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Contemporary Business Mathematics With Canadian Applications Ninth Edition S. A. HummelbrunnerChapter 1
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Copyright © 2012 Pearson Canada Inc. 1-1
Chapter 1
Review of Arithmetic
Contemporary Business Mathematics with Canadian Applications Ninth Edition S. A. Hummelbrunner/K. Suzanne CoombsPowerPoint: D. Johnston, Revised by P. Au
Copyright © 2012 Pearson Canada Inc. 1-2
ObjectivesAfter completing chapter one, the student will
be able to:• Simplify arithmetic expressions using the
basic order of operations.• Determine equivalent fractions and convert
fractions to decimals.• Convert percents to common fractions and to
decimals, and change decimals and fractions to percents.
(continued)
Copyright © 2012 Pearson Canada Inc. 1-3
Objectives(continued)
• Through problem solving, compute simple arithmetic and weighted averages.
• Determine gross earnings for employees remunerated by the payment of salaries, hourly wages, or commissions.
• Through problem solving, compute GST, HST, PST, sales taxes, and property taxes.
Basic Order of Operations
1. Perform all operations inside a bracket first (operations inside the bracket must be performed in the proper order)
2. Perform all exponents 3. Perform multiplication and division in the
order as they appear from the left to right.4. Perform addition and subtraction in order
as they appear from left to right.Copyright © 2012 Pearson Canada Inc. 1-4
Copyright © 2012 Pearson Canada Inc. 1-5
BEDMASA rule for basic order of operations
• B Brackets • E Exponents• D Division• M Multiplication• A Addition• S Subtraction
Copyright © 2012 Pearson Canada Inc. 1-6
Examples Using BEDMAS
8 – 3 x 2 = 2 Multiplication done first
20 5 + 3 x 4 = 16 Division and Multiplication done before addition
(14 + 4) 2 – 5 = 4 Operations in brackets done first followed by division and then subtraction
144 32 = 16 Exponentiation done first followed by division
Copyright © 2012 Pearson Canada Inc. 1-7
Common Fractions
Terms are 4 and 5
4 5
Numerator Denominator
Proper fraction
3 7
Numerator less than Denominator
Improper fraction
7 3
Numerator greater than Denominator
Copyright © 2012 Pearson Canada Inc. 1-8
Equivalent FractionsChange terms without changing value
Multiply numerator
and denominator by the same number.
Copyright © 2012 Pearson Canada Inc. 1-9
Equivalent Fractions in Lower Terms
Numerator and denominator are divisible by the same number
Copyright © 2012 Pearson Canada Inc. 1-10
Convert Common Fraction to Decimal Form
• Divide the numerator by the denominator
• 12/5 = 2.4• 1/8 = 0.125• 1/6 = 0.1666666….
Copyright © 2012 Pearson Canada Inc. 1-11
Convert a Mixed Number to Decimal Form
Copyright © 2012 Pearson Canada Inc. 1-12
RoundingIf the first digit in the group of digits to be dropped is 5 or greater, the last digit retained is increased by 1.
6.885 rounded to two places after the decimal point is 6.89
If the first digit in the group of digits to be dropped is less than 5, the last digit retained is not changed.
6.8543 rounded to two decimal places is 6.85.
Copyright © 2012 Pearson Canada Inc. 1-13
Complex FractionsFractions may appear in the numerator or
denominator or both.
Fraction Solution
Copyright © 2012 Pearson Canada Inc. 1-14
Using a Calculator to Evaluate a Complex Fraction
The 1 or x –1 function key can be used to x evaluate the following complex fraction:
Copyright © 2012 Pearson Canada Inc. 1-15
Using the Calculator Function 1/x
Copyright © 2012 Pearson Canada Inc. 1-16
Percent Means Hundredths% means dividing by 100
17 % = 17/100
0.8% = .8/100 = 8/1000
1/8% = .125/100
Copyright © 2012 Pearson Canada Inc. 1-17
Changing Percents to Decimals
Drop percent symbol and move decimal pointtwo places to the left.
55 % = 0.55
215 % = 2.15
0.75% = .0075
3/8 % = .375% = .00375
Copyright © 2012 Pearson Canada Inc. 1-18
Changing Decimals to Percents
Move the decimal point two places to the rightand add the % symbol.
0.00525 = .525%
0.38 = 38%
2.55 = 255%
1 3/8 = 1.375 = 137.5%
Copyright © 2012 Pearson Canada Inc. 1-19
Changing Fractions to Percents
First convert the fraction to a decimal. Then convert the decimal to a percent. 7/8 = .875 = 87.5%
1/3 = .333333 = 33.3333%
4/7 = 0.5714 = 5.71%
1 ¼ = 1.25 = 125%
Copyright © 2012 Pearson Canada Inc. 1-20
Arithmetic Averages
1) Add the values in a set of numbers to find the sum of the data values.
