McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-1 Algebr a Chapter 2.
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Transcript of McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-1 Algebr a Chapter 2.
2-2
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
After completing this chapter, you will be able to:
Learning ObjectivesLearning Objectives
LO 1.LO 1.
LO 2.LO 2.
LO 3.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
AlsoAlso
2-3
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Learning ObjectivesLearning Objectives
LO 4.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)
2-4
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Algebraic
Expression
Algebraic
Expression…indicates the mathematical operations to be carried out on a
combination of NUMBERS and VARIABLES
…indicates the mathematical operations to be carried out on a
combination of NUMBERS and VARIABLES
LO 1.LO 1.
Algebra
R & A
Algebra
R & A x(2x2 –3x – 1)Algebraic
Operations
Algebraic
Operations
2-5
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
TermsTerms…the components of an Algebraic Expression
that are separated by ADDITION or SUBTRACTION signs
…the components of an Algebraic Expression that are separated by
ADDITION or SUBTRACTION signs
x(2x2 –3x – 1)
Algebra
R & A
Algebra
R & A x(2x2 –3x – 1)Algebraic
Operations
Algebraic
Operations
2-6
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Terms
Terms
1 Term1 Term 2 Terms2 Terms 3 Terms3 Terms …any more than 1 Term!
…any more than 1 Term!
3x23x2 3x2 + xy3x2 + xy 3x2 + xy – 6y23x2 + xy – 6y2
MonomialMonomial BinomialBinomial TrinomialTrinomial PolynomialPolynomial
Algebra
R & A
Algebra
R & A x(2x2 –3x – 1)Algebraic
Operations
Algebraic
Operations
2-7
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
TermTerm…each one in an Expression consists of one or
more FACTORS separated by MULTIPLICATION or DIVISION sign
…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!
…assumed when two factors are written beside each other!
xy = x*yAlsoAlso
…assumed when one factor is written under an other!
…assumed when one factor is written under an other!
36x2y60xy2
Algebra
R & A
Algebra
R & A x(2x2 –3x – 1)Algebraic
Operations
Algebraic
Operations
2-8
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
TermTerm
Numerical Coefficient
Numerical Coefficient
Literal Coefficient
Literal Coefficient
FACTORFACTOR
3x23x2
3 x2
Algebra
R & A
Algebra
R & A x(2x2 –3x – 1)Algebraic
Operations
Algebraic
Operations
2-9
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Algebraic
Expression
Algebraic
Expression
MonomialMonomial BinomialBinomial TrinomialTrinomial PolynomialPolynomial
Numerical Coefficient
Numerical Coefficient
Literal Coefficient
Literal Coefficient
FACTORSFACTORS
Terms
Terms
Algebra
R & A
Algebra
R & A x(2x2 –3x – 1)Algebraic
Operations
Algebraic
Operations
2-10
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Division by a Monomial
Algebra
Step 1Step 1
Step 2Step 2
Identify Factors
in the numerator and denominator 36x2y
FACTORSFACTORS
3(12)(x)(x)(y)60xy2 5(12)(x)(y)(y)
Cancel Factors
in the numerator and denominator
=
= 3x5y
36 x2y60 xy2
Example
2-11
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Division by a Monomial
Algebra
Step 1Step 1
Step 2Step 2
Divide each TERM in the
numerator by the
denominatorCancel Factors
in the numerator and denominator
48a2/8a – 32ab/8a or6 4
= 48(a)(a) 32ab-8a 8a
= 6a – 4b
48a2 – 32ab8a
Example
2-12
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
What is this Expression
called?
What is this Expression
called?
