#set document(title: "6.2 Discounts, Markups, and Sales Tax", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 6.2#h(0.6em)Discounts, Markups, and Sales Tax #figure(figph[A sale board reads final sale 30 percent, 50 percent, 60 percent, and 70 percent.], alt: "A sale board reads final sale 30 percent, 50 percent, 60 percent, and 70 percent.", caption: [Sale prices are often described as percent discounts.]) === 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 #strong[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 #math.equation(block: false, alt: "part equals percentage times total")[$"part" = "percentage" × "total"$]. 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. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The formula for a discount based on a percentage is #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], with the percent discount expressed as a decimal. The price of the item after the discount is #math.equation(block: false, alt: "sale price equals original price minus discount")[$"sale price" = "original price" − "discount"$]. These are often combined into the following formula #math.equation(block: true, alt: "sale price equals original price minus percent discount times original price equals original price times open parenthesis 1 minus percent discount close parenthesis")[$"sale price" = "original price" − "percent discount" × "original price" = "original price" × ( 1 − "percent discount" )$] ] When the original price and the percent discount are known, the discount and the sale price can be directly computed. #examplebox("Example 1")[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% #solutionbox[ + Substituting the values into the formula #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], we find that the discount is #math.equation(block: false, alt: "discount equals 0 .25 times 75.80 equals 18.95")[$"discount" = upright(0) ".25" × 75.80 = 18.95$]. The discount is \$18.95. The sale price of the item is then #math.equation(block: false, alt: "sale price equals original price minus discount equals 75.80 minus 18.95 equals 56.85")[$"sale price" = "original price" − "discount" = 75.80 − 18.95 = 56.85$], or \$56.85. + Substituting the values into the formula #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], we find that the discount is #math.equation(block: false, alt: "discount equals 0 .30 times 168.90 equals 50.67")[$"discount" = upright(0) ".30" × 168.90 = 50.67$]. The discount is \$50.67. The sale price of the item is then #math.equation(block: false, alt: "sale price equals original price minus discount equals 168.90 minus 50.67 equals 118.23")[$"sale price" = "original price" − "discount" = 168.90 − 50.67 = 118.23$], 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 #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], by solving for the percent discount. #examplebox("Example 2")[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 #solutionbox[ + #strong[Step 1.] Find the discount. Using the original price and the sale price, we can find the discount with the formula #math.equation(block: false, alt: "sale price equals original price minus discount")[$"sale price" = "original price" − "discount"$]. Substituting and calculating, we find the discount to be #math.equation(block: false, alt: "900.00 equals 1 , 200.00 minus discount")[$900.00 = 1 , 200.00 − "discount"$]. Solving for the discount gives \$300.00. #strong[Step 2.] Find the percent discount. Substituting the discount of \$300.00 and the original price of \$1,200.00, into the formula #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], we can find the percent discount. #math.equation(block: true, alt: "300 .00, equals, percent discount times 1,200 .00; the fraction 300 .00 over 1 , 200.00, equals, percent discount; 0.25, equals, percent discount")[$"300" ".00" & = & "percent discount" × "1,200" ".00" \ display(display(display(frac("300" ".00", 1 "," 200.00)))) & = & "percent discount" \ 0.25 & = & "percent discount"$]Converting to percent form, the percent discount is 25%. + #strong[Step 1.] Find the discount. Using the original price and the sale price, we can find the discount with the formula #math.equation(block: false, alt: "sale price equals original price minus discount")[$"sale price" = "original price" − "discount"$]. Substituting and calculating, we find the discount to be #math.equation(block: false, alt: "29.52 equals 36.70 minus discount")[$29.52 = 36.70 − "discount"$]. Solving for the discount gives \$7.38. #strong[Step 2.] Find the percent discount. Substituting