3.E: General Business Management Applications (Exercises)

3.1: Percent Change (Are We Up Or Down?)

Mechanics

For questions 1–3, solve for the unknown (?) using Formula 3.1 (percent change).

	Old	New	∆%
1.	$109.95	$115.45	?
2.	?	$622.03	13.25%
3.	5.94%	?	−10%

4. If $9.99 is changed to $10.49, what is the percent change?
5. $19.99 lowered by 10% is what dollar amount?
6. What amount when increased by 40% is $3,500?
7. If 10,000 grows to 20,000 over a period of 10 years, what is the annual rate of change?

Applications

8. How much, including taxes of 12%, would you pay for an item with a retail price of $194.95?
9. From September 8, 2007 to November 7, 2007, the Canadian dollar experienced a rapid appreciation against the US dollar, going from $0.9482 to $1.1024. What was the percent increase in the Canadian dollar?
10. From 1996 to 2006, the “big three” automakers in North America (General Motors, Ford, and Chrysler) saw their market share drop from 71.5% to 52.7%. What is the overall change and the rate of change per year?
11. The average price of homes in Calgary fell by $10,000 to $357,000 from June 2009 to July 2009. The June 2009 price was 49% higher than the June 2005 price.
    1. What was the percent change from June 2009 to July 2009?
    2. What was the average price of a home in June 2005?
    3. What was the annual rate of change from June 2005 to June 2009?
12. On October 28, 2006, Saskatchewan lowered its provincial sales tax (PST) from 7% to 5%. What percent reduction does this represent?
13. A local Superstore sold 21,983 cases of its President's Choice cola at $2.50 per case. In the following year, it sold 19,877 cases at $2.75 per case.
    1. What is the percent change in price year-over-year?
    2. What is the percent change in quantity year-over-year?
    3. What is the percent change in total revenue year-over-year? (Hint: revenue = price × quantity)
14. A bottle of liquid laundry detergent priced at $16.99 for a 52-load bottle has been changed to $16.49 for a 48-load bottle. By what percentage has the price per load changed?

Challenge, Critical Thinking, & Other Applications

15. At a boardroom meeting, the sales manager is happy to announce that sales have risen from $850,000 to $1,750,000 at a rate of 4.931998% per year. How many years did it take for the sales to reach $1,750,000?
16. The Nova Scotia Pension Agency needs to determine the annual cost of living adjustment (COLA) for the pension payments made to its members. To do this, it averages the consumer price index (CPI) for both the previous fiscal year and the current fiscal year. It then calculates the percent change between the two years to arrive at the COLA. If CPI information is as follows, determine the COLA that the pensioners will receive.

	Previous Fiscal Year	Current Fiscal Year
November	109.2	111.9
December	109.4	112.0
January	109.4	111.8
February	110.2	112.2
March	111.1	112.6
April	111.6	113.5
May	112.1	114.6
June	111.9	115.4
July	112.0	115.8
August	111.7	115.6
September	111.9	115.7
October	111.6	114.5

17. During The Bay’s warehouse clearance days, it has reduced merchandise by 60%. As a bonus, today is Scratch ’n’ Save day, where you can receive up to an additional 25% off the reduced price. If you scratched the maximum of 25% off, how many dollars would you save off an item that is regularly priced at $275.97? What percent savings does this represent?
18. Federal Canadian tax rates for 2010 and 2011 are listed below. For example, you pay no tax on income within the first bracket, 15% on income within the next bracket, and so on. If you earned $130,000 in each year, by what percentage did your federal tax rate change? In dollars, what was the difference?

2010 Tax Brackets	Taxed at	2011 Tax Brackets
$0–$10,382	0%	$0–$10,527
$10,383–$40,970	15%	$10,528–$41,544
$40,971–$81,941	22%	$41,545–$83,088
$81,942–$127,021	26%	$83,089–$128,800
$127,022+	29%	$128,801+

19. Melina is evaluating two colour laser printers for her small business. A Brother model is capable of printing 21 colour pages per minute and operates 162.5% faster than a similar Hewlett-Packard model. She needs to print 15,000 pages for a promotion. How much less time (stated as a percentage) will it take on the Brother model?
20. A chocolate bar has been priced at $1.25 for a 52 gram bar. Due to vending machine restrictions, the manufacturer needs to keep the price the same. To adjust for rising costs, it lowers the weight of the bar to 48 grams.
    1. By what percentage has the price per gram changed?
    2. If this plan is implemented over two periods, what rate of change occurs in each period?

