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## Chapter 8

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**Chapter 8**Mathematics of Pricing START EXIT**Chapter Outline**8.1 Markup and Markdown 8.2 Profit Margin 8.3 Series and Trade Discounts 8.4 Depreciation Chapter Summary Chapter Exercises**8.1 Markup and Markdown**• A very large part of the business conducted in this world is a matter of buying things and then turning around and selling them to someone else at a profit. • The price a business pays for an item is called the wholesale price or cost. • The price a business sells the item for is known as a retail price.**8.1 Markup and Markdown**• One common method used for setting the selling price for an item is markup based on cost. • To determine a price with this method, we simply take the cost of the item and add on a pre-determined percent of the item’s cost.**8.1 Markup and Markdown**FORMULA 8.1.1 Markup Based on Cost P = C(1 + r) where P represents the SELLING PRICE C represents the COST and r represents the PERCENT MARKUP**8.1 Markup and Markdown**Example 8.1.1 • Problem • An auto mechanic charges a 40% markup based on cost for parts. What would the price be for an air filter that cost him $14.95? What is the dollar amount of his markup on this item? • Solution P = C(1 + r) P = $14.95(1 + 40%) = $20.93 Markup = $20.93 -- $14.95 = $5.98**8.1 Markup and Markdown**Example 8.1.2 • Problem • Hegel’s Bagels and Vienna Coffeehouse sells souvenir coffee mugs for $7.95. The markup based on cost is 65%. Find (a) the cost of each mug and (b) the dollar amount of the markup. • Solution P = C(1 + r) P = $7.95(1 + 65%) = $4.82 Markup = $7.95 -- $4.82 = $3.13**8.1 Markup and Markdown**Example 8.1.3 • Problem • An electronics retailer offers a computer for sale for $1,000. The retailer’s cost is $700. What is the markup percent? • Solution P = C(1 + r) $1,000 = $700(1 + r) r = 0.4285714 = 42.86%**8.1 Markup and Markdown**• We are all familiar with the idea of prices being marked down as part of a sale or some other promotion, for example. • To calculate a marked-down price, we simply apply the percent the price is to be marked down to the original price, and then subtract.**8.1 Markup and Markdown**FORMULA 8.1.2 Markdown MP = OP(1 – d) where MP represents the MARKED-DOWN PRICE OP represents the ORIGINAL PRICE d represents the PERCENT MARKDOWN**8.1 Markup and Markdown**Example 8.1.4 • Problem • At its Presidents’ Day Sale, a furniture store is offering 15% off everything in the store. What would the sale price be for a sofa that normally sells for $1,279.95? What is the dollar amount of the markdown? • Solution MP = OP(1 – d) MP = $1,279.95(1 – 15%) = $1,087.96**8.1 Markup and Markdown**Example 8.1.5 • Problem • Hal’s Hardware Haven is having a going-out-of-business sale. According to its ad, everything in the store is marked down 40%. If a set of patio lights is offered at a marked-down price of $29.97, what was the original price? How much of a dollar savings is this versus the original price? • Solution MP = OP(1 – d) $29.97 = OP(1 – 40%) OP = $49.95**8.1 Markup and Markdown**Example 8.1.6 • Problem • At the end of the summer, a backyard play set that usually sells for $599.95 is marked down to $450. What is the markdown percent? • Solution MP = OP(1 – d) $450 = $599.95(1 – d) d = 0.2499374 = 24.99%**8.1 Markup and Markdown**Example 8.1.7 • Problem • Gemma’s Gemstone Gewelry bought a necklace for $375. In the store, Gemma marked up this price by 20%. Several months later, when the necklace still had not sold, she decided to mark down the price by 20%. What was the marked down price? • Solution • Markup P = C(1 + r) P = $375(1 + 20%) = $450 • Markdown MP = OP(1 – d) MP = $450(1 – 20%) = $360**8.1 Markup and Markdown**Example 8.1.8 • Problem • If prices are calculated with a 35% markup based on cost, what is the percent that those prices should be marked down to get back to their original cost? • Solution • We don’t know what sort of things we are pricing here, much less the dollar amounts of those things. However, since we are working with percents, the actual dollar amounts don’t really matter. We choose a convenient cost of $100. P = C(1 + r) P = $100(1 + 35%) P = $135 MP = OP(1 – d) $100 = $135(1 – d) d = 25.93%**Problem 1**• Wal-Mart charges $2.35 for a carton of eggs. If the price is based on a 45% markup based on cost, how much did they pay the chicken farmer? CHECK YOUR ANSWER**Solution 1**• Wal-Mart charges $2.35 for a carton of eggs. If the price is based on a 45% markup based on cost, how much did they pay the chicken farmer? • P = C(1 + r) $2.35 = C(1 + 45%) $2.35 = C x 1.45 C = $1.62 BACK TO GAME