300 = 10,000 ( 1.000153424657534 ) 7300 = 10,000 × 3.06459091598 = 30,645.909 After 20 years, Kathy’s initial investment of $10,000 is worth $30,645.90. Compare Simple Interest to Interest Compounded Annually Compare Simple Interest and Compound Interest for Different Number of Periods Per Year To truly grasp how compound interest works over a long period of time, create a table comparing simple interest to compound interest, with different numbers of periods per year, for many years would be useful. In this situation, the principal is $10,000, and the annual interest rate is 6%. Create a table with five columns. Label the first column YEARS, the second column SIMPLE INTEREST, the third column COMPOUND ANNUALLY, the fourth column COMPOUND MONTHLY and the last column COMPOUND DAILY, as shown below. YEARS SIMPLE INTEREST COMPOUND ANNUALLY COMPOUND MONTHLY COMPOUND DAILY In the years column, enter 1, 2, 3, 5, 10, 20, and 30 for the rows. Calculate the account value for each column and each year. Compare the results from each of the values you find. How do the number of periods per year (compoundings per year) impact the account value? How does the number of years impact the account value? Redo the chart, with an interest rate you choose and a principal you choose. Are the patterns identified earlier still present? Understand and Compute Present Value When investing, there is often a goal to reach, such as “after 20 years, I’d like the account to be worth $100,000.” The question to be answered in this case is “How much money must be invested now to reach the goal?” As with simple interest, this is referred to as the present value. The money invested in an account bearing an annual interest rate of r (in decimal form), compounded n times per year for t years, is called the present value, P V , of the account (or of the money) and found using the formula P V = A ( 1 + r n ) n × t , where A is the value of the account at the investment’s end. Always round this value up to the nearest penny. Computing Present Value Find the present value of the accounts under the following conditions. A = $250,000, invested at 6.75 interest, compounded monthly, for 30 years. A = $500,000, invested at 7.1% interest, compounded quarterly, for 40 years. To reach a final account value of A = $250,000, invested at 6.75% interest, in decimal form r = 0.0675 (decimal form!), compounded monthly, so n = 12, for 30 years, substitute those values into the formula for present value. Calculating, we find the present value of the $250,000. P V = A ( 1 + r n ) n × t = 250,000 ( 1 + 0.0675 12 ) 12 × 30 = 250,000 ( 1 + 0.005625 ) 360 = 250,000 ( 1.005625 ) 360 = 250,000 7.5332454772 = 33,186.2277 In order for this account to reach $250,000 after 30 years, $33,186.23 needs to be invested. To reach a final account value of A = $500,000, invested at 7.1% interest, in decimal form r = 0.071, compounded quarterly, so n = 4, for 40 years, substitute those values into the formula for present value. Calculating, we find the present value of the $500,000. P V = A ( 1 + r n ) n × t = 500 , 000 ( 1 + 0.071 4 ) 4 × 40 = 500 , 000 ( 1 + 0.01775 ) 160 = 500 , 000 ( 1.01775 ) 160 = 500 , 000 16.6946672846 = 29 , 949.6834 In order for this account to reach $500,000 after 40 years, $29,949.69 needs to be invested. Investment Goal with Compound Interest Pilar plans early for retirement, believing she will need $1,500,000 to live comfortably after the age of 67. How much will she need to deposit at age 23 in an account bearing 6.35% annual interest compounded monthly? Knowing how much to deposit at age 23 to reach a certain value later is a present value question. The target value for Pilar is $1,500,000. The interest rate is 6.35%, which in decimal form is 0.0635. Compounded monthly means n = 12. She’s 23 and will leave the money in the account until the age of 67, which is 44 years, making t = 44. Using this information and substituting in the formula for present value, we calculate P V = A ( 1 + r n ) n × t = 1,500,000 ( 1 + 0.0635 12 ) 12 × 44 = 1,500,000 ( 1 + 0.005291 6 ¯ ) 528 = 1,500,000 ( 1.005291 6 ¯ ) 528 = 1,500,000 16.226302189 = 92,442.5037 Pilar will need to invest $92,442,51 in this account to have $1,500,000 at age 67. Compute and Interpret Effective Annual Yield As we’ve seen, quarterly compounding pays interest 4 times a year or every 3 months; monthly compounding pays 12 times a year; daily compounding pays interest every day, and so on. Effective annual yield allows direct comparisons between simple interest and compound interest by converting compound interest to its