Algebra 1 R A Algebra Chapter 2 Mc Graw-Hill

Algebra 1 R&A Algebra Chapter 2 Mc. Graw-Hill Ryerson ©

Algebra R&A 2 Learning Objectives After completing this chapter, you will be able to: LO 1. Simplify algebraic expressions LO 2. Solve linear equations in one variable and Solve "word problems" that lead to a linear equation in one unknown LO 3. Solve problems involving a single percent change Also Mc. Graw-Hill Ryerson ©

Algebra R&A LO 4. Learning Objectives 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) Mc. Graw-Hill Ryerson © 3

Algebra R&A LO 1. Algebraic x(2 x – 3 x – 1) 2 Operations Algebraic Expression …indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES Mc. Graw-Hill Ryerson © 4

Algebra R&A Algebraic x(2 x – 3 x – 1) 2 Operations Terms …the components of an Algebraic Expression that are separated by ADDITION or SUBTRACTION signs x(2 x 2 – 3 x – 1) Mc. Graw-Hill Ryerson © 5

Algebraic x(2 x – 3 x – 1) Algebra R&A 6 2 Operations Terms Monomial Binomial Trinomial Polynomial 1 Term 2 3 x Mc. Graw-Hill Ryerson © 2 Terms 2 3 Terms …any more than 1 Term! 3 x + xy 3 x 2 + xy – 6 y 2

Algebra R&A Algebraic x(2 x – 3 x – 1) 7 2 Operations 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*y Mc. Graw-Hill Ryerson © …assumed when one factor is written under an other! 36 x 2 y 60 xy 2 Also

Algebraic x(2 x – 3 x – 1) Algebra 8 2 R&A Operations Term FACTOR Numerical Coefficient 3 Mc. Graw-Hill Ryerson © 2 3 x Literal Coefficient 2 x

Algebra R&A Algebraic x(2 x – 3 x – 1) 9 2 Operations Algebraic Expression Terms Monomial Binomial Trinomial Polynomial FACTORS Numerical Literal Coefficient Mc. Graw-Hill Ryerson ©

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

Algebra 11 Algebra R&A Division by a Monomial 48 a 2 – 32 ab Example 8 a Divide each TERM Step in the numerator by 1 the denominator Cancel Factors Step in the numerator and 2 denominator Mc. Graw-Hill Ryerson © 48 a 2/8 a – 32 ab/8 a or 6 4 = 48(a)(a) - 32 ab 8 a 8 a = 6 a – 4 b

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

Algebra 13 Algebra R&A Substitution Example P = $100 Evaluate P(1 + rt) r = 0. 09 t = 7/12 7 Months = $100 [1 + 0. 09*(7/12)] = $100[1. 0525] = $105. 25 Mc. Graw-Hill Ryerson © Think 105. 25 Use Calculator BEDMAS . 09 * 7/12 +1 = * 100 =

Algebra 14 Algebra R&A Exponents 34 Base 32 *33 3 = 32 + 3 5 =3 Exponent 3 4 i. e. 3*3*3*3 Power = 81 Mc. Graw-Hill Ryerson © = 243 Rule of (1 + i)20 (1 + i)8 =(1+ = (1+ i)20 -8 i) 12 (32)4 = 32*4 = 38 = 6561 More

Algebra 15 Algebra R&A Exponents 3 x 6 y 3 x 2 z 3 3 x 6 y 3 2 3 x 4 y 3 2 z 3 = x z 3 Mc. Graw-Hill Ryerson © 2 Square each factor Simplify inside the brackets first X 4 Rule of 2 = 2 x 4*2 y 3* 3 3*2 Z 2 Simplify = 9 x 8 y 6 z 6

Algebra 16 Algebra R&A Exponents Rule of …to (a) evaluate (1. 62)5 Use Calculator 1. 62 5 11. 56 …to (b) evaluate (1. 62)-5 1. 62 5 0. 0896 Mc. Graw-Hill Ryerson ©

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

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

Algebra 19 Algebra R&A Solving Linear Equations in one Unknown Solve for x from the following: x = 341. 25 + 0. 025 x Collect like Terms x = 341. 25 + 0. 025 x x - 0. 025 x = 341. 25 1 – 0. 025 0. 975 x = 341. 25 Divide both sides by 0. 975 Mc. Graw-Hill Ryerson © x = 341. 25 0. 975 x = 350

Algebra 20 Algebra R&A for the Unknown Barbie and Ken sell cars at the Auto World. Barbie sold twice as many cars as Ken. In April they sold 15 cars. How many cars did each sell? Mc. Graw-Hill Ryerson ©

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

Algebra 22 Algebra R&A 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? Mc. Graw-Hill Ryerson ©

Algebra 23 Algebra R&A 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 Mc. Graw-Hill Ryerson ©