3.2: Averages (What Is Typical?)

Mechanics

Calculate a simple average for questions 1 and 2.

1. 8, 17, 6, 33, 15, 12, 13, 16
2. $1,500 $2,000 $1,750 $1,435 $2,210

Calculate a weighted average for questions 3 and 4.

3. 4, 4, 4, 4, 12, 12, 12, 12, 12, 12, 12, 15, 15

Data	$3,600	$3,300	$3,800	$2,800	$5,800
Weight	2	5	3	6	4

Calculate a geometric average for exercises 5 and 6. Round all percentages to four decimals.

5. 5.4%, 8.7%, 6.3%
6. 10%, −4%, 17%, −10%

Applications

7. If a 298 mL can of soup costs $2.39, what is the average price per millilitre?
8. Kerry participated in a fundraiser for the Children's Wish Foundation yesterday. She sold 115 pins for $3 each, 214 ribbons for $4 each, 85 coffee mugs for $7 each, and 347 baseball hats for $9 each. Calculate the average amount Kerry raised per item.
9. Stephanie’s mutual funds have had yearly changes of 9.63%, -2.45%, and 8.5%. Calculate the annual average change in her investment.
10. In determining the hourly wages of its employees, a company uses a weighted system that factors in local, regional, and national competitor wages. Local wages are considered most important and have been assigned a weight of 5. Regional and national wages are not as important and have been assigned weights of 3 and 2, respectively. If the hourly wages for local, regional, and national competitors are $16.35, $15.85, and $14.75, what hourly wage does the company pay?
11. Canadian Tire is having an end-of-season sale on barbecues, and only four floor models remain, priced at $299.97, $345.49, $188.88, and $424.97. What is the average price for the barbecues?
12. Calculate the grade point average (GPA) for the following student. Round your answer to two decimals.

Course	Grade	Credit hours
Economics 100	D	5
Math 100	B	3
Marketing 100	C	4
Communications 100	A	2
Computing 100	A+	3
Accounting 100	B+	4

Grade	Grade Point
A+	4.5
A	4.0
B+	3.5
B	3.0
C+	2.5
C	2.0
D	1.0
F	0.0

13. An accountant needs to report the annual average age (the length of time) of accounts receivable (AR) for her corporation. This requires averaging the monthly AR averages, which are listed below. Calculate the annual AR average.

Month	Monthly AR Average
January	$45,000
February	$70,000
March	$85,000
April	$97,000
May	$145,000
June	$180,000
July	$260,000
August	$230,000
September	$185,000
October	$93,000
November	$60,000
December	$50,000

14. From January 2007 to January 2011, the annual rate of inflation has been 2.194%, 1.073%, 1.858%, and 2.346%. Calculate the average rate of inflation during this period.

Challenge, Critical Thinking, & Other Applications

15. Gabrielle is famous for her trail mix recipe. By weight, the recipe calls for 50% pretzels, 30% Cheerios, and 20% peanuts. She wants to make a 2 kg container of her mix. If pretzels cost $9.99/kg, Cheerios cost $6.99/kg, and peanuts cost $4.95/kg, what is the average cost per 100 g rounded to four decimals?
16. Caruso is the marketing manager for a local John Deere franchise. He needs to compare his average farm equipment sales against his local Case IH competitor’s sales. In the past three months, his franchise has sold six $375,000 combines, eighteen $210,000 tractors, and fifteen $120,000 air seeders. His sales force estimates that the Case IH dealer has sold four $320,000 combines, twenty-four $225,000 tractors, and eleven $98,000 air seeders. Express the Case IH dealer's average sales as a percentage of the John Deere dealer's average sales.
17. You are shopping for shampoo and consider two brands. Pert is sold in a bundle package of two 940 mL bottles plus a bonus bottle of 400 mL for $13.49. Head & Shoulders is sold in a bulk package of three 470 mL bottles plus a bonus bottle of 280 mL for $11.29.
    1. Which package offers the best value?
    2. If the Head & Shoulders increases its package size to match Pert at the same price per mL, how much money do you save by choosing the lowest priced package?
18. The following are annual net profits (in millions of dollars) over the past four years for three divisions of Randy’s Wholesale:

\[\text { Cosmetics: } \$ 4.5, \$ 5.5, \$ 5.65, \$ 5.9\nonumber \]

\[\text { Pharmaceutical: } \$ 15.4, \$ 17.6, \$ 18.5, \$ 19.9\nonumber \]

\[\text { Grocery: } \$ 7.8, \$ 6.7, \$ 9.87, \$ 10.75\nonumber \]Rank the three divisions from best performing to worst performing based on average annual percent change.