equivalent simple interest rate. We can even directly compare different compound interest situations. This gives information that can be used to identify the best investment from a yield perspective. Using a formula, we can interpret compound interest as simple interest. The effective annual yield formula stems from the compound interest formula and is based on an investment of $1 for 1 year. Effective annual yield is Y = ( 1 + r n ) n - 1 where Y = effective annual yield, r = interest rate in decimal form, and n = number of times the interest is compounded in a year. Y is interpreted as the equivalent annual simple interest rate. Determine and Interpret Effective Annual Yield for 6% Compounded Quarterly Suppose you have an investment paying a rate of 6% compounded quarterly. Determine and interpret that effective annual yield of the investment. Here, n = 4 (quarterly) and r = 0.06 (decimal form). Substituting into the formula we find that the effective annual yield is Y = ( 1 + 0.06 4 ) 4 - 1 = ( 1.015 ) 4 - 1 = 1.06136 - 1 = 0.0614 = 6.14 % Therefore, a rate of 6% compounded quarterly is equivalent to a simple interest rate of 6.14%. Determine and Interpret Effective Annual Yield for 5% Compounded Daily Calculate and interpret the effective annual yield on a deposit earning interest at a rate of 5% compounded daily. In this case, the rate is r = 0.05 and n = 365 (daily). Using the formula Y = ( 1 + r n ) n - 1 , we have Y = ( 1 + 0.05 365 ) 365 - 1 = ( 1.0001369863 ) 365 - 1 = 1.051267 - 1 = 0.0513 This tells us that an account earning 5% compounded daily is equivalent to earning 5.13% as simple interest. Choosing a Bank Minh has a choice of banks in which he will open a savings account. He will deposit $3,200 and he wants to get the best interest he can. The banks advertise as follows: Bank Interest Rate ABC Bank 2.08% compounded monthly 123 Bank 2.09% compounded annually XYZ Bank 2.05% compounded daily Which bank offers the best interest? To compare these directly, Minh could change each interest rate to its effective annual yield, which would allow direct comparison between the rates. Computing the effective annual yield for all three choices gives: ABC Bank: Y = ( 1 + 0.0208 12 ) 12 - 1 = 0.0210 = 2.10 % 123 Bank: Y = ( 1 + 0.0209 1 ) 1 - 1 = 0.0209 = 2.09 % XYZ Bank: Y = ( 1 + 0.0205 365 ) 365 - 1 = 0.0207 = 2.07 % ABC Bank has the highest effective annual yield, so Minh should choose ABC bank. Check Your Understanding Key Terms Compound interest Effective annual yield Key Concepts Compound interest means that the interest earned during one period will earn interest in later periods. Essentially, the amount of the principal grows from period to period. The important values in computing compound interest are the interest rate, the principal, the length of time the investment, and the number of times the investment is compounded. Compound interest has minimal impact early, but later has a very large impact. You can determine how much to invest today in order to reach a goal for some time later. Compound interest can be translated into an effective annual yield, which allows for comparison between investment options. Videos Compound Interest Compare Simple Interest to Interest Compounded Annually Compare Simple Interest and Compound Interest for Different Number of Periods Per Year Formulas A = P ( 1 + r n ) n t P V = A ( 1 + r n ) n × t Y = ( 1 + r n ) n - 1

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Making a Personal Budget Calculating a budget is important to your financial health. (credit: “Budget planning concept on white desk” by Marco Verch Professional Photographer/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Create a personal budget with the categories of expenses and income. Apply general guidelines for a budget. “That doesn’t fit in the budget.” “We didn’t budget for that.” “We need to figure out our budget and stick to it.” A budget is an outline of how money and resources should be spent. Companies have them, individuals have them, your college has one. But do you have one? Creating a realistic budget is an important step in careful stewardship of your financial health. Designing your budget will help understand the financial priorities you have, and the constraints on your life choices. You want to have enough income to pay not only for the necessities, but also for things that represent your wants, like trips or dinner out. You also may want to save money for large purchases or retirement. You do not want to just get by, and you do not want the problems associated with overdue balances, rising