1. BUSINESS MATHEMATICS
&
STATISTICS

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

3. 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

4. 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

5. 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

6. 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

7. ACCUMULATION FACTOR (AF)
= 26.24
R = 10,000
Accumulated value = 10,000x 26.24
= 260,240

8. 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

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

10. Algebraic Operations x(2x2 –3x – 1) Algebraic Expression
…indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES

11. Algebraic Operations x(2x2 –3x – 1) x(2x2 –3x – 1) Terms
…the components of an Algebraic Expression that are separated
by ADDITION or SUBTRACTION signs
x(2x2 –3x – 1)

12. Algebraic Operations Monomial Polynomial Binomial Trinomial
x(2x2 –3x – 1)
Algebraic
Operations
Terms
Monomial
Polynomial
Binomial
Trinomial
…any more than 1 Term!
1 Term
2 Terms
3 Terms
3x2
3x2 + xy
3x2 + xy – 6y2

13. Algebraic Operations x(2x2 –3x – 1) 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!
…assumed when one factor is written under an other!
xy = x*y
36x2y
60xy2
Also

14. Numerical Coefficient
x(2x2 –3x – 1)
Algebraic
Operations
Term
FACTOR
Numerical Coefficient
Literal Coefficient
3x2 x2 3

15. Numerical Coefficient
x(2x2 –3x – 1)
Algebraic
Operations
Algebraic Expression
Terms
Monomial
Binomial
Trinomial
Polynomial
FACTORS
Numerical Coefficient
Literal Coefficient

16. Division by a Monomial 36 x2y 60 xy2 Example
Identify Factors in the numerator and denominator
Step 1
FACTORS
36x2y
3(12)(x)(x)(y) =
60xy2
5(12)(x)(y)(y)
Step 2
Cancel Factors in the numerator and denominator = 3x 5y

17. = - Division by a Monomial = 6a – 4b 48a2/8a – 32ab/8a or 6 4 48(a)(a)
Example
48a2/8a – 32ab/8a or
Divide each TERM in the numerator by the denominator
Step 1 6 4 =
48(a)(a)
32ab - 8a 8a
Step 2
Cancel Factors in the numerator and denominator
= 6a
– 4b

18. What is this Expression called?
Multiplying Polynomials
What is this Expression called?
Example
-x(2x2 – 3x – 1)
Multiply each term in the TRINOMIAL by (–x) = ) ( -x ( )
2x2 + ) ( -x ) (
-3x + ) ( -x ) ( -1
The product of two negative quantities is positive. =
-2x3 +
3x2 + x

19. 4 8 12 34 32 *33 (32)4 3 = 32 + 3 = 32*4 3 = 3 5 = 3 Exponents Rule of
(1 + i)20
(1 + i)8 34
32 *33
(32)4
Base 3 =
= 32*4
=(1+ i)20-8
Exponent 3 4
= 3 5
= 3 8
i.e. 3*3*3*3
= (1+ i) 12
Power
= 81
= 243
= 6561

20. Exponents Rule of
3x6y3 2
x2z3
Simplify inside the brackets first
Step 1
Square each factor
Step 2
Simplify
Step 3 X4
3x6y3 2
x2z3 =
3x4y3 2 z3
32x4*2y3* 2
9x8y6 = = z6
Z3*2

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

22. x = 350 Step Step Solving Linear Equations in one Unknown
Solve for x from the following: x = x
Collect like Terms
Step
x = x
x x =
1 – 0.025
0.975x
0.975x =
Divide both sides by 0.975
Step
x =
0.975
x = 350

23. BUSINESS MATHEMATICS
&
STATISTICS

24. for the Unknown Solving Word Problems Data
Barbie and Ken sell cars at the Auto World.
Barbie sold twice as many cars as Ken.
Data
In April they sold 15 cars.
How many cars did each sell?

25. Review and Applications
Algebra
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
Variable(s)
2C + C = 15
3C = 15
Barbie = 2 C = 10 Cars
Ken = C = 5 Cars
C = 5

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

27. Expressed In colon notation format
Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of $7800, $5200 and $6500 respectively.
Colleen
Heather
Mark
Expressed In colon notation format
7800 :
5200 :
6500
Equivalent ratio (each term divided by 100) : : 78 52 65
Equivalent ratio with lowest terms
Divide 52 into each one
: :

28. Expressing a Ratio in Equivalent Forms Q
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?

29. Expressing a Ratio in Equivalent Forms Q
A 560 bed hospital operates with registered nurses and 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?

30. Divide both sides of the equation by 4
Expressing
a Ratio in
Equivalent Forms
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 4 =
Cross - multiply Y 3
Divide both sides of the equation by 4
4Y = 1800 * 3
Y = 1800 * 3
= $1350 4

31. S Expressing a Ratio in Equivalent Forms
A 560 bed hospital operates with registered nurses and 185 other support staff The hospital is about to open a new 86-bed wing.
560 : 232 : 185 = 86 : RN : SS S
R N
560 86
560 86
Hire or RN’s =
Hire or SS =
185
232 RN SS
560RN = 232*86
560SS = 185*86
560RN = 19952
560SS = 15910
RN = / 560
SS = / 560

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

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

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

35. Q
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

36. = 1 litre = .667 litre = 667 ml. M J = = 500 ml M J G A 3 2 G A
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
500 ml
M J
G A 3
M J 2
G A = =
Cross - multiply
Cross - multiply
1.5
0.5
1.5
0.5
Mango Juice = 3 * 0.5 /1.5
Ginger Ale = 2 * 0.5 /1.5
= 1 litre
= .667 litre = 667 ml.