19. You are shopping for a Nintendo Wii gaming console and visit www.shop.com, which finds online sellers and lists their prices for comparison. Based on the following list, what is the average price for a gaming console (rounded to two decimals)?

NothingButSoftware.com	$274.99
eComElectronics	$241.79
NextDayPC	$241.00
Ecost.com	$249.99
Amazon	$169.99
eBay	$165.00
Buy.com	$199.99
HSN	$299.95
Gizmos for Life	$252.90
Toys ‘R’ Us	$169.99
Best Buy	$169.99
The Bay	$172.69
Walmart	$169.00

20. Juanita receives her investment statement from her financial adviser at Great-West Life. Based on the information below, what is Juanita’s average rate of return on her investments?

Investment Fund	Proportion of Entire Portfolio Invested in Fund	Fund Rate of Return
Real Estate	0.176	8.5%
Equity Index	0.073	36.2%
Mid Cap Canada	0.100	−1.5%
Canadian Equity	0.169	8.3%
US Equity	0.099	−4.7%
US Mid Cap	0.091	−5.7%
North American Opportunity	0.063	2.5%
American Growth	0.075	−5.8%
Growth Equity	0.085	26.4%
International Equity	0.069	−6.7%

3.3: Ratios, Proportions, and Prorating (It Is Only Fair)

Mechanics

For questions 1–4, reduce the ratios to their smallest terms.

1. 1. \(66 : 12\)
   2. \(48 : 112 : 80\)
   3. \(24 : 36 : 84 : 108\)
2. 1. \(7.2 : 6\)
   2. \(0.03 : 0.035 : 0.02\)
   3. \(0.27 : 0.18 : 0.51 : 0.15\)
3. 1. \(5 \dfrac{1}{4} : 6\)
   2. \(\dfrac{3}{2} : \dfrac{9}{5} : \dfrac{21}{10}\)
   3. \(2 \dfrac{1}{8} : \dfrac{9}{8} : 3 \dfrac{3}{8} : 1\)
4. 1. \(\dfrac{8}{3} : \dfrac{5}{9}\)
   2. \(\dfrac{5}{6} : \dfrac{1}{3} : 2\)
   3. \(\dfrac{1}{7} : \dfrac{1}{3} : \dfrac{1}{4} : \dfrac{1}{6}\)
5. Reduce the following ratios to the smallest term of one. Round to two decimals as needed.
   1. \(48 : 53\)
   2. \(5 \dfrac{7}{8} : 2 \dfrac{3}{4} : 3 \dfrac{1}{12}\)
   3. \(\dfrac{2}{3} : 11 \dfrac{1}{11} : 5 \dfrac{5}{9} : 2.08\)
6. Solve the following proportions for the unknown variable. Round to two decimals as needed.
   1. \(7 : 12 = 109 : y\)
   2. \(3.23 : 4.07 : 2.12 = x : 55.9625 : 29.15\)
   3. \(q : \dfrac{5}{3}=13.75 : 25\)
7. Solve the following proportions for all of the unknown variables.
   1. \(\$ 12.15 : \$ 38.30 : r=x : \$ 59,184.53 : \$ 26,998.93\)
   2. \(16 : 5 : 9 : 30=g : h : i : \$ 397,767\)

For questions 8 and 9, prorate the total as indicated by the ratio.

8. \(\text { Ratio }=3 : 2\), \(\text { Total }=\$ 11,368.25\)
9. \(\text { Ratio }=7 : 5 : 3, \text { Total }=\$ 46,923.90\)