debt, and possibly losing something you have worked hard to obtain. While creating a budget may seem intimidating at first, coming up with your basic budget outline is the hardest part. Over time, you will adjust not only the numbers, but the categories. Creating a Budget You should view creating a budget as a financial tool that will help you achieve your long-term goals. A budget is an estimation of income and expenses over some period of time. You will be able to track your progress, which will help you to prepare for the future by making smart investment decisions. There are several budget-creating tools available, such as the apps Good budget and Mint, and Google Sheets. Getting started, though, begins well before you find an app. The following are steps that can be used to create your monthly budget. Track your income and expenses Review your income and expenses for the past 6 months to a year. This will give you an idea of your current habits. Set your income baseline Determine all the sources of income you will have. This income may from paychecks, investments, or freelance work. It even includes child support and gifts. Be sure to use income after taxes. This allows you to determine your maximum expenditures per month. For income that is not steady, such as gig work or freelance work, use the previous 6 to 12 months of income to find an average income from that gig or freelance work. Use this average in the budgeting process. Determine your expenses Review your bills from the past 6 months. You should include mortgage payments or rent, insurance, car payments, utilities, groceries, transportation expenses, personal care, entertainment, and savings. Using your credit card statements and bank statements will help you determine these amounts. Be aware that some of the expenses will not change over time. These are referred to as fixed expenses , like rent, car payments, insurance, internet service, and the like. Other expenses may vary widely from month to month and are appropriately called variable expenses , and include such expenses as gasoline, groceries. Some expenses are yearly, such as insurance or property taxes. Other expanses may be quarterly (four times per year) or semiannual (twice per year). To budget for such bills by month, divide the bill total by the number of months the bill covers. Categorize your expenses These categories may be housing, transportation, or food, for broad categories, or may get more specific, where you categorize car payments, car insurance, and gasoline separately. The categories are your choices. Be sure to account for the cost of maintaining a vehicle or home. The more specific you are, the better you’ll understand your spending needs and habits. Total your monthly income and monthly expenses and compare These values should be compared. If your expenses are higher than your income, then adjustments have to be made. Decisions of what to do with any extra income is part of the planning process also. Make plans for unplanned expenses Ask anyone, an unexpected car repair can ruin a carefully crafted budget. Have a plan for how you can be ready for these random expenses. This often means creating a cushion in your budget. Use your budget to make decisions and adjust for any changes Your budget is a changeable document. Add to it when you wish, refer to it when special purchases are to be made. Keeping your budget up to date helps accommodate changes in income and expenses. Creating a Budget In this section, we will focus on income and expenses. One of the easiest ways to manage a budget is to create a table, with one column containing income sources, another with income values, a third with expense categories, and a last containing expenses. An example is shown in . Income Source Amount Expense Amount Full-time job $3,565 Rent $975 Uber $185 Car Payment $355 TOTAL $3,750 Student Loan $418 Electric $76 Food $400 Gasoline $250 Car Insurance $165 Clothing $100 Entertainment $100 TOTAL $2,839 Table with Budget Gross Pay and Take Home Pay If you’ve ever had a paycheck, you know that taxes are taken out of your pay before you get your check. This amount of money varies from state to state, and sometimes even city to city. For a person making $50,000 per year gross salary in Salt Lake City, Utah, take home pay is about 75.6% of gross salary. In Detroit, Michigan, take home pay is about 74.5% of gross salary. Lakeland, Florida, take home pay is about 80.5% of gross salary. These also change based on how much a person earns! Before choosing a place to live, it makes sense to determine how much deductions