Chapter 1
Review and
Applications of
Basic
Mathematics
Copyright © 2020 McGraw-Hill Education Limited.
Microsoft® PowerPoint® Presentation by Julie Howse, St. Lawrence College.
Previous edition updated by Rob Sorenson, Camosun College.
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2-2
Why do you
need math
at all?
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Introduction
Basic math skills are
essential for success
in your business
courses, and the
world of business,
and in life!
14
Introduction
You may ask: is most of the
calculations I need not done by the
financial calculator and excel?
Here is what a financial calculator
cannot do for you:
• Decide what information is
relevant
• Analyse and interpret the
results/answers
• Explain the results to clients
These are the math
competencies expected
by employers
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Chapter 1:
Overview
This chapter begins
with a review of
basic concepts and
then explores some
business
applications.
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Learning Objectives
LO1 Perform arithmetic operations in their
proper order. Order of operations
LO2 Convert fractions to their percent and
decimal equivalents.
LO3 Maintain the proper number of digits in
calculations. Rounding
LO4 Perform calculations using fractions,
decimals and percentages.
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Learning Objectives (cont’d)
LO5 Calculate gross earnings of employees
paid a salary, an hourly wage or
commissions.
LO6 Calculate simple or weighted
ation averages.
LO7 Perform basic calculations for the
GST, HST, PST
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Arithmetic Operations
• Order of operations: BEDMAS
1.
2.
3.
4.
Perform all operations within brackets.
Evaluate the powers (exponents).
Perform multiplication and division.
Perform addition and subtraction.
• BEDMAS - Brackets, Exponents, Division,
Multiplication, Addition, Subtraction
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Order of Operations Example
• Consider the following expression.
• BEDMAS tells us to complete the bracket first,
then the exponent and then the
addition/subtractions.
10 + (2 × 3)2 −5
= 10 + 62 − 5
= 10 + 36 − 5
= 41
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Order of Operations Example
• Consider the following expression.
• BEDMAS tells us to complete the division first and
then the addition/subtractions.
• If no brackets – do the addition and subtraction
from left to right. Or be very mindful of the sign
30 − 6 ÷ 3 + 5
= 30 − 2 + 5
= 33
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Skill Check
• Evaluate the following using Order of
Operations
i.
(−3)3 + 4 × 5 + 3 − 6
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Skill Check
• Evaluate the following using Order of
Operations
i.
(−3)3 + 4 × 5 + 3 − 6
= (-3 X -3 X -3) + (20 + 3) – 6
= – 27 + 23 – 6
= – 10
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Skill Check
• Evaluate the following using Order of
Operations
5 + 12 − 24
4 + 34
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Skill Check
• Evaluate the following using Order of
Operations
5 + 12 − 24
4 + 34
=
5+12−16
4+81
=
17−16
85
=
1
85
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Skill Check
• Evaluate the following using Order of
Operation
− −5
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2
3
+6×2+4
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Skill Check
• Evaluate the following using Order of
Operation
− −5 2 + 6 × 2 + 43
=
– 25 + 12 + 64
= 51
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One more example
• Do Exercise question 1.1 # 14 on page 4 in
your textbook
(4 x 3 – 2)2 ÷ (4 – 3 x 22)
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One more example
• Do Exercise question 1.1 # 14 on page 4
(4 x 3 – 2)2 ÷ (4 – 3 x 22)
= (12 – 2)2 ÷ ( 4 – 12)
= (10)2 ÷ (– 8)
= 100 ÷ – 8 = - 12.50
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See page 8
• Example 1.2G. Optimal use of your
calculator.
• Show Appendix 1A. AOS versus Chn see
page 35
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Example 1.2H – Evaluating Complex
Fractions
Evaluate each of the following complex
fractions accurate to the cent.
a.
b.
$425
0.09 24
1+
12
87
$1265 1+0.115×
365
43
1+0.125×
365
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Skill Check
Let’s try these two examples with our calculators
i.
$5600
1+0.175×
46
365
Tip: Either use the two-step process or brackets
for the denominator. Also, be careful not to press
= on your calculator too soon.
ii.
$480 1 + 0.05 ×
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271
365
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Fractions, Decimals & Percent
• A faction consists of a numerator 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
and a denominator.
𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
• Proper faction – the numerator is
smaller than the denominator.
• Improper faction – the numerator
is larger than the denominator.
