McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-1 Algebr a Chapter 2

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-2 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 variable and Solve "word problems" that lead to a linear equation in one unknown Solve problems involving a single percent change Also

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-3 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 )

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-4 A lgebraic E xpression …indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES LO 1. Algebra R & A Algebra R & A x(2x 2 –3x – 1) A lgebraic O perations A lgebraic O perations

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-5 T erms …the components of an Algebraic Expression that are separated by ADDITION or SUBTRACTION signs x(2x 2 –3x – 1) Algebra R & A Algebra R & A x(2x 2 –3x – 1) A lgebraic O perations A lgebraic O perations

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-6 T erms 1 Term 2 Terms 3 Terms …any more than 1 Term! 3x 2 3x 2 + xy 3x 2 + xy – 6y 2 Monomial Binomial Trinomial Polynomial Algebra R & A Algebra R & A x(2x 2 –3x – 1) A lgebraic O perations A lgebraic O perations

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-7 T erm …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*y Also …assumed when one factor is written under an other! 36x 2 y 60xy 2 Algebra R & A Algebra R & A x(2x 2 –3x – 1) A lgebraic O perations A lgebraic O perations

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-8 T erm Numerical Coefficient Literal Coefficient FACTOR 3x 2 3x2x2 Algebra R & A Algebra R & A x(2x 2 –3x – 1) A lgebraic O perations A lgebraic O perations

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-9 A lgebraic E xpression Monomial Binomial Trinomial Polynomial Numerical Coefficient Literal Coefficient FACTORS T erms Algebra R & A Algebra R & A x(2x 2 –3x – 1) A lgebraic O perations A lgebraic O perations

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-10 Division by a Monomial Algebra Step 1 Step 2 Identify Factors in the numerator and denominator 36x 2 y FACTORS 3(12)(x)(x)(y) 60xy 2 5(12)(x)(y)(y) Cancel Factors in the numerator and denominator = = 3x 5y 36 x 2 y 60 xy 2 Example

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-11 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 48a 2 /8a – 32ab/8a or 64 = 48(a)(a)32ab - 8a = 6a– 4b 48a 2 – 32ab 8a Example

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-12 What is this Expression called? Multiplying Polynomials Algebra Example -x(2x 2 – 3x – 1) Multiply each term in the TRINOMIAL by (– x ) ++= )( -x-x () 2x22x2 )( -x-x )( -3x )( -x-x )( = -2x 3 +3x 2 + The product of two negative quantities is positive. x

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-13 Think BEDMAS Think BEDMAS Algebra Example Evaluate P(1 + rt) Substitution P = $100r = 0.09t = 7/12 = $100[ *(7/12)] 7 Months = $100[1.0525] = $ Use Calculator.09 * 7/12 +1 = * 100 =

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-14 Algebra Exponents Rule of = 3 2*4 3 4 Base 3 Exponent 4 3 i.e. 3*3*3*3 Power = *3 3 = = 3 5 = 243 (1 + i) 20 (1 + i) 8 (3 2 ) 4 = (1+ i) = (1+ i) 12 = 3 8 = 6561 More

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-15 Algebra Exponents Rule of X4X4 3x 6 y 3 2 x 2 z 3 Simplify inside the brackets first = 3x 4 y 3 2 z 3 Square each factor = 3 2 x 4*2 y 3* 2 Z 3*2 Simplify z6 z6 9x8y69x8y6 = 3x 6 y 3 2 x 2 z 3

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-16 Algebra Use Calculator Exponents Rule of …to (a) evaluate (1.62) 5 …to (b) evaluate (1.62)

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-17 Solving Linear Equations Algebra LO 2. Algebra R & A Algebra R & A

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-18 Solving Linear Equations in one Unknown Algebra Equality in Equations A Expressed as: A + 9 = 137 A = 137 – 9 A = 128

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-19 Algebra Solving Linear Equations in one Unknown Solve for x from the following : x = x Collect like Terms x x = x = x = x 1 – x Divide both sides by x = x = 350

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-20 Algebra for the Unknown Barbie 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?

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-21 Algebra How many cars did each sell? Unknown(s) Cars Barbie Ken 2C + C = 15 3C = 15 C = 5 Barbie = 2 C = 10 Cars Ken = C = 5 Cars 2 C C Variable(s) Barbie sold twice as many cars as Ken. In April they sold 15 cars.

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-22 Algebra You pay 38% income tax on any additional earnings. You have an opportunity to work 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?

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-23 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 =.62 Overtime Rate =$35.25 *.62 = $21.86 Net Earnings If Canoe costs $2,750, then $2,750/ $21.86 = Overtime Hours

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-24 Algebra LO 3. Algebra R & A Algebra R & A

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-25 Algebra Monday’s Sales were $1000 and grow to $2500 the next day. Find the percent change. Initial Value1000 Final Value2500 % 100 ( / 100) Change % = x X = 2500*100% / 1000 = 250% Cross - multiply …Also $

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-26 % 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

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-27 In the making of dried fruit, 15kg. of fruit shrinks to 3 kg. Find the percent change. Initial Value 15 Final Value 3 % 100 ( / 100) Difference %15 = X = 3*100% / 15 = 20% Cross - multiply x Kg. Algebra 3

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-28 % 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.

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-29 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

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-30 = $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) = $ (1.01) = $ or $ st Year 2 nd Year 3 rd Year

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-31 Algebra LO 4. Investment

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-32 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

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-33 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 1Year 2Year 3Year 4 $1040$ $ $1000 * (1 +.04) = $1040 $1040 * (1 +.08) = $ = $ = $ $ * (1 -.10) $ * (1 +.09) Includes Interest earned during the period………………………. …Alternative

McGraw-Hill Ryerson © Algebra R & A Algebra R & A (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? = $

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-35 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 = = = or 22.22% Decrease Increase in Interest Rate

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-36 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 = = =.2857 or 28.57% Increase

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-37 … no voting rights but … has preference over Common Stock in getting dividends! … no voting rights but … has preference over Common Stock in getting dividends! Shareholders’ Equity Capital Stock Preferred Stock Common Stock X Company Balance Sheet Stocks … has voting rights …gets dividends AFTER preference shareholders! … has voting rights …gets dividends AFTER preference shareholders! Shares of Ownership of a company

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-38 Stocks Gross Profit Operating Expenses Net Profit X Company Income 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

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-39 McGraw-Hill Ryerson© A Stock Exchange … an orderly trading place for stocks! A Stock Exchange … an orderly trading place for stocks! Investors …people who buy and sell stock on the floor of the exchanges. They charge a commission for trading stocks.

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-40 Investors Stock Quotation Calculations Earnings per share? PE Ratio? Stock Yield? Closing price per share Annual dividend per share = $.10 $ =.3% PE Ratio Stock Yield Closing price per share Annual earnings per share = $ $1.11 = 37 Earnings per share Total # of shares outstanding Annual earnings = $100,000 1,000,000 = $0.10

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-41 … 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, * $6,250 = 125 $6, * $62.50 = $8, * $8,125 = * $1,875 = $8, * $62.50 = $1,875 Orders in multiples of 100, 200, 400, etc.

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-42 Bought 100 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? Sold 100 shares at $68 = Commission at 1% = Net Receipts Total Costs Net Gain Dividends Total Gain $1, $5, % ROI $5, $5, $6, $6,732 $6, , $1, (100 * $.10) $1,554.75

McGraw-Hill Ryerson © Algebra R & A Algebra R & A 2-43 This completes Chapter 2