from pay will impact your income. Creating a Budget Heather has graduated college and currently works as a nurse for a rural medical group. Her net monthly income from that job is $3,765.40. She also works part-time on the weekends, earning another $672.00 per month. Her monthly expenses are rent at $1,050, car payments at $489, student loan payments at $728, car insurance at $139, utilities at $130, clothing at $150, entertainment (going out with friends, Netflix, Amazon Prime, movies) at $300, credit card debt at $200, food at $360, and gasoline at $275. Create her budget in a table, compare the total income to total expenses, and determine how much excess income per month she has or how much she falls short by each month. Step 1: To begin, we create the table with appropriate headings. Income Source Amount Expense Amount Step 2: Her income categories are her nursing job, with $3,765.40 per month, and her part-time job, with $672.00 per month. Entering these into the table, we have the following. Income Source Amount Expense Amount Nursing $3,765.40 Part-time $672.00 Step 3: Her monthly expenses are listed above. Entering the categories and the amount for each of those expenses, the table is now Income Source Amount Expense Amount Nursing $3,765.40 Rent $1,050 Part-time $672.00 Car Payment $489 Student Loan $728 Car Insurance $139 Utilities $130 Clothing $150 Entertainment $300 Credit Card $200 Food $400 Gasoline $250 Step 4: Totaling the income and expenses, we see that her total income is $4,437.40 per month, and her total expenses are $3,836 per month. Comparing these, we see that Heather has $601.40 in excess income per month. This provides a cushion in her budget. Creating a Budget Carol is working in a dental lab, creating dentures and bridges. Monthly her take home pay is $2,816 (based on $22 per hour minus payroll taxes). She also receives $320 per month in child support for her one daughter. Her monthly expenses are rent at $700, car payments at $229, student loan payments at $250, car insurance at $119, health insurance at $225, utilities at $80, clothing at $75, entertainment at $200, food at $275, and gasoline at $275. Create Carol’s budget in a table, compare the total income to total expenses, and determine how much excess income per month she has or how much she falls short by each month. Step 1: To begin, we create the table with appropriate headings. Income Source Amount Expense Amount Step 2: Her income categories are from work, $2,816, and child support, $320, per month. Entering these into the table, we have the following. Income Source Amount Expense Amount Job $2,816.00 Child support $320.00 Step 3: Her monthly expenses are listed above. Entering the categories and the amount for each of those expenses, the table is now Income Source Amount Expense Amount Job $2,816.00 Rent $700 Child support $320.00 Car Payment $229 Student Loan $250 Car Insurance $119 Utilities $80 Health insurance $225 Clothing $75 Entertainment $200 Food $275 Gasoline $275 Step 4: Totaling the income and expenses, we see that her total income is $3,136.00 per month, and her total expenses are $2,428.00 per month. Comparing these, we see that Carol has $708.00 in excess income per month. This is the cushion in her budget. Using the budget process, we can make decisions on adding expenses to the budget. To do so, check the cushion of the budget to see if there is room in the budget for the new expense. Adding to an Existing Budget In the example above, Carol had excess income of $708.00. She looks up the cost of before-school care for her daughter. She finds that, monthly, the cost would be $252.00 per month. Is this an affordable program for Carol? Add this expense to her budget table. She can afford this, as the cost for the before school program is $252.00 and she had extra income of $708.00. Adding this to her budget, her budget table is now Income Source Amount Expense Amount Job $2,816.00 Rent $700 Child support $320.00 Car Payment $229 Student Loan $250 Car Insurance $119 Utilities $80 Health insurance $225 Clothing $75 Entertainment $200 Food $275 Gasoline $275 Before-school care $252 Now, she has $456.00 in excess income per month. The 50-30-20 Budget Philosophy It isn’t clear, obvious, or easy to decide how much of your income to allocate to various categories of expenses. Many people pay their bills and then consider all the leftover money to be spending money. However, when developing your own budget, you may want to follow the 50-30-20 budget philosophy , which provides a basic guideline for how your income could be allocated. Fifty