BUSINESS MATHEMATICS & STATISTICS

Module 2 Exponents and Radicals Linear Equations (Lectures 7) Investments (Lectures 8) Matrices (Lecture 9) Ratios & Proportions and Index Numbers (Lecture 10)

LECTURE 7 Review of lecture 6 Exponents and radicals Simplify algebraic expressions Solve linear equations in one variable Rearrange formulas to solve for any of its contained variables

Annuity Value = 4,000 Down payment = 1,000 Rest in 20 installments of 200 Sequence of payments at equal interval of time Time = Payment Interval

NOTATIONS R = Amount of annuity N = Number of payments I = Interest rater per conversion period S = Accumulated value A = Discounted or present worth of an annuity

ACCUMULATED VALUE S = r ((1+i)^n – 1)/i A = r ((1- 1/(1+i)^n)/i) Accumulated value= Payment x Accumulation factor Discounted value= Payment x Discount factor

ACCUMULATION FACTOR (AF) i = 4.25 % n = 18 AF = (( )^18-1) = R = 10,000 Accumulated value = 10,000x = 260,240

DISCOUNTED VALUE Value of all payments at the beginning of term of annuity = Payment x Discount Factor (DF) DF = ((1-1/(1+i)^n)/i) = ((1-1/( )^8)/0.045) = 6.595

ACCUMULATED VALUE = 2,000 x ((1-1/( )^8)/0.055) = 2,000 x11.95 =23,900.77

Algebraic Expression …indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES x(2x 2 –3x – 1) Algebraic Operations Algebraic Operations

Terms …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(2x 2 –3x – 1) Algebraic Operations Algebraic Operations

Terms 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 x(2x 2 –3x – 1) Algebraic Operations Algebraic 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 Also …assumed when one factor is written under an other! 36x 2 y 60xy 2 x(2x 2 –3x – 1) Algebraic Operations Algebraic Operations

Term Numerical Coefficient Literal Coefficient FACTOR 3x 2 3 x2x2 x(2x 2 –3x – 1) Algebraic Operations Algebraic Operations

Algebraic Expression Monomial Binomial Trinomial Polynomial Numerical Coefficient Literal Coefficient FACTORS Terms x(2x 2 –3x – 1) Algebraic Operations Algebraic Operations

Division by a Monomial 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

Division by a Monomial 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

What is this Expression called? Multiplying Polynomials 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

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

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

Solving Linear Equations in one Unknown Equality in Equations A Expressed as: A + 9 = 137 A = 137 – 9 A = 128

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

BUSINESS MATHEMATICS & STATISTICS

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?

Algebra How many cars did each sell? Unknown(s) Cars Barbie Ken 2C + C = 15 3C = 15 C = 5 Barbie = 2 C = 10 Cars Barbie = 2 C = 10 Cars Ken = C = 5 Cars Ken = C = 5 Cars 2 C C Variable(s) Barbie sold twice as many cars as Ken. In April they sold 15 cars.

Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of $7800, $5200 and $6500 respectively. What is the ratio of Colleen’s to Heather’s to Marks’s partnership interest? What is the ratio of Colleen’s to Heather’s to Marks’s partnership interest?

Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of $7800, $5200 and $6500 respectively : 6500 : ColleenHeatherMark Expressed In colon notation format Equivalent ratio (each term divided by 100) :: Equivalent ratio with lowest terms Divide 52 into each one 1.5 : 1 : 1.25

The ratio of the sales of Product X to the sales of Product Y is 4:3. The sales of product X in the next month are forecast to be $1800. What will be the sales of product Y if the sales of the two products maintain the same ratio?

A 560 bed hospital operates with 232 registered nurses and 185 other support staff. The hospital is about to open a new 86-bed wing. Assuming comparable staffing levels, how many more nurses and support staff will need to be hired?

The ratio of the sales of Product X to the sales of Product Y is 4:3. The sales of product X in the next month are forecast to be $1800. Since X : Y = 4 : 3, then $1800 : Y = 4 : 3 $1800 Y = 3 4 Cross - multiply 4Y = 1800 * 3 Y = 1800 * 3 4 Divide both sides of the equation by 4 = $1350

A 560 bed hospital operates with 232 registered nurses and 185 other support staff. The hospital is about to open a new 86-bed wing. 560 : 232 : 185 = 86 : RN : SS 560 = RN 560RN = 232*86 560RN = RN = / 560 Hire or 36 RN’s 560 = SS 560SS = 185*86 560SS = SS = / 560 Hire or 29 SS S S

A punch recipe calls for fruit juice, ginger ale and vodka in the ratio of 3:2:1. If you are looking to make 2 litres of punch for a party, how much of each ingredient is needed? LO 2. & 3.

A punch recipe calls for mango juice, ginger ale and orange juice in the ratio of 3:2: = 6 Total Shares 2 litres / 6 = 333 ml per share M JG AO * 3 * 2 * 1 = 1 litre = 667 mls = 333 mls 333 ml per share

A punch recipe calls for mango juice, ginger ale and orange juice in the ratio of 3:2:1. If you have 1.14 litres of orange juice, how much punch can you make? = 6 Total Shares Punch 6 = Cross - multiply Punch = 6 * 1.14 litres = 6.84 litres

You check the frige and determine that someone has been drinking the orange juice. You have less than half a bottle, about 500 ml. How much fruit juice and ginger ale do you use if you want to make more punch using the following new punch recipe? Mango juice: ginger ale: orange juice = 3 : 2 : 1.5

How much fruit juice and ginger ale do you use if you want to make more punch using the following new punch recipe?: Mango juice: ginger ale: Orange juice = 3 : 2 : 1.5 M JG A 1.5 = 3 M J ml Cross - multiply Mango Juice = 3 * 0.5 /1.5 = 1 litre = G A 0.5 Ginger Ale = 2 * 0.5 /1.5 =.667 litre = 667 ml. Cross - multiply