Multiplying Polynomials
Algebra
Example -x(2x2 – 3x – 1)
Multiply each term in the TRINOMIAL by (–x)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
2-13
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
ThinkBEDMAS
ThinkBEDMAS
Algebra
Example Evaluate P(1 + rt)
SubstitutionSubstitution
P = $100 r = 0.09 t = 7/12
= $100 [1 + 0.09*(7/12)]
7 Months7 Months
= $100 [1.0525]
= $105.25 = $105.25
Use Calculator
.09 *7/12+1 =
* 100 =
105.25
2-14
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Algebra
Exponents Rule of
= 32*4
34 34
Base 3Exponent 4 3i.e. 3*3*3*3
Power = 81= 81
32 *33 32 *33
= 32 + 3
= 3 5
= 243= 243
(1 + i)20 (1 + i)20 (1 + i)8 (32)4(32)4
=(1+ i)20-8
= (1+ i)12= 3 8
= 6561= 6561
MoreMore
2-15
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Algebra
Exponents Rule of
X4
3x6y3 2
x2z3
Simplify inside the brackets first
Simplify inside the brackets first
= 3x4y3 2
z3
Square each factorSquare each factor
= 32x4*2y3* 2 Z3*2
SimplifySimplify
z69x8y6 =
3x6y3 2
x2z3
2-16
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Algebra
Use Calculator
Exponents Rule of
…to (a) evaluate (1.62)5
…to (b) evaluate (1.62)-5
1.6211.5611.56
1.62 5
0.0896 0.0896
5
2-17
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Solving Linear EquationsSolving Linear Equations
Algebra
LO 2.LO 2.
Algebra
R & A
Algebra
R & A
2-18
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Solving Linear Equations in one Unknown
Algebra
Equality in Equations
A + 9 137
Expressed as: A + 9 = 137 A = 137 – 9
A = 128
2-19
McGraw-Hill Ryerson ©
Algebra
R & A
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 – 0.0251 – 0.025 0.975x
Divide both sides by 0.975
x = 341.25 0.975
x = 350x = 350
2-20
McGraw-Hill Ryerson ©
Algebra
R & A
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?
2-21
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Algebra
How many cars did each sell?
Unknown(s) CarsBarbieKen
2C + C = 15 3C = 15
C = 5
Barbie = 2 C = 10 CarsBarbie = 2 C = 10 Cars
Ken = C = 5 CarsKen = C = 5 Cars
2 CC
Variable(s)
Barbie sold twice as many cars as Ken.
In April they sold 15 cars.
2-22
McGraw-Hill Ryerson ©
Algebra
R & A
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?
2-23
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Algebra
Tax Rate 38% on any additional earnings
Overtime 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 Earnings
If Canoe costs $2,750, then $2,750/ $21.86 = 125.8 Overtime Hours
2-24
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A Algebra
LO 3.LO 3.
Algebra
R & A
Algebra
R & A
2-25
McGraw-Hill Ryerson ©
Algebra
R & A
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
$
2-26
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
% change = Difference Base
This method is referred to as
the Base Method
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
2-27
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
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
2-28
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
% change = Difference Base
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.
2-29
McGraw-Hill Ryerson ©
Algebra
R & A
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
2-30
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
= $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
2-32
McGraw-Hill Ryerson ©
Algebra
R & A
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
2-33
McGraw-Hill Ryerson ©
Algebra
R & A
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
2-34
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
1000(1.04) (1.08)(.90)(1.09)
Alternative Solution
Solve for all 4 years at once!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
2-35
McGraw-Hill Ryerson ©
Algebra
R & A
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?
% change = Difference Base
= .07 - .09 .09
= -.02 .09
= -.2222 or 22.22% DecreaseIncrease in Interest Rate
2-36
McGraw-Hill Ryerson ©
Algebra
R & A
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?
% change = Difference Base
= .09 - .07 .07
= .02 .07
=.2857 or 28.57% Increase
2-37
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
… no voting rightsbut
… has preference over Common Stock in getting dividends!
… no voting rightsbut
… has preference over Common Stock in getting dividends!
Shareholders’ EquityCapital Stock
Preferred StockCommon Stock
X CompanyBalance Sheet
StocksStocks
… has voting rights
…gets dividends AFTER preference
shareholders!
… has voting rights
…gets dividends AFTER preference
shareholders!
Shares of Ownership of a company
Shares of Ownership of a company
2-38
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A StocksStocks
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
2-39
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
McGraw-Hill Ryerson©
A Stock Exchange … an orderly trading place for stocks!
A Stock Exchange … an orderly trading place for stocks!
Investors
Stock BrokersStock Brokers
…people who buy and sell stock on the floor
of the exchanges. They charge
a commission for trading stocks.
2-40
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
Investors
Stock Quotation CalculationsStock Quotation Calculations
Earnings 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 outstanding
Annual earnings= $100,000
1,000,000= $0.10
2-41
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
… 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 = $1,87530* $62.50 = $1,875
Orders in multiples of 100, 200, 400, etc.
2-42
McGraw-Hill Ryerson ©
Algebra
R & A
Algebra
R & A
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 Receipts
Total 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