percent of your budget is allotted to your needs, 30% of your budget is allotted to pay for your wants, and 20% of your budget is allotted for savings and debt service (paying off your debts). Knowing what expenses are necessary and what expenses are wants is important, since wants and needs are often confused. The following are necessary expenses that represent basic living requirements and debt services. This list isn’t complete: mortgage/rent, utilities, car, car insurance, health care, groceries, gasoline, child care (for working parents), and minimum debt payments. The 50-30-20 budget philosophy suggests that 50%, or half, your income go to these necessities. Wants, though, are things you could live without but still wish to have, such as Amazon Prime, restaurant dinners, coffee from Starbucks, vacation trips, and hobby costs. Even a gym membership or that new laptop are wants. Creating the room to afford these wants is important to our mental health. Not budgeting for things we want will negatively impact our quality of life. The remaining 20% should be set aside, either in retirement funds, stocks, other investments, an emergency fund (recommendations are that an emergency fund have 3 months of income), and perhaps extra spent to pay down debt. This 20% is very useful for addressing those unexpected costs, such as repairs or replacement of items that no longer work. Without budgeting this cushion, any expense that is a surprise can cause us to miss necessary payments. The list of necessary expenses was not complete. There are other expenses that could be included. Necessary Expenses and Expenses that are Wants For some people, an expense will be necessary while the same expense for someone else will be a want. A good example of this is internet service. Many people consider internet service as a need, especially those who work from home or who are not able to leave their homes. One could also call internet service a need if they have children in school. For others, internet service is a want. If a person’s job doesn’t require them to be online, if they are not in school, if they do not have kids, then internet service can be dropped. There are public options for internet service. One could even use their phone as a hot spot. Cars often fall into the category of need, but could also fall into the want category, depending on where and how you live. Bikes, public transportation, and walking are all options that could replace a car. This would then remove the cost of gasoline and car insurance. Another consideration when deciding if an item on your budget is a need or a want is about your choices and priorities. A car is a need for many. But the need for a car is not the same as the need for a specific car. If you choose to buy a car with payments that exceed your budgeted amount for the car, then that car is a want. The amount you exceed the budget now belongs in the want category. The same can be said for housing. If you want an apartment that costs $1,250 per month, but your budget only allows for an apartment that costs $900, then $350 of the rent is a want. The point of that is to carefully consider if an expense is a need as opposed to a want. When your expenses exceed your income, you may want to change how you budget your income to line up with these guidelines. This may mean cutting back, finding less-expensive living arrangements, finding a less-expensive (and more fuel-efficient) car, or sacrificing some specialty groceries. Using these guidelines keeps your financial life manageable. Better still, they can guide you as you begin your life after graduation. Evaluate a Budget Using 50-30-20 model In the example above, after Carol added before school care for her daughter to the budget, her budget was as shown below. Evaluate Carol’s budget using the 50-30-20 budget philosophy. Income Source Amount Expense Amount Job $2,816.00 Rent $700 Child support $320.00 Car Payment $229 Student Loan $250 Car Insurance $119 Utilities $80 Health insurance $225 Clothing $75 Entertainment $200 Food $275 Gasoline $275 Before-school care $252 Carol’s total income is $3,136.00. Applying the 50-30-20 budget philosophy to this income requires the calculation of each of those percentages. For the necessities, Carol should budget 50% of her income, or 0.5 × $ 3,136.00 = $ 1,568.00 . For her wants, she should budget 30% of her income, or 0.3 × $ 3,136.00 = $ 940.80 . For savings and extra debt service, she should budget 20% of her income, or 0.2 × $ 3,136.00 = $ 627.20 . In her budget, her necessities include all expenses except for