Applications

10. EB Games sells gaming consoles. Last month, it sold $22,500 worth of Nintendo Wii consoles, $31,500 worth of Microsoft Xbox consoles, and $18,000 worth of Sony PlayStation consoles. Express the ratio of Wii to Xbox to PlayStation sales in its lowest terms.
11. The manufacturing cost of a deluxe candle is made up of $2.40 in paraffin, $1.20 in dye, $1.60 in overhead, and $4.40 in direct labour. Express the ratio between these costs respectively in their lowest terms.
12. If one Canadian dollar buys $1.0385 of US dollars, how much US money can you spend so that you do not exceed the maximum duty-free amount of $800 Canadian on your next US vacation?
13. For every $100 of retail spending, the average Canadian spends approximately $19.12 at motor vehicle dealerships and $18.35 at grocery stores. If the average Canadian spends $49,766 per year on retail spending, how much more does a Canadian spend annually at motor vehicle dealerships than at grocery stores?
14. Marina has a three-sevenths interest in a partnership. If she sells one-quarter of her interest for $8,250, what is the implied value of the partnership?
15. You are making a punch for an upcoming party. The recipe calls for 2½ parts ginger ale to ⅘ parts grenadine to 1⅜ parts vodka. If your punch bowl can hold 8.5 litres, how much (in litres) of each ingredient is needed? Round the answers to two decimals as needed.

Challenge, Critical Thinking, & Other Applications

16. A manufacturing facility requires one member of the board of directors for every 10 executives. There are five managers for every executive and eight workers for every manager. The average salary paid to directors is $105,000 along with $80,000 to executives, $55,000 to managers, and $30,000 to workers. If the facility has 1,383 employees, what is the total labour cost?
17. It is common for strip mall owners to allocate to their tenants general overhead costs such as snow clearing, security, and parking lot maintenance in one of two ways. The first method involves allocating these costs on the basis of the square metres each tenant operates. The second method involves allocating these costs on the basis of the number of annual transactions made by each tenant. At Charleswood Square, there are three tenants. The 7-Eleven takes up 150 square metres, Tim Hortons takes up 235 square metres, and Quiznos takes up 110 square metres. Last year, 7-Eleven had 350,400 transactions, Tim Hortons had 657,000 transactions, and Quiznos had 197,100 transactions. Total overhead costs for Charleswood Square last year were $25,980. Determine which method of allocation each tenant would prefer and how much of a savings in percentage its preferred method would represent relative to the other allocation method.
18. The average annual Canadian cellphone plan covers 780 minutes of voice time, 600 text messages, and 8 video messages with a total annual cost of $500.63. However, some Canadians are high-usage consumers and average 439 minutes per month.
    1. Assuming the same proportion of usage and that each component shares the cost equally on a per-unit basis, how many text messages and video messages do these high-usage Canadian consumers average per year? How much should they pay annually for their cellphone usage?
    2. A similar typical cellphone plan in the Netherlands has a total annual cost of $131.44. How much would a high-usage plan cost annually in the Netherlands?
19. Procter & Gamble (P&G) is analyzing its Canadian sales by region: 23% of sales were on the West coast; 18% were in the Prairies, 45% were in central Canada, and 14% were in Atlantic Canada. Express and reduce the ratio of P&G's respective sales to the smallest term of one. Round to two decimals.
20. A company has 14 managers, 63 supervisors, and 281 workers. After a record profit year, the company wants to distribute an end-of-year bonus to all employees. Each worker is to get one share, each supervisor gets twice as many shares as workers, and each manager gets twice as many shares as supervisors. If the bonus to be distributed totals $1,275,000, how much does each manager, supervisor, and worker receive?

Review Exercises

Mechanics

1. Compute an average for the following data set: $1,368.95 $747.45 $5,362.25 $699.99 $2,497.97
2. Compute an average mark for the following quiz results.

Mark	0	1	2	3	4	5	6	7	8
# of Students	0	1	0	3	4	12	12	6	2

3. Solve each of the following for the unknown.

	Last Year's Sales	This Year's Sales	Percent Change
a.	$325,000	$377,000	?
b.	?	$5,460,000	225%
c.	$12,675	?	−38%

4. The following problems involve a series of percent changes. Calculate the annual rate of change.

	Initial Value	Number of % Changes	Final Value
a.	$4,300	2	$4,568.75
b.	$300	3	$114

5. Reduce the following ratios to their smallest terms.
   1. \(112 : 32 : 144\)
   2. \(1.44 : 0.84\)
   3. \(\dfrac{1}{4} : \dfrac{2}{5} : \dfrac{5}{8}\)
   4. \(\dfrac{17}{27} : \dfrac{3}{4}\)
6. Reduce the following ratios to the smallest term of one. Round to two decimals as needed.
   1. \(\dfrac{3}{13} : \dfrac{3}{7}\)
   2. \(0.05325 : 0.0615 : 0.0445\)
   3. \(38 : 95 : 27 : 47\)
7. Solve the following proportions for the unknown variables.
   1. \(6 : x=27 : 18\)
   2. \(10 : 12.5 : 20=q : r : 50\)
8. Prorate the total as indicated by the ratio. Ratio = \(9 : 8 : 3\), \(\text { Total }=\$ 25,000\)