• Mixed number – a whole number
plus a fraction.
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3
4
5
2
1
2
2
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Fractions, Decimals & Percent
𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
• The numerator signifies the number of parts
taken out of the whole.
• Denominator signifies the whole.
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Equivalent Fractions page 5
• Equivalent fractions are equal in value even
though their respective numerators and
denominators may differ.
1 2
5
• For example: = =
2 4 10
• To create an equivalent fraction, both the
numerator and denominator must be
multiplied by the same number.
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Decimal and Percent Equivalents
• The decimal equivalent of a fraction is
obtained by dividing the numerator by the
denominator.
• The percent equivalent is found by
multiplying the decimal equivalent by 100
and adding the % sign.
3
= 0.75 = 75%
4
Fraction
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Decimal Equivalent
Percent Equivalent
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Calculating Percent of a Number
To find the percent of a number, convert the
percent to its decimal equivalent by diving the
percent by 100
For example:
22% 𝑜𝑓 $185 = 0.22 × $185 = $40.70
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Example 1.2J – A Problem Using Percents
A battery manufacturer encloses a rebate coupon for
15% off in a package of AAA batteries retailing for $6.29.
What rebate does the coupon represent?
Solution: Convert the percentage to a decimal and then
find the rebate.
𝑅𝑒𝑏𝑎𝑡𝑒 = 0.15 × $6.29 = $0.94
The 15% rebate is equivalent to a cash rebate of $0.94.
Show the easy way to do this on your calculator
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Skill Check
Jenica puts 15% of every paycheque into a
savings plan and 45% of the money in the
savings plan is in a Tax-Free Savings Account
(TFSA). If she earns $1,560 every paycheque,
how much money is she putting into the
TFSA?
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Skill Check: answer
Jenica puts 15% of every paycheque into a
savings plan and 45% of the money in the
savings plan is in a Tax-Free Savings Account
(TFSA). If she earns $1,560 every paycheque,
how much money is she putting into the
TFSA?
$1,560 x 0.15 = $234 x 0.45 = $105.30
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Skill Check
Convert each of the following to their
equivalent decimal, percent and fractional
form.
Decimal
Fraction
Percent
0.38
3
8
64%
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Rounding Decimals
• Some decimal equivalents have an endless
series of digits (nonterminating decimal).
• Some may even be repeating decimals.
• A shorthand notation for a repeating
decimal is to place a bar over the repeating
portion.
2
= 0.222222 = 0. 2
9
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Rules for Rounding Decimals
1. If the first digit dropped is 5 or greater, increase
the last retained digit by 1.
2. If the first digit dropped is less than 5, leave the
last retained digit unchanged.
3. In intermediate results, keep one more figure
than the number of figures required in the final
result.
See page 7 in your textbook for “common rounding
errors”
* Calculator Tip: Use the memory registers to save intermediate
results. This will eliminate rounding errors.
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Payroll. We only do gross salary
An employee’s remuneration may be based
on:
• a salary,
• hourly wage,
• a piecework rate or
• a rate of commission
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Salaries
• A salary is a fixed amount of pay per period (i.e.
$45,000 per year).
• We can convert from annual to another period:
Monthly
Semi-monthly
Bi-weekly
12 periods/year 24 periods/year 26 periods/year
Weekly
52 periods/year
An annual salary of $45,000 would become:
Monthly
Semi-monthly
Bi-weekly
Weekly
$45,000
12
$45,000
24
$45,000
26
$45,000
52
= $3,750
= $1,875
= $1,730.77
= $865.38
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Why does Bi-weekly have 26 payment periods while semimonthly only has 24?
Exhibit 3.6 Example of a Biweekly Pay Period Schedule
See example 1.3A 0n page 14
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• In jobs where the amount of work
varies, employees may be paid an
hourly wage.
Hourly
Wages
• A premium rate may apply when
more than a certain number of
hours per week are worked
(overtime) or when an employee
works on a statutory holiday.
• Overtime is normally paid at: 1.5x
the normal wage (called time and
a half)
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Hourly Wages
For example, consider an employee who
works 45 hours in one week (40 regular hours
and 5 overtime hours) and is paid a wage of
$12.00/hr and 1.5 times that for overtime.