2) Divide the sum of the data values by the number of values in the set. (continued)
Copyright © 2012 Pearson Canada Inc. 1-21
Arithmetic Averages(continued)
A student obtains the following scores on five mid-term exams: 78%, 80%, 50%, 65%, 72%
The average score on the five equally weighted mid-terms is:
Copyright © 2012 Pearson Canada Inc. 1-22
Weighted Average
• Use a weighting factor to indicate the number of items in a group or the relative importance of data items.– Multiply each item by the numbers involved or
by a weighting factor representing its importance.
(continued)
Copyright © 2012 Pearson Canada Inc. 1-23
Weighted Average Grade A=4, B= 3, C=2, D=1
Credit
Courses Grade Hours
Accounting B 3 Mathematics C 4 English A 3 Elective A 2 Total = 12 Weighted Average
(3x3+2x4+4x3+4x2) 12
= 3.08
Payroll ApplicationsSalaries
• Compensation of employees by salary is usually on a monthly or a yearly basis.– Monthly salaried personnel get paid either monthly
or semi-monthly– Personnel on a yearly salary basis may get paid
monthly, semi-monthly, every two weeks, weekly, or according to special schedules such as those used by some boards of education to pay their teachers.
• If salary is paid weekly or every two weeks, the year is assumed to consist of exactly 52 weeks
Copyright © 2012 Pearson Canada Inc. 1-24
Copyright © 2012 Pearson Canada Inc. 1-25
Commissions
• Straight - Percent on net sales for a given time period
• Graduated - Increase in percent paid for higher sales levels for a given time period
• Salary Plus Commission - Guarantees minimum income for a given time period
Copyright © 2012 Pearson Canada Inc. 1-26
Straight Commission
Monthly Sales $25,000
Monthly commissionrate
6%
Monthly commission $25,000 x .06 = $1500
Copyright © 2012 Pearson Canada Inc. 1-27
Graduated CommissionRate $50,000 monthly sales
5% First $10,000 6% Next $10,000 9% Above $20,000 .05 x 10,000 500 .06 x 10,000 600 .09 x (50,000-20,000) 2700 Commission 500+600+2700 =
3800
Copyright © 2012 Pearson Canada Inc. 1-28
Wages
• Compensation paid to hourly rated employees
• Gross earnings are calculated by multiplying the number of hours worked by the hourly rate of pay plus any overtime pay
Gross Earnings = Gross Pay for regular workweek +
Overtime Pay
Copyright © 2012 Pearson Canada Inc. 1-29
Calculating Gross Pay
• Regular workweek - 40 hours• Overtime is most often paid at time-and-a-half
the regular hourly rate for any hours exceeding an established number of regular hours per week or per day
Overtime hourly rate = 1.5 x regular hourly rate
Copyright © 2012 Pearson Canada Inc. 1-30
Calculating Gross EarningsRegular Workweek 40 hours Regular hourly rate $16.50 / hour Overtime rate 1.5 x regular
rate/hour Total Weekly Hours 52 hours worked
Regular Pay 40 x $16.50 = 660
Overtime Pay (52-40) x $16.50 x 1.5
Gross Earnings $660 + 297 = $957
Copyright © 2012 Pearson Canada Inc. 1-31
Overtime Premium
The overtime premium on the excess iscalculated separately.
52 X $16.50 = 858
12 x 0.5 x $16.50 = 99 (Excess)
Gross Pay = $858 + 99 = $957
Application Taxes
• A tax is a fee charged on sales, services, property, or income by a government to pay for services provided by the government. As consumers, we encounter the;– provincial sales tax (PST),– goods and services tax (GST), – or the harmonized sales tax (HST).
Copyright © 2012 Pearson Canada Inc. 1-32
Copyright © 2012 Pearson Canada Inc. 1-33
Application Taxes
• The Goods and Services Tax (GST) is a federal tax charged on almost all goods and services.
• PST and GST are blended to form the Harmonized Sales Tax or HST in some provinces. For British Columbia and Ontario, the HST came into effect on July 1, 2010. British Columbia 12% HST– Newfoundland 13% HST– New Brunswick 13% HST– Nova Scotia 15% HST– Ontario 13% HST
• Purchase a VCR for $109.• The HST (On) = $109 x 13% = 109 x 0.13 = $14.17
Copyright © 2012 Pearson Canada Inc. 1-34
Provincial Sales Tax
• Some Provinces such as Manitoba levy a PST (7%) and GST (5%)– Alberta does not levy a PST– Quebec levies a GST + QST
• Purchase a VCR for $109 in Manitoba• GST = $109 x .07 = $7.63• PST = $109 x .05 = $5.45• Total Sales Tax = GST + PST = $13.08.
Copyright © 2012 Pearson Canada Inc. 1-35
Property Tax• Municipal tax charged on the assessed value of real
estate.• Based on the assessed value of real estate, both
residential and commercial. • Some education taxes are also calculated using this
method. The property tax is determined by applying a percent to the assessed value of the property.
• In some municipalities, the assessed value is divided by 1000, and then the percent is applied.
Calculating Property Tax
Copyright © 2012 Pearson Canada Inc. 1-36
Copyright © 2012 Pearson Canada Inc. 1-37
Summary
• Common business applications such as payroll, commissions, and taxes involve the use of basic arithmetical operations, fractions, percents, and decimals.
• Averages are useful in analyzing data.