BOARD**Problem 2**• You bought an evening gown for $35, what a deal! If the markdown percent was 70%, what was the original price? CHECK YOUR ANSWER**Solution 2**• You bought an evening gown for $35, what a deal! If the markdown percent was 70%, what was the original price? • $35 = OP(1 – 70%) $35 = OP x 0.30 OP = $116.67 BACK TO GAME BOARD**8.2 Profit Margin**• The gross profit on an item is the difference between what the item cost and what it sold for. • Of course, the business of buying and selling is not that simple. Every store has to take on plenty of other overhead expenses such as rent, utilities, salaries, finance costs, advertising expenses, etc. Gross profit does not account for those. • The net profit, on the other hand, is the profit made after taking into account all of the expenses of doing business. • The profit margin is the profit expressed as a percent of the selling price.**8.2 Profit Margin**Example 8.2.1 • Problem • Sally’s Fashion Paradise sells a dress that cost $45 for $65. Find the gross profit margin from this sale. • Solution • Gross Profit = $65 -- $45 = $20 • Gross Profit Margin = $20/$65 = 0.3077 = 30.77% Example 8.22 • Problem • Sally’s Fashion Paradise sells women’s purses, pricing them with a 35% gross profit margin. If a purse is priced at $72, what is the gross profit in that price? • Solution • Gross Profit = 35% x $72 = $25.20**8.2 Profit Margin**Example 8.2.3 • Problem • Last year, sales at Sally’s Fashion Paradise totaled $219,540. The cost of the items sold was $147,470. What was the overall gross profit margin for the year? • Solution • Total Gross Profit = $219,540 -- $147,470 = $72,070 • Gross Profit Margin = $72,070/$219,540 = 32.83%**8.2 Profit Margin**Example 8.2.4 • Problem • Last year, Sally’s Fashion Paradise had overhead expenses totaling $63,073. Find (a) expenses as a percent of sales and (b) the net profit margin. • Solution • $63,073/$219,540 = 28.73% • Net Profit = $72,070 -- $63,073 = $8,997 • Net Profit Margin = $8,997/$219,540 = 4.10%**8.2 Profit Margin**Example 8.2.5 • Problem • Two years ago, Sally’s shop had sales totaling $153,670. The cost of the goods sold was $118,945, and her expenses totaled $57,950. Find her overall (a) gross profit margin and (b) net profit margin for that year. • Solution • Gross Profit = $153,670 -- $118,945 = $34,725 • Gross Profit Margin = $34,725/$153,670 = 22.60% • Net Profit = $34,725 -- $57,950 = -$23,225 • Net Profit Margin = -$23,225/$153,670 = -15.11%**8.2 Profit Margin**Example 8.2.6 • Problem • In the year in which the dress sold for $65, the total sales were $219,540 and expenses were $63,073. If expenses are allocated in proportion to sales, how much of the store’s expenses is attributable to that dress? • Solution • $65/$219,540 = 0.02961% • 0.02961% x $63,073 = $18.67**8.2 Profit Margin**• Determining a price by using a target gross margin is called markup based on selling price, in contrast to markup based on cost. • If we know the item’s cost, and if we know our markup percent based on cost, calculating the selling price is fairly straightforward. • Profit margin, though, is a percent of the selling price. • Obviously, we don’t know the selling price before we know the selling price!**8.2 Profit Margin**FORMULA 8.2.1 Markup Based on Selling Price C = SP(1 – r) where C represents the ITEM’S COST SP represents the SELLING PRICE and r represents the GROSS PROFIT MARGIN**8.2 Profit Margin**Example 8.2.7 • Problem • Determine the selling price of an item costing $45 in order to have a 35% gross profit margin. • Solution • C = SP(1 – r) $45 = SP(1 – 35%) SP = $69.23**8.2 Profit Margin**Example 8.2.8 • Problem • A cooperative market allows its members to place special orders for items they want to buy in bulk. The price the member pays is based on an 8% markup on selling price. • Lynne ordered a case of protein bars, for which the market’s cost was $24.17. How much will she pay for this order? • C = SP(1 – r) $24.17 = SP(1 – 8%) SP = $26.27**Problem 1**• Harvey’s sells watermelons for $4.99 each, even though farmers sell them for $2.00 each. What is the gross profit margin? CHECK YOUR ANSWER**Solution 1**• Harvey’s Supermarket sells watermelons for $4.99 each, even though farmers sell them for $2.00 each. What is the gross profit margin? • Gross Profit = $4.99 -- $2.00 = $2.99 • Gross Profit Margin = $2.99/$4.99 = 59.92% BACK TO GAME BOARD**Problem 2**• Harvey’s Supermarket had overhead expenses totaling $49,265, the cost of items sold was $153,076, and total sales were $240,543. • What is the net profit margin? CHECK YOUR ANSWER**Solution 2**• Harvey’s Supermarket had overhead expenses totaling $49,265, the cost of