the discount of \$7.38 and the original price of \$36.70, into the formula #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], we can find the percent discount. #math.equation(block: true, alt: "29 .52, equals, percent discount times 36 .70; the fraction 29 .52 over 36.70, equals, percent discount; 0.2, equals, percent discount")[$"29" ".52" & = & "percent discount" × "36" ".70" \ display(display(display(frac("29" ".52", 36.70)))) & = & "percent discount" \ 0.2 & = & "percent discount"$]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 #math.equation(block: false, alt: "sale price equals original price times (1 minus percent discount)")[$"sale price" = "original price" × "(1" − "percent discount)"$]. The original price can be found by solving this equation for the original price. #examplebox("Example 3")[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 #solutionbox[ + Using the percent discount and the sale price, we can find the original price with the formula #math.equation(block: false, alt: "sale price equals original price times (1 minus percent discount)")[$"sale price" = "original price" × "(1" − "percent discount)"$]. Substituting and solving for the original price, we find #linebreak() #math.equation(block: true, alt: "sale price, equals, original price times (1 minus percent discount); 450 .00, equals, original price times (1 minus 0 .10); 450 .00, equals, original price times (0 .90); 500 .00, equals, original price")[$"sale price" & = & "original price" × "(1" − "percent discount)" \ "450" ".00" & = & "original price" × "(1" − upright(0) ".10)" \ "450" ".00" & = & "original price" × "(0" ".90)" \ "500" ".00" & = & "original price"$] #linebreak() The original price of the item was \$500.00. #linebreak() + Using the percent discount and the sale price, we can find the original price with the formula#math.equation(block: false, alt: "sale price equals original price times (1 minus percent discount)")[$"sale price" = "original price" × "(1" − "percent discount)"$]. Substituting and solving for the original price, we find #linebreak() #math.equation(block: true, alt: "sale price, equals, original price times (1 minus percent discount); 90 .00, equals, original price times (1 minus 0 .75); 90 .00, equals, original price times (0 .25); 360 .00, equals, original price")[$"sale price" & = & "original price" × "(1" − "percent discount)" \ "90" ".00" & = & "original price" × "(1" − upright(0) ".75)" \ "90" ".00" & = & "original price" × "(0" ".25)" \ "360" ".00" & = & "original price"$] #linebreak() 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). #examplebox("Example 4")[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? #solutionbox[ 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 #math.equation(block: false, alt: "discount equals percent discount times original price equals 0 .30 times 80 equals 24")[$"discount" = "percent discount" × "original price" = upright(0) ".30" × "80" = "24"$]. 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. ] ] #examplebox("Example 5")[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? #solutionbox[ 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 #math.equation(block: false, alt: "$ 240 minus $ 168 equals $ 72")[$"$" 240 − "$" 168 = "$" 72$]. Substituting these values into the formula #math.equation(block: false, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$], we can find the percent discount. #math.equation(block: true, alt: "discount, equals, percent discount times original price; 72, equals, percent discount times 240; the fraction 72 over 240, equals, the fraction percent discount times 240 over 240; 0.3, equals, percent discount")[$"discount" & = & "percent discount" × "original price" \ 72 & = & "percent discount" × 240 \ display(display(display(frac(72, 240)))) & = & display(display(display(frac("percent discount" × 240, 240)))) \ 0.3 & = & "percent discount"$] The student percent discount on the bus pass is 30%. ] ] #examplebox("Example 6")[Finding the Original Price of a New Pair of Tires][ Kendra’s car developed a flat, and the tire store told her that two tires had to be replaced. She got a 10% discount on the pair of tires, and the sale price came to \$189.00. What was the original price of the tires? #solutionbox[ Using the percent discount and the sale price, we can find the original price with the formula#math.equation(block: false, alt: "sale price equals original price times (1 minus percent discount)")[$"sale price" = "original price" × "(1" − "percent discount)"$]. Substituting and solving for the original price, we find #math.equation(block: true, alt: "sales price, equals, original price times (1 minus percent discount); 189 .00, equals, original price times (1 minus 0 .10); 189 .00, equals, original price times (0 .90); 210 .00, equals, original price")[$"sales price" & = & "original price" × "(1" − "percent discount)" \ "189" ".00" & = & "original price" × "(1" − upright(0) ".10)" \ "189" ".00" & = & "original price" × "(0" ".90)" \ "210" ".00" & = & "original price"$] The original price of the two tires Kendra bought was \$210.00. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Computing_Price_Based")[Computing Price Based on a Percent Off Coupon] ] #notebox("Work It Out", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ 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 #strong[cost]. The retailer then charges more than that amount for the goods. This increase is called the #strong[markup]. This selling price, or #strong[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. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ The formula for a markup based on a percentage is #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], with the percent markup expressed as a decimal. The price of the item after the markup is #math.equation(block: false, alt: "retail price equals cost plus markup")[$"retail price" = "cost" + "markup"$]. These are often combined into the following formula #math.equation(block: true, alt: "retail price equals cost plus percent markup times cost equals cost times open parenthesis 1 plus percent markup)")[$"retail price" = "cost" + "percent markup" × "cost" = "cost" × \( 1 + "percent markup)"$] ] It should be noted that the formulas used for a markup are very similar to those for a discount, with addition replacing the subtraction. #examplebox("Example 7")[Determining the Retail Price Based on the Cost and the Percent Markup][ Calculate the markup for the given cost and markup percentage. Then calculate the retail price. + Cost = \$62.00; percent markup is 15% + Cost = \$750.00; percent markup is 45% #solutionbox[ + Substituting the values into the formula #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], we find that the markup is #math.equation(block: false, alt: "markup equals 0.15 times 62.00 equals 9.30")[$"markup" = 0.15 × 62.00 = 9.30$]. The markup is \$9.30. #linebreak() The retail price of the item is then #math.equation(block: false, alt: "retail price equals cost plus markup")[$"retail price" = "cost" + "markup"$], or \$62.00 + \$9.30 = \$71.30. + Substituting the values into the formula #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], we find that the markup is #math.equation(block: false, alt: "markup equals 0.45 times 750.00 equals 337.50")[$"markup" = 0.45 × 750.00 = 337.50$]. The markup is \$337.50. #linebreak() The retail price of the item is then #math.equation(block: false, alt: "retail price equals cost plus markup")[$"retail price" = "cost" + "markup"$], or \$750.00 + \$337.50 = \$1,087.50. ] ] Sometimes the cost and the retail price of an item are known. From this, the percent markup can be computed using the formula #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], by solving for the percent markup. #examplebox("Example 8")[Calculating the Percent Markup from the Cost and Retail Price][ Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places. + Cost = \$90.00; retail price = \$103.50 + Cost = \$5.20; retail price = \$9.90 #solutionbox[ + #strong[Step 1:] Using the cost and the retail price, we can find the markup with the formula #math.equation(block: false, alt: "retail price equals cost plus markup")[$"retail price" = "cost" + "markup"$]. Substituting and calculating, we find the markup to be #math.equation(block: false, alt: "103.50 equals 90.00 plus markup")[$103.50 = 90.00 + "markup"$]. Solving for the markup gives \$13.50. #linebreak() #strong[Step 2:] After substituting the markup, \$13.50, and the original price, \$90.00, into the formula #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], we can find the percent markup. #linebreak() #math.equation(block: true, alt: "13 .50, equals, percent markup times 90.00; the fraction 13 .50 over 90.00, equals, percent markup; 0.15, equals, percent markup")[$"13" ".50" & = & "percent markup" × 90.00 \ display(display(display(frac("13" ".50", 90.00)))) & = & "percent markup" \ 0.15 & = & "percent markup"$] #linebreak() Converting to percent form, the percent markup is 15%. #linebreak() + #strong[Step 1:] Using the cost and the retail price, we can find the