Business Mathematics Jerome Chapter 02

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McGraw-Hill Ryerson ©

Algebra

R & A

Algebra

Chapter 2

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Algebra

R & A

After completing this chapter, you will be able to:

Learning Objectives

LO 1.

LO 2.

LO 3.

Simplify algebraic expressions

Solve linear equations in one variableand

Solve "word problems" that lead to a linear equation in one unknown

Solve problems involving a single percent change

Also

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Algebra

R & A Learning Objectives

LO 4. Calculate returns from investments in terms of both dollars and percentages

Solve problems involving a series of compounding percent changes and Determine the overall effect of compounding a series of percent changes or rates of

return on investment(ROI)

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Algebra

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Algebraic Expression…indicates the mathematical operations to be

carried out on a combination of NUMBERS and VARIABLES

LO 1.

Algebra

R & A x(2x2 –3x – 1)Algebraic

Operations

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Algebra

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Terms…the components of an Algebraic Expression

that are separated by ADDITION or SUBTRACTION signs

x(2x2 –3x – 1)

Algebra


Operations

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Algebra

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Terms

1 Term 2 Terms 3 Terms …any more than 1 Term!

3x2 3x2 + xy 3x2 + xy – 6y2

Monomial Binomial Trinomial Polynomial

Algebra


Operations

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Algebra

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Term…each one in an Expression consists of one or

more FACTORS separated by MULTIPLICATION or DIVISION sign

…assumed when two factors are written beside each other!

xy = x*yAlso

…assumed when one factor is written under an other!

36x2y60xy2

Algebra


Operations

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Algebra

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Term

Numerical Coefficient

Literal Coefficient

FACTOR

3x2

3 x2

Algebra


Operations

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Algebra

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Algebraic Expression

Monomial Binomial Trinomial Polynomial

Numerical Coefficient

Literal Coefficient

FACTORS

Terms

Algebra


Operations

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Algebra

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Division by a Monomial

Algebra

Step 1

Step 2

Identify Factors in the numerator

and denominator 36x2yFACTORS

3(12)(x)(x)(y)60xy2 5(12)(x)(y)(y)

Cancel Factors in the numerator

and denominator

=

= 3x5y

36 x2y60 xy2

Example

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Algebra

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Division by a Monomial

Algebra

Step 1

Step 2

Divide each TERM in the

numerator by the denominator

Cancel Factors in the numerator

and denominator

48a2/8a – 32ab/8a or6 4

= 48(a)(a) 32ab-8a 8a

= 6a – 4b

48a2 – 32ab8a

Example

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Algebra

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What is this Expression

called?

Multiplying Polynomials

Algebra

Example -x(2x2 – 3x – 1)

Multiply each term in the TRINOMIAL by (–x)++= )(-x ( )2x2 )(-x )(-3x )(-x )(-1

= -2x3 + 3x2 +The product of two negative quantities is positive.

x

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Algebra

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ThinkBEDMAS

Algebra

Example Evaluate P(1 + rt)

SubstitutionSubstitution

P = $100 r = 0.09 t = 7/12

= $100 [1 + 0.09*(7/12)]

7 Months

= $100 [1.0525]= $105.25

Use Calculator

.09 *7/12+1 =

* 100 =

105.25

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Algebra

R & A Algebra

Exponents Rule of

= 32*4

34

Base 3Exponent 4 3i.e. 3*3*3*3

Power = 81

32 *33

= 32 + 3

= 3 5

= 243

(1 + i)20 (1 + i)8 (32)4

=(1+ i)20-8

= (1+ i)12= 3 8

= 6561

More

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Algebra

R & A Algebra

Exponents Rule of

X4

3x6y3 2

x2z3

Simplify inside the brackets first

= 3x4y3 2

z3

Square each factor

= 32x4*2y3* 2 Z3*2

Simplify

z69x8y6 =

3x6y3 2

x2z3

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Algebra

R & A Algebra

Use Calculator

Exponents Rule of

…to (a) evaluate (1.62)5

…to (b) evaluate (1.62)-5

1.6211.56

1.62 5 0.0896

5

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Algebra

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Solving Linear Equations

Algebra

LO 2.