Algebra 24 Algebra R&A LO 3. Mc. Graw-Hill Ryerson ©

Algebra 25 Algebra R&A Monday’s Sales were $1000 and grow to $2500 the next day. Find the percent change. $ % (/100) Initial Value 1000 1 250 Final Value 2500 2. 5 Change 1500 150 1. 5 100% x Cross - multiply = 1000 2500 X = 2500*100% / 1000 = 250% …Also Mc. Graw-Hill Ryerson ©

Algebra 26 Algebra R&A Monday’s Sales were $1000 and grow to $2500 the next day. Find the percent change. This method is referred to as the Base Method % change = Difference Base Mc. Graw-Hill Ryerson © Initial(Base)Value $ 1000 Final Value 2500 Difference $ 1500 % change = $ 1000 = 1. 5 or 150% Increase

Algebra 27 Algebra R&A In the making of dried fruit, 15 kg. of fruit shrinks to 3 kg. Find the percent change. Kg. Initial Value Final Value Difference x 100% Mc. Graw-Hill Ryerson © = 3 15 3 12 % 100 20 80 (/100) 1. 2. 8 Cross - multiply 15 X = 3*100% / 15 = 20%

Algebra 28 Algebra R&A In the making of dried fruit, 15 kg. of fruit shrinks to 3 kg. Find the percent change. Initial(Base)Value Final Value 15 3 % change = Difference Base 12 % change = 12 15 =. 8 or 80% Decrease Mc. Graw-Hill Ryerson ©

Algebra 29 Algebra R&A 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 term of the contract? This can be solved in a single step! = $16(1 + 3%)(1 + 2%)(1 + 1%) Mc. Graw-Hill Ryerson © Solve

Algebra 30 Algebra R&A = $16(1 + 3%)(1 + 2%)(1 + 1%) 1 st Year 2 nd Year 3 rd Year Mc. Graw-Hill Ryerson © = $16(1 +. 03)(1 +. 02)(1 +. 01) = $16(1. 03)(1. 02)(1. 01) = $16. 48(1. 02)(1. 01) = $16. 8096(1. 01) = $16. 977 or $16. 98

Algebra 31 Algebra R&A LO 4. Investment Mc. Graw-Hill Ryerson ©

Algebra 32 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? Solve Mc. Graw-Hill Ryerson ©

Algebra 33 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? Year 1 Year 2 Year 3 Year 4 $1000 $1040 $1000 * (1 +. 04) $1040 * (1 +. 08) $1123. 20 * (1 -. 10) $1010. 88 * (1 +. 09) = $1101. 86 = $1040 = $1123. 20 = $1010. 88 Includes Interest earned during the period……………. Mc. Graw-Hill Ryerson © …Alternative

Algebra 34 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? Alternative Solution Solve for all 4 years at once! 1000(1. 04) (1. 08)(. 90)(1. 09) = $1101. 86 1 -10% Mc. Graw-Hill Ryerson ©

Algebra 35 Algebra R&A 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 = -. 02. 09 - =. 2222 or 22. 22% Decrease Increase in Interest Rate Mc. Graw-Hill Ryerson ©

Algebra 36 Algebra R&A 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 =. 02. 07 =. 2857 or 28. 57% Increase Mc. Graw-Hill Ryerson ©

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

Algebra R&A Stocks X Company Income Statement Gross Profit Operating Expenses Net Profit Dividends - Payments to shareholders from Net Profits Dividends in arrears - Payments owed to *Cumulative Preferred shareholders … are entitled to a specific amount of dividends each year Mc. Graw-Hill Ryerson © 38

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

Algebra Stock Quotation Calculations 40 R&A k Stoc ield? Y k Stoc d Yiel PE Ratio Earnings per share Mc. Graw-Hill Ryerson © PE Ratio? Earnings per share ? Investors Annual dividend per share Closing price per share = $. 10 =. 3% $41. 125 Closing price per share Annual earnings per share = $41. 125 $1. 11 Annual earnings Total # of shares outstanding = = 37 $100, 000 = $0. 10 1, 000

Algebra 41 R&A Orders in multiples of 100, 200, 400, etc. … 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 = Mc. Graw-Hill Ryerson © 125 $6, 375 130 * $62. 50 = $8, 125. 00. 02 * $8, 125 = 162. 50. 01 * $1, 875 = 18. 75 $8, 306. 25 30* $62. 50 = $1, 875

Algebra 42 R&A 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? Bought 100 shares at $52. 25 = $5, 225. 00 52. 25 Commission at 1% = $5, 277. 25 Total Costs Net Receipts Total Costs Net Gain Dividends + Total Gain Mc. Graw-Hill Ryerson © Sold 100 shares at $68 Commission at 1% Net Receipts $6, 732. 00 - 5, 277. 25 $1, 454. 75 100. 00 (100 * $. 10) $1, 554. 75 = $6, 800 = - 68 $6, 732 $1, 554. 75 $5, 277. 25 29. 46% ROI