markup with the formula #math.equation(block: false, alt: "retail price equals cost plus markup")[$"retail price" = "cost" + "markup"$]. Substituting and calculating, we find the markup to be #math.equation(block: false, alt: "9.90 equals 5.20 plus markup")[$9.90 = 5.20 + "markup"$]. Solving for the markup gives \$4.70. #linebreak() #strong[Step 2:] After substituting the markup, \$4.70, and the original price, \$5.20, into the formula #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], we can find the percent markup. #linebreak() #math.equation(block: true, alt: "4 .70, equals, percent markup times 5.20; the fraction 4.70 over 5.20, equals, percent markup; 0.9038, equals, percent markup")[$upright(4) ".70" & = & "percent markup" × 5.20 \ display(display(display(frac(4.70, 5.20)))) & = & "percent markup" \ 0.9038 & = & "percent markup"$] #linebreak() 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 #math.equation(block: false, alt: "retail price equals cost times (1 plus percent markup)")[$"retail price" = "cost" × "(1" + "percent markup)"$]. The cost can be found by solving this equation for the cost. #examplebox("Example 9")[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 #solutionbox[ + Using the percent markup and the retail price, we can find the cost with the formula #math.equation(block: false, alt: "retail price equals cost times (1 plus percent markup)")[$"retail price" = "cost" × "(1" + "percent markup)"$]. Substituting and solving for the cost, we find #linebreak() #math.equation(block: true, alt: "retail price, equals, cost times (1 plus percent markup); 10 .62, equals, cost times open parenthesis 1 plus 0.2 close parenthesis; 10 .62, equals, cost times (1 .2); 8 .85, equals, cost")[$"retail price" & = & "cost" × "(1" + "percent markup)" \ "10" ".62" & = & "cost" × ( 1 + 0.2 ) \ "10" ".62" & = & "cost" × "(1" ".2)" \ upright(8) ".85" & = & "cost"$] #linebreak() The cost of the item was \$8.85. #linebreak() + Using the percent markup and the retail price, we can find the cost with the formula#math.equation(block: false, alt: "retail price equals cost times (1 plus percent markup)")[$"retail price" = "cost" × "(1" + "percent markup)"$]. Substituting and solving for the original price, we find #linebreak() #math.equation(block: true, alt: "retail price, equals, cost times (1 plus percent markup); 26 .55, equals, cost times (1+2 .25); 26 .55, equals, cost times (3 .25); 11 .80, equals, cost")[$"retail price" & = & "cost" × "(1" + "percent markup)" \ "26" ".55" & = & "cost" × "(1+2" ".25)" \ "26" ".55" & = & "cost" × "(3" ".25)" \ "11" ".80" & = & "cost"$] #linebreak() 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. #examplebox("Example 10")[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? #solutionbox[ 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 #math.equation(block: false, alt: "markup equals percent markup times cost equals 0.60 times 1.30 equals 0.78")[$"markup" = "percent markup" × "cost" = 0.60 × 1.30 = 0.78$] . The markup is \$0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is #math.equation(block: false, alt: "retail price equals cost plus markup equals 1.30 plus 0.78 equals 2.08")[$"retail price" = "cost" + "markup" = 1.30 + 0.78 = 2.08$]. The retail price is \$2.08. ] ] #examplebox("Example 11")[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? #solutionbox[ 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 #math.equation(block: false, alt: "$ 840.00 minus $ 480.00 equals $ 360.00")[$"$" 840.00 − "$" 480.00 = "$" 360.00$]. Substituting these values into the formula #math.equation(block: false, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$], we can find the percent markup. #math.equation(block: true, alt: "markup, equals, percent markup times cost; 360, equals, percent markup times 480; the fraction 360 over 480, equals, the fraction percent markup times 480 over 480; 0.75, equals, percent markup")[$"markup" & = & "percent markup" × "cost" \ 360 & = & "percent markup" × 480 \ display(display(display(frac(360, 480)))) & = & display(display(display(frac("percent markup" × 480, 480)))) \ 0.75 & = & "percent markup"$] The markup on the phone is 75%. ] ] #examplebox("Example 12")[Finding the Cost of a T-Shirt][ Bob decided to order a t-shirt for his gaming friend online for \$29.50. He knows the markup on such t-shirts is 18%. What was the t-shirt’s cost before the markup? #solutionbox[ Using the percent markup and the retail price, \$29.50, we can find the cost with the formula #math.equation(block: false, alt: "retail price equals