Algebra

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Algebra

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Solving Linear Equations in one Unknown

Algebra

Equality in Equations

A + 9 137

Expressed as: A + 9 = 137 A = 137 – 9

A = 128

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Algebra

R & A Algebra

Solving Linear Equations in one Unknown

Solve for x from the following: x = 341.25 + 0.025x

Collect like Termsx - 0.025x = 341.25

0.975x = 341.25

x = 341.25 + 0.025x

1 – 0.025 0.975x

Divide both sides by 0.975

x = 341.25 0.975 x = 350

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Algebra

R & A Algebra

for the UnknownBarbie and Ken sell cars at the Auto World.

In April they sold 15 cars.

Barbie sold twice as many cars as Ken.

How many cars did each sell?

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Algebra

R & A Algebra

How many cars did each sell?Unknown(s) Cars

BarbieKen

2C + C = 15 3C = 15

C = 5

Barbie = 2 C = 10 Cars

Ken = C = 5 Cars

2 CC

Variable(s)

Barbie sold twice as many cars as Ken.

In April they sold 15 cars.

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Algebra

R & A Algebra

You pay 38% income tax on any additional earnings. You have an opportunity to work overtime

@ 1.5 times your base wage of $23.50 per hour.

Rounded to the nearest quarter-hour, how much overtime must you work to earn enough money(after tax) to buy a canoe that

costs $2,750, including sales tax?

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Algebra

R & A Algebra

Tax Rate 38% on any additional earningsOvertime Rate =$23.50 * 1.5 times = $35.25

Total Earnings Less 38% of Total Earnings = After Tax Income 1 - .38 = .62

Overtime Rate =$35.25 * .62 = $21.86 Net EarningsIf Canoe costs $2,750, then $2,750/ $21.86 = 125.8 Overtime Hours

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Algebra

R & A Algebra

LO 3.

Algebra

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Algebra

R & A Algebra

Monday’s Sales were $1000 and grow to $2500 the next day. Find the percent change.

Initial Value 1000Final Value 2500

%100

(/100)1

250 2.5 Change 1500 150 1.5

100%25001000 = x

X = 2500*100% / 1000 = 250%

Cross - multiply

…Also

$

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Algebra

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% change = Difference Base

This method is referred to as

the Base Method

$ 1000

Initial(Base)Value $ 1000 Final Value 2500

Difference $ 1500

% change = $ 1500

= 1.5 or 150% Increase

Monday’s Sales were $1000 and grow to $2500 the next day. Find the percent change.

Algebra

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Algebra

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In the making of dried fruit, 15kg. of fruit shrinks to 3 kg. Find the percent change.

Initial Value 15Final Value 3

%100

(/100)1

20 .2 Difference 12 80 .8

100% 15=

X = 3*100% / 15 = 20%

Cross - multiply x

Kg.

Algebra

3

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Algebra

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15

Initial(Base)Value 15 Final Value 3

Difference 12

% change = 12

= .8 or 80% Decrease

Algebra

In the making of dried fruit, 15kg. of fruit shrinks to 3 kg. Find the percent change.

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Algebra

R & A Algebra

A union signed a three year collective agreement that provided for wage increases of 3%, 2%, and 1% in successive years.An employee is currently earning

$16 per hour. What will the wage per hour be at the end of

the of the term of the contract?This can be solved in a single step!

= $16(1 + 3%)(1 + 2%)(1 + 1%)

Solve

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Algebra

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= $16(1 + .03)(1 + .02)(1 + .01)

Algebra

= $16(1 + 3%)(1 + 2%)(1 + 1%)

= $16(1.03)(1.02)(1.01)= $16.48(1.02)(1.01)= $16.8096(1.01)= $16.977 or $16.98

1st Year

2nd Year

3rd Year

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Algebra

R & A Algebra

LO 4.