items sold was $153,076, and total sales were $240,543. • What is the net profit margin? • Net Profit = $240,543 -- $153,076 -- $49,265 = $38,202 • Net Profit Margin = $38,202/$240,543 = 15.88% BACK TO GAME BOARD**Problem 3**• Determine the selling price of an item costing $395.72 in order to have a 40% gross profit margin. CHECK YOUR ANSWER**Solution 3**• Determine the selling price of an item costing $395.72 in order to have a 40% gross profit margin. • C = SP(1 – r) $395.72 = SP(1 – 40%) $395.72 = SP x 0.60 SP = $659.53 BACK TO GAME BOARD**Problem 4**• You purchased a set of dishes for $67.45. If the markup based on selling price is 41%, what was the item’s cost? CHECK YOUR ANSWER**Solution 4**• You purchased a set of dishes for $67.45. If the markup based on selling price is 41%, what was the item’s cost? • C = SP(1 – r) C = $67.45(1 – 41%) C = 39.80 BACK TO GAME BOARD**8.3 Series and Trade Discounts**• Merchants are normally free to set prices as they see fit, basing their pricing decisions on costs, profit targets, and competition. • Still, many manufacturers do set suggested prices for their products. The suggested price for an item is known as a list price or manufacturer’s suggested retail price (MSRP). • When a product has a list price, it is not uncommon for the item to be sold to a merchant on the basis of a discount to the list price. This is known as a trade discount.**8.3 Series and Trade Discounts**Example 8.3.1 • Problem • Ampersand Computers bought 12 computers from the manufacturer. The list price is $895.00 and the manufacturer offered a 25% trade discount. How much did Ampersand pay for the computers? • Solution $895.00 x 25% = $671.25 12 x $671.25 = $8,055.00**8.3 Series and Trade Discounts**Example 8.3.2 • Problem • Samir’s House of Gadgets placed an order for 500 thingmies (list price $4.95), 350 jimmamathings (list price $8.95), and 800 hoozamawhatzits (list price $17.99). The manufacturer offers a 27 ½% trade discount and includes shipping in its prices. Find the total due on the invoice for this order.**8.3 Series and Trade Discounts**Example 8.3.2 Cont. • Solution**8.3 Series and Trade Discounts**FORMULA 8.3.1 Trade Discounts NP = LP(1 – d) where NP represents the NET (DISCOUNTED) PRICE LP represents the LIST PRICE d represents the PERCENT DISCOUNT**8.3 Series and Trade Discounts**Example 8.3.4 • Problem • What is the net cost for each jimmamathing in the previous example? • Solution NP = LP(1 – d) NP = $8.95(1 – 27.5%) NP = $6.49**8.3 Series and Trade Discounts**Example 8.3.5 • Problem • Samir realized that he forgot to order 400 doohickeys. He called the manufacturer and was given a price of $2,652 for them. He did not ask for the list price but realizes that he needs to know it now. What is the list price for a doohickey? • Solution $2,652/400 = $6.63 NP = LP(1 – d) $6.63 = LP(1 – 27.5%) LP = $9.14**8.3 Series and Trade Discounts**Example 8.3.6 • Problem • A manufacturer offers a 30% trade discount. If the merchant sells items at a 10% discount to list, what is the gross profit margin? What is the markup based on cost? • Solution • We will use $100 for convenience. • The cost would be 70% x $100 = $70 • The selling price would be 90% x $100 = $90 • Gross Profit Margin C = SP(1 – r) $70 = $90(1 – r) r = 22.22% • Markup Based on Cost $90 = $70(1 + r) r = 28.57%**8.3 Series and Trade Discounts**• Series discounts are multiple discounts to a price in succession. • Sometimes, a manufacturer may offer multiple trade discounts. • For example, a company might normally offer a 25% trade discount but during a special promotion or to match a competitor’s pricing, might offer an additional 15% discount. • Despite appearances, it’s incorrect to conclude that success discounts of 25% and 15% are equivalent to a single discount of 40%. • The single discount equivalent to a series of discounts is referred to as the single equivalent discount.**8.3 Series and Trade Discounts**Example 8.3.7 • Problem • The list price for a herbal weight loss supplement is $39.95. The manufacturer normally offers a 25% trade discount, but during a special promotion it offers an additional 15% discount. Find the net price for this item. • Solution 75% x $39.95 = $29.96 85% x $29.96 = $25.47 This could also be calculated more simply as (75%)(85%)($29.96) = $25.47**8.3 Series and Trade Discounts**Example 8.3.8 • Problem • Find the single equivalent discount for successive 25% and 15% discounts. • Solution • We will work from an assumed price of $100. These discounts will reduce the price to 75% x 85% x $100 = $63.75 This is a total discount of $100 -- $63.75 = $36.25 As a percent of the list price, it’s $36.25/$100 = 36.25%