original price times (1 plus percent markup)")[$"retail price" = "original price" × "(1" + "percent markup)"$]. Substituting and solving for cost we find #math.equation(block: true, alt: "retail price, equals, cost times (1 plus percent markup); 29 .50, equals, cost times (1+0 .18); 29 .50, equals, cost times (1 .18); 25 .00, equals, cost")[$"retail price" & = & "cost" × "(1" + "percent markup)" \ "29" ".50" & = & "cost" × "(1+0" ".18)" \ "29" ".50" & = & "cost" × "(1" ".18)" \ "25" ".00" & = & "cost"$] The cost of the t-shirt was \$25.00. ] ] === Compute Sales Tax Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. None of these sales tax revenues go to the federal government. For example, North Carolina has a state sales tax of 4.75% while New Mexico has a state sales tax of 5%. Additionally, many counties in North Carolina charge an additional 2% sales tax, bringing the total sales tax for most (72 of the 100) counties in North Carolinians to 6.75%. However, in Durham, the county sales tax is 2.25% plus an additional 0.5% tax used to fund public transportation, bringing Durham County’s sales tax to 7%. To find the sales tax in a particular place, then, add other locality sales taxes to the base state sales tax rate. How much we pay in sales tax depends on where we are, and what we are buying. To determine the amount of sales tax on taxable purchase, we need to find the product of the purchase price, or marked price, and the sales tax rate for that locality. #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ To calculate the amount of sales tax paid on the purchase price in a locality with sales tax given in decimal form, calculate #math.equation(block: false, alt: "sales tax equals purchase price times tax rate")[$"sales tax" = "purchase price" × "tax rate"$] The total price is then #math.equation(block: false, alt: "Total price equals purchase price plus purchase price times tax rate equals purchase price times open parenthesis 1 plus tax rate)")[$"Total price" = "purchase price" + "purchase price" × "tax rate" = "purchase price" × \( 1 + "tax rate)"$] ] #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[ #emph[When the sales tax calculation results in a fraction of a penny, then normal rounding rules apply, round up for half a penny or more, but round down for less than half a penny.] ] You should notice that this the same as markup, except using a different term. Sales tax plays the role of markup, the purchase price plays the role of cost, and the tax rate plays the role of percent markup. This means all the strategies developed for markups apply to this situation, with the changes indicated. #examplebox("Example 13")[Sales Tax in Kankakee Illinois][ The sales tax in Kankakee, Illinois, is 8.25%. Find the sales tax and total price of items based on the purchase price listed. + Purchase price = \$428.99 + Purchase price = \$34.88 #solutionbox[ + The sales tax is found using #math.equation(block: false, alt: "sales tax equals purchase price times tax rate")[$"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 #math.equation(block: false, alt: "sales tax equals $ 428.99 times 0.0825 equals $ 35.391675")[$"sales tax" = "$" 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 #math.equation(block: false, alt: "$ 428.99 plus $ 35.88 equals $ 464.87")[$"$" 428.99 + "$" 35.88 = "$" 464.87$]. + The sales tax on the item is found using #math.equation(block: false, alt: "sales tax equals purchase price times tax rate")[$"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 #math.equation(block: false, alt: "sales tax equals $ 34.88 times 0.0825 equals $ 2.8776")[$"sales tax" = "$" 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 #math.equation(block: false, alt: "$ 34.88 plus $ 2.88 equals $ 37.76")[$"$" 34.88 + "$" 2.88 = "$" 37.76$]. ] ] As before, the information available might be different than only the purchase price and the sales tax rate. In these cases, use either #math.equation(block: false, alt: "sales tax equals purchase price times tax rate")[$"sales tax" = "purchase price" × "tax rate"$] or #math.equation(block: false, alt: "Total price equals purchase price times (1 plus tax rate)")[$"Total price" = "purchase price" × "(1" + "tax rate)"$] and solve for the indicated tax, price, or rate. These problems mirror those for percent markup. Be aware, almost all sales tax rates are structured as full percentages, or half percent, or one-quarter percent, or three-quarter percent. This means the decimal value of the sales tax rate, written as a percent, will be either 0, as in 5.0%, 5 as in 7.5%, 25 as in 3.25%, or 75 as in 4.75%. When rounding for the sales tax percentage, be sure to use this guideline. #examplebox("Example 14")[Calculating the Sales Tax from the Purchase Price and the Total Price][ Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax percentages. + Purchase price = \$329.50; total price = \$354.21 + Purchase Price = \$13.77; total price = \$14.39 #solutionbox[ + #strong[Step 1.] Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is \$24.71. #linebreak() #strong[Step 2.] Find the sales tax rate. Using the purchase price, the sales tax, and the formula #math.equation(block: false, alt: "sales tax equals purchase price times tax rate")[$"sales tax" = "purchase price" × "tax rate"$], the sales tax rate can be found. Substituting and solving yields #linebreak() #math.equation(block: true, alt: "Sales Tax, equals, purchase price times tax rate; $ 24.71, equals, $ 329.50 times tax rate; the fraction $ 24.71 over $ 329.50, equals, tax rate; 0 .07499, equals, tax rate")[$"Sales Tax" & = & "purchase price" × "tax rate" \ "$" 24.71 & = & "$" 329.50 × "tax rate" \ display(display(display(frac("$" 24.71, "$" 329.50)))) & = & "tax rate" \ upright(0) ".07499" & = & "tax rate"$] #linebreak() Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%. #linebreak() + #strong[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. #linebreak() #strong[Step 2.] Find the sales tax rate. Using the purchase price, the sales tax, and the formula #math.equation(block: false, alt: "sales tax equals purchase price times tax rate")[$"sales tax" = "purchase price" × "tax rate"$], the sales tax rate can be found. Substituting and solving yields #linebreak() #math.equation(block: true, alt: "sales tax, equals, purchase price times tax rate; $ 0.62, equals, $ 13.77 times tax rate; the fraction $ 0.62 over $ 13.77, equals, tax rate; 0 .04503, equals, tax rate")[$"sales tax" & = & "purchase price" × "tax rate" \ "$" 0.62 & = & "$" 13.77 × "tax rate" \ display(display(display(frac("$" 0.62, "$" 13.77)))) & = & "tax rate" \ upright(0) ".04503" & = & "tax rate"$] #linebreak() Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%. ] ] #examplebox("Example 15")[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 #solutionbox[ + When the sales tax rate and the total price are known, the formula #math.equation(block: false, alt: "total price equals purchase price times (1 plus tax rate)")[$"total price" = "purchase price" × "(1" + "tax rate)"$] can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find #linebreak() #math.equation(block: true, alt: "Total price, equals, purchase price times (1 plus tax rate); $ 36.56, equals, purchase price times (1 plus 0 .0575); the fraction $ 36.56 over 1.0575, equals, purchase price; $ 34.57, equals, purchase price")[$"Total price" & = & "purchase price" × "(1" + "tax rate)" \ "$" 36.56 & = & "purchase price" × "(1" + upright(0) ".0575)" \ display(display(display(frac("$" 36.56, 1.0575)))) & = & "purchase price" \ "$" 34.57 & = & "purchase price"$] #linebreak() The purchase price, the price before tax, was \$34.57. #linebreak() + When the sales tax rate and the total price are known, the formula #math.equation(block: false, alt: "total price equals purchase price times (1 plus tax rate)")[$"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 #linebreak() #math.equation(block: true, alt: "Total price, equals, purchase price times (1 plus tax rate); $ 97.17, equals, purchase price times (1 plus 0 .0425); the fraction $ 97.17 over 1.0425, equals, purchase price; $ 93.21, equals, purchase price")[$"Total price" & = & "purchase price" × "(1" + "tax rate)" \ "$" 97.17 & = & "purchase price" × "(1" + upright(0) ".0425)" \ display(display(display(frac("$" 97.17, 1.0425)))) & = & "purchase price" \ "$" 93.21 & = & "purchase price"$] #linebreak() The purchase price, the price before tax, was \$93.21. ] ] === Solve Application Problems Involving Sales Tax Solving problems involving sales tax follows the same ideas and steps as solving problems for markups. But here we will use the following formula: #math.equation(block: true, alt: "total price equals purchase price plus sales tax")[$"total price" = "purchase price" + " sales tax "$] We can also use the formula: #math.equation(block: true, alt: "total price equals purchase price times (1 plus sales tax rate)")[$"total price" = "purchase price" × "(1" + "sales tax rate)"$] This can be seen in the following examples. #examplebox("Example 16")[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? #solutionbox[ 