Investment

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Algebra

R & A Algebra

You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10%

and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the

end of the last year?

Solve

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Algebra

R & A

You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

Algebra

$1000

Year 1 Year 2 Year 3 Year 4

$1040 $1123.20 $1010.88$1000 * (1 + .04)

= $1040

$1040 * (1 + .08)

= $1123.20 = $1010.88 = $1101.86

$1123.20 *(1 - .10)

$1010.88 *(1 +.09)

Includes Interest earned during the period……………………….…Alternative

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Algebra

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1000(1.04) (1.08)(.90)(1.09)

Alternative Solution

Solve for all 4 years at once!

Algebra

1 -10%

You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

= $1101.86

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Algebra

R & A Algebra

A chartered bank is lowering the interest rate on its loans from 9% to 7%.

What will be the percent decrease in the interest rate on a given balance?


= .07 - .09 .09

= -.02 .09

= -.2222 or 22.22% DecreaseIncrease in Interest Rate

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Algebra

R & A Algebra

A chartered bank is increasing the interest rate on its loans from 7% to 9%

What will be the percent increase in the interest rate on a given balance?


= .09 - .07 .07

= .02 .07

=.2857 or 28.57% Increase

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Algebra

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… no voting rightsbut

… has preference over Common Stock in getting dividends!

Shareholders’ EquityCapital Stock

Preferred StockCommon Stock

X CompanyBalance Sheet

Stocks

… has voting rights

…gets dividends AFTER preference

shareholders!

Shares of Ownership of a company

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Algebra

R & A Stocks

Gross ProfitOperating Expenses

Net Profit

X CompanyIncome Statement

Dividends - Payments to shareholders from Net Profits

Dividends in arrears - Payments owed to *Cumulative Preferred shareholders

*Cumulative Preferred shareholders … are entitled to a specific amount of dividends each year

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Algebra

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McGraw-Hill Ryerson©

A Stock Exchange … an orderly trading place for stocks!

Investors

Stock Brokers

…people who buy and sell stock on the floor

of the exchanges. They charge

a commission for trading stocks.

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Algebra

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Investors

Stock Quotation CalculationsEarnings per share?

PE Ratio?Stock

Yield?

Closing price per shareAnnual dividend per share = $.10

$41.125 = .3%

PE Ratio

Stock

Yield

Closing price per share Annual earnings per share = $41.125

$1.11 = 37

Earnings per share

Total # of shares outstandingAnnual earnings = $100,000

1,000,000= $0.10

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Algebra

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… fewer than 100 shares. (…there is an additional commission fee of 1%!)If you buy 100(or 130) shares at $62.50 per share with a 2% commission, calculate your total cost.

100 * $62.50 = $6,250

.02 * $6,250 = 125 $6,375

130 * $62.50 = $8,125.00.02 * $8,125 = 162.50.01 * $1,875 = 18.75

$8,306.2530* $62.50 = $1,875

Orders in multiples of 100, 200, 400, etc.

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Algebra

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Bought100 shares at $52.25 = Commission at 1% = Total Costs

Suppose you bought 100 shares at $52.25 and sold them 1 year later at $68. With a 1% commission rate

buying and selling the stock and a current $.10 dividend per share in effect, what was your return on investment?

Sold100 shares at $68 =Commission at 1% =Net Receipts

Net ReceiptsTotal Costs Net GainDividendsTotal Gain

$1,554.75$5,277.25

29.46% ROI

$5,225.0052.25

$5,277.25

$6,800- 68

$6,732

$6,732.00- 5,277.25$1,454.75

+ 100.00 (100 * $.10)$1,554.75

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Algebra

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This completes Chapter 2

Education

Business Mathematics Jerome Chapter 02