Applications

9. In the year 2000, Canada's population was 31,496,800, and 40.3% of Canadians used the Internet. By 2008, Canada's population was 33,212,696, and 84.3% of Canadians used the Internet. Calculate the annual rate of change, rounded to two decimals, in the number of Canadians using the Internet.
10. Maurice, Jennifer, Natasha, and Sinbad just graduated from their business program at Mohawk College and decided to go into business together. Each individual invested funds in the ratio of 4 : 7 : 3 : 6, respectively. If Natasha invested $24,000, what was the total investment in the business?
11. An Extra Foods store needs to allocate overhead expenses of $336,000 across its three departments based on their sales. If refrigerated foods accounted for $765,000 in sales, non-refrigerated foods accounted for $981,500 in sales, and non-food products accounted for $694,000 in sales, what amount of overhead expenses should be allocated to each department?
12. Boris's neighbourhood just had a neighbourhood garage sale. He decided to make some money by operating a food stand. If he sold 115 hot dogs for $2.50 each, 260 bags of chips for $0.75 each, and 412 soft drinks for $1.25 each, what were his total sales for the day, and how much did he average per transaction (rounded to two decimals)?
13. Julian is trying to understand his investment portfolio better. He currently has $40,000 in Canadian stocks, $7,500 in US stocks, and $2,500 in international stocks. Express the respective ratio of his stocks in their lowest terms.
14. Abdul is at the end of the third year of a five-year investment. His account has a current balance of $1,687.14. Over the past three years, his account has increased 3%, 4%, and 5% respectively. In the next two years, his account is supposed to increase 5.5% and then 6%. How much did Abdul initially invest? How much will he have at the end of the five years?
15. You subscribed to 90 issues of The Hockey News magazine at a total cost of $418.50. After receiving 57 issues, you decide to cancel your subscription. What refund should you receive on your subscription?
16. If one Canadian dollar can purchase $0.7328 euros, 0.6379 pounds, or 83.3557 yen, how much of each currency could you purchase with $925 Canadian?

Challenge, Critical Thinking, & Other Applications

17. Little Caesars Pizza just delivered two large pizzas. Each has been cut up into 12 slices. If Sergei can eat only one-fifth of what Richard can eat, Basilio eats twice as much as Sergei, and Alphonso eats 100% more than Basilio, how many slices of pizza should each receive (assuming they'll eat all of the pizza)? If the total bill including tip came to $15.00, how much does each person owe based on how much he ate?
18. Vincent Manufacturing has two divisions with three different product lines in each. Sales for the current year are as follows:

	Division One		Division Two
Product	Unit Price	Unit Sales	Unit Price	Unit Sales
A	$19.95	3,456	$25.31	3,543
B	$31.56	6,852	$17.41	8,524
C	$22.32	1,894	$29.99	2,005

1. The company wants to set up a ratio of the weighted average price in Division One compared to the weighted average price in Division Two. Calculate the ratio and reduce the ratio to the smallest term of one. Round all numbers to two decimals throughout.
2. Division One's weighted average price is forecasted to increase by 6% next year and 3% in the third year. Division Two's weighted average price is forecasted to increase by 4% next year and 6% in the third year. Calculate the ratios in each of the second and third years, and reduce each ratio to the smallest term of one. Round all numbers to two decimals throughout.

19. In aviation, one of the measures of success is known as the load factor. This is a percentage that is calculated by taking the number of air miles that consumers flew and dividing it by the total air miles that were available for consumers to purchase. In 2008, WestJet Airlines had 17.139 billion available miles, of which consumers purchased 13.731 billion miles. In 2009, it had 17.558 billion available miles, of which consumers purchased 13.835 billion miles. In 2010, it had 19.535 billion available miles, of which consumers purchased 15.613 billion miles. Calculate the percent change in the load factors for each year and the average annual rate of change. Round your answers to four decimal places.
20. As you prepare for vacation, you need to take your spending money of $750 Canadian and convert it to various currencies. On your vacation, you will be spending seven days in New Zealand, eleven days in India, and six days in South Africa. You figure that you will spend your money in proportion to the number of days in each country. If one Canadian dollar equals 1.3136 New Zealand dollars, 46.3478 Indian rupees, and 7.2401 South African rand, how much of each currency will you need to leave home with?

Contributors and Attributions

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