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 #math.equation(block: false, alt: "purchase price times tax rate equals $499 times 0.0881 equals $ 43.96")[$"purchase price" × "tax rate" = "$499" × 0.0881 = "$" 43.96$] in sales tax for the TV. The total price that Keven will pay is the purchase price plus the sales tax, or #math.equation(block: false, alt: "$ 499.00 plus $ 43.96 equals $ 542.96")[$"$" 499.00 + "$" 43.96 = "$" 542.96$]. ] ] #examplebox("Example 17")[Compute Sales Tax for Austin, Texas][ Jillian visits Austin, Texas, and purchases a new set of weights for her home. She spends, including sales tax, \$467.64. The sales tax rate in Austin Texas is 8.25%. How much of the total price is sales tax? #solutionbox[ The sales tax paid for this purchase is the difference in the total price and the purchase price. We know the total price is \$467.64. We also know the sales tax rate, which is 8.25%. In decimal form, this is 0.0825. Using these values and the formula #math.equation(block: false, alt: "total price equals purchase price times (1 plus tax rate)")[$"total price" = "purchase price" × "(1" + "tax rate)"$] to find the purchase price. #math.equation(block: true, alt: "total price, equals, purchase price times (1 plus tax rate); $ 467.64, equals, purchase price times open parenthesis 1 plus 0.0825 close parenthesis; $ 467.64, equals, purchase price times open parenthesis 1.0825 close parenthesis; $ 432, equals, 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"$] Knowing both the total price and the now the purchase price, we can find the difference, which is the sales tax. The total price was \$467.64. The purchase price was \$432. The difference of the total price and the purchase price, or the sales tax, is then \$467.64 − \$432.00, which is \$35.64. Jillian pays \$35.64 in sales tax. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Finding_Sales_Tax")[Finding Sales Tax Percentage] ] #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ West Virginia was the first state to impose a sales tax. This happened on May 3, 1921. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Look up your locality on #link("https://openstax.org/r/resources_rates")[this website that lists standard state-level sales tax rates] and compare the sales tax structure in your state to two nearby states (for the lower 48) and for any two states (Alaska and Hawaii). ] === Key Terms - Discount - Cost - Markup - Retail price === Key Concepts - Discounts are markdowns from an original price. - Mark-ups are increases to the price paid by a retailer to cover their costs. - be able to calculate the markup based on a percentage of the cost - Sales taxes vary from state to state and often county to county. - Retail prices, sales prices and percent discounts can be calculated if the other two values are known. - Original costs, retail prices, and percent markup can be calculated if the other two values are known. - In calculations, sales tax acts like a markup. === Videos - #link("https://openstax.org/r/Computing_Price_Based")[Computing Price Based on a Percent Off Coupon] - #link("https://openstax.org/r/Finding_Sales_Tax")[Finding Sales Tax Percentage] === Formulas #math.equation(block: true, alt: "discount equals percent discount times original price")[$"discount" = "percent discount" × "original price"$] #math.equation(block: true, alt: "sale price equals original price minus discount")[$"sale price" = "original price" − "discount"$] #math.equation(block: true, alt: "sale price equals original price minus percent discount times original price equals original price times (1 minus percent discount)")[$"sale price" = "original price" − "percent discount" × "original price" = "original price" × "(1" − "percent discount)"$] #math.equation(block: true, alt: "markup equals percent markup times cost")[$"markup" = "percent markup" × "cost"$] #math.equation(block: true, alt: "retail price equals cost plus markup")[$"retail price" = "cost" + "markup"$] #math.equation(block: true, alt: "retail price equals cost plus percent markup times cost equals cost times (1 plus percent markup)")[$"retail price" = "cost" + "percent markup" × "cost" = "cost" × "(1" + "percent markup)"$] #math.equation(block: true, alt: "sales tax equals purchase price times tax rate")[$"sales tax" = "purchase price" × "tax rate"$] #math.equation(block: true, alt: "Total price equals purchase price plus purchase price times tax rate equals purchase price times (1 plus tax rate)")[$"Total price" = "purchase price" + "purchase price" × "tax rate" = "purchase price" × "(1" + "tax rate)"$]