• The gross wages for the week would be:
(40 × $12.00) + (5 × $18.00) = $570.00
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See problem 1 on page 17
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$58,800/52 weeks = $1,130.77 per week
How much is that per hour: 1,130.77/35 hours = $32.31 per
hour
Overtime pay is: 32.31 x 1.5 = $48.46
Gross salary for the week in which she worked 39 hours:
$1,130.77 + ($48.46 x 4 hours) = $1,324.62
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Skill Check
Ming is paid an annual salary of $60,500
based on a 44-hour workweek. What is his
gross pay for a biweekly pay period if he
works 46 hours in the first week and 48.5
hours in the second week? Overtime is paid at
time and a half.
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Skill Check: Solution
Ming is paid an annual salary of $60,500 based on a
44-hour workweek. What is his gross pay for a
biweekly pay period if he works 46 hours in the first
week and 48.5 hours in the second week? Overtime
is paid at time and a half.
60,500/26 = $2,326.92
Regular pay : 2,326.92/2 weeks = $1,163.46/44 hours
=$26.44 per hour
Overtime pay: $26.44 x 1.5 = $39.66 per hour
6.5 hours overtime x $39.66 = $257.79
Pay: $257.79 + $2,326.92 = $2,584.71
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Piecework pay
• Piecework pay is often paid in
manufacturing jobs. Employees receive
money for each item (i.e. piece) they
produce.
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Piecework pay
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See page 17
• Question 8 on page 17 is another good
example of piecework pay.
…Herb packs fish
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See page 17
• Question 8 on page 17 is another good
example of piecework pay.
Weight packed per day = 7.5(250)(0.500kg) = 937.5
kg.
Earnings per day = 7.5($8.25) + (937.5 – 500)($0.18)
= $140.63
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Commissions
• For sales positions, it is typical to base at
least a portion of the salesperson’s
remuneration on sales volume.
• Straight commission – earnings are based
entirely on a percentage of sales.
• Graduated commission – the commission
becomes higher with higher levels of sales.
• Salary plus commission – earnings include a
base salary plus a commission.
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Example 1.3D – Salary Plus Commission
James manages a men’s clothing store. His monthly remuneration has
three components: a $2500 base salary, plus 2% of the amount by
which the store’s total sales volume for the month exceeds $40,000,
plus 8% of the amount by which his personal sales exceed $4000.
Calculate his gross compensation for a month in which his sales
totalled $9900 and other staff had sales amounting to $109,260.
Solution:
Base salary
$2,500.00
Commission on total 0.02($109,260+$9,900-$40,000)
store’s volume:
Commission on personal
sales:
Total compensation
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0.08($9,900-$4,000)
1,583.20
472.00
$4,555.20
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$15,000 + 37,000 = 52,000
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Skill Check
Georgia sells cosmetics from her part-time
home-based business. She receives a straight
commission of 15% from her supplier. At the
year-end, she also receives a 4% bonus on
sales exceeding her annual quota of
$100,000. What will her gross annual
earnings be for a year in which her average
monthly sales are $10,000?
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Answer to: Georgia sells cosmetics
• Annual salary before bonus:
• $10,000 per month x 12 = $120,000
• Commission on $120,000 x 0.15 = $18,000
Bonus: $120,000 - $100,000 = $20,000 x 0.04 =
$800
Gross annual salary: $18,000 + $800 = $18,800
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Let’s just look at
• Question 4 on page
24.
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Question 4 on page 24:
answer
a) The average commission rate will be
$30,000 3%  $20,000 4%  $10,000 6%
$60,000
average commission rate will be:
$30,000(3%)+$20,000(4%)+$50,000(6%)
$100,000
=
3.83%
b. The
= 4.70%
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Simple and Weighted Averages
• Determining the average of a set of numbers is a
useful business calculation.
• A simple average is the sum of a set of values divided
by the number of values in the set.
• It is useful in cases where each item in the set has the
same importance. Example: class average on a test
• A weighted average attaches a weighting factor to
each value to represent its relative importance.
• This is useful in cases where each item in the set has a
different level of importance.
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Simple Averages
• To calculate a simple average, we take the sum of
a set of values and divide by the number of values
in the set.
𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠
𝑆𝑖𝑚𝑝𝑙𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
• For example, the simple average age of three
people ages 20, 23 and 24 would be:
20 + 23 + 24
= 22.33
3
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Let’s look at an
example
Question 6 on
page 24
15
7
4 nights x $158 = $632
2 nights x $199 = $398
1 night x $239 = $239
5 nights x $130 = $650
2 nights x $118 = $236
14 nights
$2,155
Solution to
Question 6 on
page 24
15
8
Look at
Just look at question 9 on top of page 25.
(No need to do this in class. I just wanted
students to see another example where
simple average is the appropriate method to
use)
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The simple average is not always the
best number to use. For example
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Weighted
Averages
A weighted
average includes
the relative
importance, or
“weight”, of each
item in the set.
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Weighted average
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Example 1.4B – Investment Portfolio
One year ago, Mrs. Boyd divided her savings among four mutual funds as follows:
20% was invested in a bond fund, 15% in a money market fund, 40% in a Canadian
equity fund, and 25% in a global equity fund. During the past year, the rates of return
on the individual mutual funds were 10%, 4%, −2%, and 15%, respectively. What was
the overall rate of return on her portfolio?
Solution:
Rate of
return(Value)
Fraction of
money
invested
(Weighting
factor)
(Weighting factor)
×(Value)
10%
0.20
2.0%
4%
0.15
0.6%
Canadian equity fund
−2%
0.40
−0.8%
Global equity fund
15%
0.25
3.75%
1.00
5.55%
Mutual fund
Bond fund
Money market fund
Total:
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Simple vs. Weighted?
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Let us do: Question 12 on page 25
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Let us do
• Question 12 on page 25
0.1514%  0.2010%  0.10 13%  0.3512%  0.2027%
1.00
= 12.40%
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Answer to question 12 on page 25
Each rate of return should be weighted by the fraction of the money
invested in the respective security. The rate of return on the portfolio is:
0.150.14 0.200.10 0.10 0.13 0.350.12 0.200.27
1.00
= 0.1240 x 100 = 12.40%
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Question 14 at the bottom of page 25
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Question 14 at the bottom of page 25
The weighted average price increase was
0.3010%  0.20 5%  0.5015%
1.00
= 9.50%
Stats Canada: How to think about
inflation
Personal Inflation calculator
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One more example to consider:
Question 17 on page 34
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Weighted average answer to
Question 17 on page 34
Security
Amount Invested ($)
Rate of
return (%)
Company U
shares
$5000
30% = 0.30 5,000/62,500
= 0.08
0.30x0.08
=0.0240
Province V
bonds
$20,000
-3% =
-0.03
20,000/62,5000
=0.32
- 0.03x0.32 =
- 0.0096
Company W
shares
$8,000
-15% = 0.15
8,000/62,500=
0.128
- 0.15x0.129 =
- 0.0194
Units in Funds X
$25,000
13% = 0.13 25,000/62,500=
0.40
0.13x0.40=
0.0520
Company Y
shares
$4,500
45% = 0.45 4,500/62,500 =
0.720
0.45x0.720=
0.03240
Total amount
invested
5,000 +
20,000+8,000+25,000
+4,500 = $62,500
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Amount
Invested as (%)
Weighted
average
0.08+0.32+0.128 0.0242+(+0.40+0.720 = 1 0.0096)+(0.0192)+0.052+
0.03240 = 7.96%
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Goods and Services Tax (GST);
Harmonized Sales Tax (HST)
The Goods and Services Tax (GST) is a federal
sales tax. The tax is collected by the seller
from the purchaser at the point of sale.
Consequently, a business collects the GST on
the prices of things it sells to customers, and
it pays the GST on the prices things it buys
from suppliers.
The GST rate is 5%.
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Goods and Services Tax (GST);
Harmonized Sales Tax (HST)
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Taxes Paid on Consumer Purchases
Sales Taxes and Rates in Canadian Provinces and Territories
EXAMPLE 1.5B-Calculating the PST,
GST, and HST
Calculate the total sales taxes on a $100 item
in
a. Manitoba
b. Quebec
c. Prince Edward Island
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Skill testing
Calculate the total amount, including both GST
and PST, that an individual will pay for a car priced
at $32,500 in
i.
Alberta
ii.
Saskatchewan
iii.
Quebec
Like question 3 on page 31
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Exclude:
• Exclude property Tax on page 29 onward
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Chapter 1
End of Chapter
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Microsoft® PowerPoint® Presentation by Julie Howse, St. Lawrence College.
Previous edition updated by Rob Sorenson, Camosun College.