4.4: The Cost of Business

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INTRODUCTION

In this collaboration, we will be discussing price markups and markdowns using both spreadsheet software (Google Sheets or Excel) and algebra to calculate prices before or after a markup or markdown.

Markup: the amount added to the original price of an item, usually calculated as a percentage of the original price
Markdown: the amount subtracted from the original price of an item, usually calculated as a percentage of the original price

Both markups and markdowns are defined as the amount added to or subtracted from the original cost. However, to calculate the cost after a markup or markdown, we will be using multiplication, as shown in Preparation 4.4.

According to fastcompany.com⁹, “Amazon has finally implemented what was long considered a given after it bought Whole Foods for $13.7 billion last year. Now Amazon Prime members get 10% off hundreds of sales items in Whole Foods stores. In addition, they’ll receive steep discounts on other items each week.

The discounted items will be labeled in-store with yellow “10% off” sale signs and “Prime Member Deal” signs. In order to get the savings, customers will just need to scan the Whole Foods mobile app’s “Prime Code” barcode at checkout or give the cashier their phone number. One caveat: Right now, the Prime Member discounts only apply to stores in Florida, but Amazon says the new program will quickly roll out to all its 463 U.S. stores this summer.”

If a Whole Foods customer who is not an Amazon Prime member spent $250 on groceries labeled with “Prime Member Deal,” how much would an Amazon Prime member pay for the same groceries? Discuss in your group, and try to come up with two ways of calculating your answer.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

using algebra to generalize calculations is a valuable tool.

By the end of this collaboration, you should be able to

write or interpret an expression that models a percentage increase or decrease.
use expressions and equations to solve contextual problems about percentage increases or decreases.

PROBLEM SITUATION: WRITING FORMULAS WITH PERCENTAGES

In this collaboration, you will find ways in which Eduardo and Paula can use spreadsheet formulas and percentages to calculate data for their business.

(1) The information from Preparation 4.4 told you that Eduardo and Paula were writing a formula to calculate prices for their merchandise. Their formulas are given below (B2 = the amount they pay to buy the item in dollars).

Paula: B2 + 0.65*B2

Eduardo: B2*1.65

(a) In your group, discuss if both of these formulas are equivalent to one another. Be prepared to justify your decision.

(b) Calculate the price for each item shown in the spreadsheet below. Round your answers to two decimal places.

Screen capture of data in an excel sheet, indicating the following in cells:

Column C = B2 plus 0.65 multiplied by B2

A1 = Item
A2 = Women’s t-shirt
A3 = Men’s t-shirt
A4 = Youth’s t-shirt
A5 = Skirt
A6 = Dress

B1 = Cost of Item
B2 = $8.79
B3 = $10.13
B4 = $6.59
B5 = $23.45
B6 = $35.67

C1 = Price

ADJUSTING PRICING: SIX MONTHS LATER

Six months later, Eduardo and Paula are doing very well with their business. They learned at the first event that prices to the nearest cent are not very practical because it takes so much time to make change for customers. One advantage of a spreadsheet is that you can set it up to give the results of calculations in different forms. Paula revised the spreadsheet to round the prices to the nearest dollar. The rounded results and the formula she used are shown below.

(2) Although their products have sold very well, Paula and Eduardo have some items that have not sold. They decide to put these items on sale for 15% off.

(a) Estimate the sale price of the items. Try not to use calculators or paper or pencil! Discuss your estimation strategies with your group.

Screen capture of data in an excel sheet, indicating the following in cells:

Column C formula = ROUND(B2 plus 0.65 multiplied by B2,0)

A1 = Item
A2 = Women’s t-shirt
A3 = Men’s t-shirt
A4 = Youth’s t-shirt
A5 = Skirt
A6 = Dress

B1 = Cost of Item
B2 = $8.79
B3 = $10.13
B4 = $6.59
B5 = $23.45
B6 = $35.67

C1 = Price
C2 = $15.00
C3 = $17.00
C4 = $11.00
C5 = $39.00
C6 = $59.00

D1 = 15% Discount

(b) Write a formula that Paula can use in Cell D2 to find the sale price of a women’s t-shirt. Try to find more than one formula that works. You may choose whether to include the Round function.

(c) Paula and Eduardo have a few skirts that do not sell even after being discounted. They decide to take another 15% off. Eduardo says this is the same as a 30% discount, but Paula disagrees. Who is right? What is the sale price?

(3) At some events, Eduardo and Paula have to charge sales tax. This means that the total charge is no longer rounded to the nearest dollar, and they have to take the time to make change again. For example, for a women’s t-shirt they need to charge about $15 to make a profit. With 7.5% tax, the total cost to the customer is $16.13. Eduardo and Paula decide to set their prices so that the price comes out to an even dollar amount after the tax is added. So, for the women’s t-shirt, they want the total after tax to be $17. (Eduardo and Paula decide to round the amount up to the next dollar so they do not lose any money.)

(a) The following are the amounts Eduardo and Paula want to charge after tax. Calculate the price before tax that gives these amounts. The tax rate is 7.5%.

Screen capture of data in an excel sheet, indicating the following in cells:

A1 = Item
A2 = Women’s t-shirt
A3 = Men’s t-shirt
A4 = Youth’s t-shirt
A5 = Skirt
A6 = Dress

B1 = Price Before Tax

C1 = Price after Tax
C2 = $17.00
C3 = $19.00
C4 = $12.00
C5 = $42.00
C6 = $64.00

(b) What formula could Paula use in Cell B2 to calculate the price before tax for a women’s t-shirt?

(4) The following question gives you more practice with using percentages in expressions and equations. Column A gives expressions that either increase or decrease the quantity x by a given percentage. Column B gives verbal descriptions of the changes. Write the letter of the description next to the correct expression. A verbal description may be used more than once or not at all.

Answer	Column A	Column B
	x + 0.45x	a) increase x by 45%
	0.45x	b) increase x by 55%
	x – 0.55x	c) increase x by 145%
	1.45x	d) increase x by 245%
	3.45x	e) increase x by 345%
	2.45x	f) decrease x by 55%
	x + 0.55x	g) decrease x by 45%
	x – 0.45x	h) decrease x by 145%
	0.55x

FURTHER APPLICATIONS

(5) Organizations and groups often receive grants that help pay for activities over several years. When planning a budget over several years, people often increase the amounts budgeted each year by a small percentage to account for an increase in prices or inflation. A neighborhood activity center is applying for a three-year grant to provide an after-school program. The amounts budgeted for different items for the first year of a grant are shown below. The grant writer wants to increase all the amounts by 2.5% for Year 2 and by another 2.5% for Year 3.

Salaries: $120,000
Benefits: $21,600
Insurance: $10,000
Materials and supplies: $6,000

Make a spreadsheet that calculates the amounts. You will either make an actual spreadsheet or complete the table below as if it were a spreadsheet and record your formulas as indicated. Your instructor will tell you how to complete the assignment.

Screen capture of data in an excel sheet, indicating the following in cells:
A1 = Item
B1 = Year 1
C1 = Year 2
D1 = Year 3

(a) Formula for Year 2 salaries:

(b) Formula for Year 3 salaries:

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

_____________________________________

⁹ https://www.fastcompany.com/40573252/amazon-prime-members-now-get-10-off-at-whole-foods

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