BUSINESS MATHEMATICS

&

STATISTICS

Module 2Exponents and Radicals

Linear Equations (Lectures 7)Investments (Lectures 8)

Matrices (Lecture 9)Ratios & Proportions and Index Numbers (Lecture 10)

LECTURE 7Review 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

AnnuityValue = 4,000

Down payment = 1,000Rest in 20 installments of 200

Sequence of payments at equal interval of timeTime = Payment Interval

NOTATIONSR = Amount of annuity

N = Number of paymentsI = Interest rater per conversion period

S = Accumulated valueA = Discounted or present worth of an annuity

ACCUMULATED VALUES = 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 = 18AF = ((1 + 0.0425)^18-1)

= 26.24R = 10,000

Accumulated value = 10,000x 26.24= 260,240

DISCOUNTED VALUEValue of all payments at the beginning of term of annuity

= Payment x Discount Factor (DF) DF = ((1-1/(1+i)^n)/i)

= ((1-1/(1+0.045)^8)/0.045)= 6.595

ACCUMULATED VALUE= 2,000 x ((1-1/(1+0.055)^8)/0.055)

= 2,000 x11.95

=23,900.77

Algebraic ExpressionAlgebraic 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

x(2x2 –3x – 1)Algebraic

Operations

AlgebraicOperations

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)

x(2x2 –3x – 1)Algebraic

Operations

AlgebraicOperations

TermsTerms

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

x(2x2 –3x – 1)Algebraic

Operations

AlgebraicOperations

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*y

AlsoAlso

…assumed when one factor is written under an

other!

…assumed when one factor is written under an

other!

36x2y

60xy2

x(2x2 –3x – 1)Algebraic

Operations

AlgebraicOperations

TermTerm

Numerical Coefficient

Numerical Coefficient

Literal Coefficient

Literal Coefficient

FACTORFACTOR

3x23x2

3 x2

x(2x2 –3x – 1)Algebraic

Operations

AlgebraicOperations

Algebraic ExpressionAlgebraic Expression

MonomialMonomial BinomialBinomial TrinomialTrinomial PolynomialPolynomial

Numerical Coefficient

Numerical Coefficient

Literal Coefficient

Literal Coefficient

FACTORSFACTORS

TermsTerms

x(2x2 –3x – 1)Algebraic

Operations

AlgebraicOperations

Division by a Monomial

Step 1

Step 1

Step 2

Step 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

=

=3x

5y

36 x2y60 xy2Example

Division by a Monomial

Step 1Step 1

Step 2Step 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 – 32ab8aExample

What is this Expression

called?

What is this Expression

called?

Multiplying Polynomials

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

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

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

Solving Linear Equations in one Unknown

Equality in Equations

A + 9 137

Expressed as: A + 9 = 137 A = 137 – 9

A = 128

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

BUSINESS MATHEMATICS

&

STATISTICS

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?

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.

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.

7800 5200 : 6500:

Colleen Heather Mark Expressed In colon

notation format

Expressed In colon

notation format

Equivalent ratio (each term divided by 100)

78 52 65: :Equivalent ratio with lowest terms Divide 52

into each oneDivide 52

into each one

1.5 : 1 : 1.25 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

$1800Y

=34 Cross - multiply

4Y = 1800 * 3

Y = 1800 * 34

Divide both sides of the equation by 4

= $1350= $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 =

86232 RN

R N

560RN = 232*86

560RN = 19952

RN = 19952 / 560

Hire 35.63 or 36

RN’s

Hire 35.63 or 36

RN’s

560 =185

86

SS

560SS = 185*86

560SS = 15910

SS = 15910 / 560

Hire 28.41 or 29

SS

Hire 28.41 or 29

SS

SS

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.LO 2. & 3.

A punch recipe calls for mango juice, ginger ale and orange juice in the ratio

of 3:2:1.

3+2+1 = 6

Total Shares

2 litres / 6 = 333 ml per share

M J G A O

* 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?3+2+1 = 6Total Shares

1 1.14Punch6 = Cross - multiply

Punch = 6 * 1.14 litres = 6.84 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 J G A

1.5 =3 M J

0.5

500 ml500 ml

Cross - multiply

Mango Juice = 3 * 0.5 /1.5

= 1 litre= 1 litre

21.5

= G A0.5

Ginger Ale = 2 * 0.5 /1.5

= .667 litre = 667 ml.= .667 litre = 667 ml.

Cross - multiply