Transcript BusinessMathematicsStatistics

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 = ((1 + 0.0425)^18-1)
= 26.24
R = 10,000
Accumulated value = 10,000x 26.24
= 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/(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
x(2x2 –3x – 1)
Operations
Algebraic Expression
…indicates the mathematical operations to be carried out on a
combination of NUMBERS and VARIABLES
Algebraic
x(2x2 –3x – 1)
Operations
Terms
…the components of an Algebraic Expression that are separated
by ADDITION or SUBTRACTION signs
x(2x2 –3x – 1)
Algebraic
x(2x2 –3x – 1)
Operations
Terms
Monomial
Binomial
1 Term
2 Terms
3x2
3x2 + xy
Trinomial
3 Terms
3x2 + xy – 6y2
Polynomial
…any more
than 1 Term!
Algebraic
x(2x2 –3x – 1)
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
…assumed when one
factor is written under an
other!
36x2y
60xy2
Also
Algebraic
x(2x2 –3x – 1)
Operations
Term
FACTOR
Numerical
Coefficient
Literal
Coefficient
3x2
3
x2
Algebraic
x(2x2 –3x – 1)
Operations
Algebraic Expression
Terms
Monomial
Binomial
Trinomial
Polynomial
FACTORS
Numerical
Coefficient
Literal
Coefficient
Division by a Monomial
Example
Step
1
Step
2
Identify Factors
in the numerator and
denominator
36 x2y
60 xy2
FACTORS
36x2y
60xy2
Cancel Factors
in the numerator and
denominator
=
3(12)(x)(x)(y)
5(12)(x)(y)(y)
3x
=
5y
Division by a Monomial
Example
Divide each TERM
Step in the numerator by the
denominator
1
Step
2
Cancel Factors
in the numerator and
denominator
48a2 – 32ab
8a
48a2/8a – 32ab/8a or
6
4
= 48(a)(a) - 32ab
8a
8a
= 6a – 4b
Multiplying Polynomials
Example
-x(2x2
What is this
Expression
called?
– 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
Exponents Rule of
Base
34
32 *33
3
= 32 + 3
5
=3
Exponent
3
4
i.e. 3*3*3*3
Power = 81
= 243
(1 + i)20
(1 + i)8
=(1+
= (1+
i)20-8
i)
12
(32)4
= 32*4
= 38
= 6561
Exponents Rule of
3x6y3
x2z3
Square each factor
Simplify inside the
brackets first
X4
3x6y3 2 3x4y3
=
2
3
xz
z3
2
2
=
4*2
3*
2
2
3x y
3*2
Z
Simplify
=
6
8
9x y
z
6
Solving Linear Equations
in one Unknown
Equality in Equations
A+9
Expressed as:
137
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 Terms x = 341.25 + 0.025x
x - 0.025x = 341.25
1 – 0.025 0.975x = 341.25
Divide both
sides by 0.975
x = 341.25
0.975
x = 350
BUSINESS MATHEMATICS
&
STATISTICS
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?
Algebra
Barbie sold twice as many cars as Ken.
In April they sold 15 cars.
How many cars did each sell?
Unknown(s) Cars
Variable(s)
2C
C
Barbie
Ken
2C + C = 15
3C = 15
C = 5
Barbie = 2 C = 10 Cars
Ken
=
C = 5 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?
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
7800
Heather
:
5200
Mark
: 6500
Equivalent ratio (each term divided by 100)
78
:
52
:
:
1
:
notation
format
65
Equivalent ratio with lowest terms
1.5
Expressed
In colon
1.25
Divide 52
into each one
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
Cross - multiply
$1800 = 4
Y
3
Divide both sides of
4Y = 1800 * 3
the equation by 4
Y = 1800 * 3
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
R
= 86 : RN : SS
N
560
86
232 = RN
560RN = 232*86
560RN = 19952
RN = 19952 / 560
Hire
35.63
or
36 RN’s
560
185 =
86
SS
560SS = 185*86
560SS = 15910
SS = 15910 / 560
S
Hire
28.41
or
29 SS
LO 2. & 3.
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?
A punch recipe calls for mango juice,
ginger ale and orange juice in the ratio
of 3:2:1.
M J
GA
O
Total Shares 3+2+1 = 6
333 ml per share
2 litres / 6
= 333 ml per share
* 3 * 2 * 1
= 1 litre
= 667 mls
= 333 mls
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?
Total Shares 3+2+1 = 6
1
1.14
Cross - multiply
=
Punch
6
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 J
3
MJ
= 0.5
1.5
500 ml
Cross - multiply
Mango Juice = 3 * 0.5
/1.5
= 1 litre
GA
2
GA
=
1.5
0.5
Ginger Ale = 2 * 0.5 /1.5
= .667 litre = 667 ml.