entertainment. These expenses total $2,480, which exceeds the suggested budget amount of $1568.00. To follow the guidelines, Carol would have to cut back on these necessities. For her wants, she spends $200.00 on entertainment, which is well below the suggested budget amount of $940.80. If she modifies how much she spends on needs, she may be able to increase the spending on her wants. Her excess income is $456.00, which is below what she should be saving and using to pay down extra debt. If she does adjust how much she spends on needs, she could increase the amount for savings. Creating a Budget Based on the 50-30-20 Budget Philosophy Carmen is about to graduate and has been offered a job at a bank as a data scientist. She estimates her monthly take home pay to be $5,662.50. Apply the 50-30-20 philosophy to that monthly income. How should Carmen use this information? Step 1. To apply the 50-30-20 budget philosophy to Carmen’s income, she needs to calculate 50%, 30%, and 20% of her income. Fifty percent of her income is 0.5 × $ 5,662.50 = $ 2,831.25 . Thirty percent of her income is 0.3 × $ 5,662.50 = $ 1,698.75 . Twenty percent of her income is 0.2 × $ 5,662.50 = $ 1,132.50 . Step 2. She would then budget $2,831.25 for her needs, $1,698.75 for her wants, and $1,132.50 for savings and debt service. Step 3. When choosing where to live, what to eat, and what to drive, she should make choices that keep those costs, combined with her debt service costs, gasoline, and utilities, below $2,831.25. This means she will have to make decisions about what her priorities are. Step 4. She should then figure out what she wants to do with her money, and stay within the limits, that is, keep those costs below $1,698.75. Step 5. Finally, she can begin building her savings with the remaining $1,132.50. Using the 50-30-20 Budget Philosophy to Analyze Affordability Steve is thinking of moving out of his family’s home. He currently works at a full-time job making $18 per hour, which will give him, approximately, a net annual income of $29,180 (working 40 hours per week for 52 weeks per year). He has student debt that he pays off at $218.00 per month, and already owns a car that he pays $162.00 per month for. Apply the 50-30-20 budget philosophy to Steve’s income. If he follows the budget, how much does he have, after paying his car payment and student loan, to spend on necessities. If he follows the budget, how much will he set aside for wants? For savings? Discuss the affordability of moving out, based on Steve’s budget. Before the 50-30-20 philosophy can be applied, Steve’s monthly income needs to be determined. This is found by dividing his annual income by 12. This gives $ 29,180 / 12 = $ 2,431.67 . This will be used for his monthly budget. To apply the 50-30-20 philosophy to Steve’s income, find 50%, 30%, and 20% of his monthly income. Needs (50%): 50% of his income is 0.50 × $ 2,431.67 = $ 1,215.83 . Wants (30%): 0.30 × $ 2,431.67 = $ 729.50 Savings (20%): 0.20 × $ 2,431.67 = $ 486.33 The total for Steve’s needs is $1,215.83. From this, he already pays $218.00 for his student loans, and $162.00 for his car payment. Together that is $380.00. Subtracting from the amount he should budget for his needs, he can spend $835.83 on other needs. Steve budgeted $729.50 for wants, and $486.33 for savings and other debt servicing. Steve will have other needs to pay for, including rent, utilities, food, heath care, gasoline, and car insurance. It is difficult to imagine Steve being able to afford to move out, unless he reallocates money that he would want to save, or use for entertainment and other wants, or takes on another job. Even if Steve uses all the money that the 50-30-20 budget sets aside for savings, he still only has $1,322.16 to spend on those necessities. It does not appear he can afford to move out. 50-30–20 Budget Philosophy Check Your Understanding Key Terms Budget Necessary expenses Fixed expenses Variable expenses 50-30-20 budget philosophy Key Concepts A budget is a set of guidelines for how to allocate your income. Budgeting helps to plan for many of life’s expenses Budgets are used to compare income to expenses. When expenses exceed income, changes have to be made. Budgets can help evaluate the affordability of life changes. One guideline for setting a budget is the 50-30-20 budget philosophy. The guidelines suggest that 50% of income is allocated to necessary expenses, 30% to expenses that wants, and 20% to savings and other debt reduction. Videos Creating a Budget 50-30–20 Budget Philosophy

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