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Transcript of Math Prsntation (Autosaved)
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Mathematics is used in most aspects of daily life. Many ofthe top jobs such as business consultants, computer
consultants, company directors and a host of others
require a solid understanding of basic mathematics.
It also play important role in business, like Business
mathematics by commercial enterprises to record and
manage business operations. Mathematics typically used
in commerce includes elementary arithmetic, such as
fractions, decimals, and percentages, elementary algebra,
statistics and probability.
Business management can be made more effective in
some cases by use of more advanced mathematics such
as calculus, matrix algebra and linear programming.
Commercial organizations use mathematics in accounting,
inventory management, marketing, sales forecasting, and
financial analysis.
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LOGARITHM
Advanced tools to optimization and
planning in economic system.
Logarithms are simply the powers or the
indices to a given base. It is the clever
method of reducing long multiplicationsinto much simpler additions and
reduction divisions into subtraction
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The earliest records of mathematics
show it arising in response to particular
needs in business and industries. It is the
applications of advanced tools to
optimization and planning in.
Logarithm plays an important role of
business mathematics and decision
analysis. So, now describing logarithm as
a tool in business planning and decision
making.
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Practical implementation in
business decisional works
Calculating due date, interest, and maturity
value of simple interest when the loan is for
monthly, daily or yearly for a business.
Computing compound interest and resulting
compound amount at the maturity date for
saving and investment
Calculating salary, commission and annuity
Calculating the outstanding amount for
installment purchase
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Business firm must calculate its
depreciation to know the actualfinancial condition of that particular
firm by using logarithm.
Budgetary planning is the most
important planning. Without knowing
cost of salary, machinery cost,depreciation cost budget cant be
prepared
Calculating the amount of sinking fund,
pension, annuity, retain earning,
compound interest, accumulation are
easier and accurate with logarithm.
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MATHEMATICAL APPLICATION
An example is given below to show it that
how important logarithm is and what role itplays in proper planning and decision
making:-
A limited company at which intends to create a
depreciation fund to replace at the end of the
25th
year assets costing taka 100000. The
amount to be retained out of profits every year
if the interest rate is 3% is to be calculated.
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Solution :-
Here, m= a/I {(1+i)^n -1)}
m=100000, i =0.03, n =25
Where,
100000= a/0.03{(1+0.03)^25 -1)}
=a/0.03{(1.03)^25-1}
=a/0.03(20.089-1)
A=100000*0.03/1.089
=27755 (Ans)
X=(1.03)^25
log x=25 log (1.03)
=25*0.01288
=0.3200
X =antilog (0.3200)
X=2.089
Hence, the amount to be retained is Tk.27755.
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PERMUTATION
In mathematics, the notion of permutation is
used with several slightly different meanings,
all related to the act of permuting
(rearranging) objects or values.
A permutation of a set of objects is an
arrangement of those objects into a
particular order.
The number of permutations of n distinct
objects is n (n 1) (n 2)...21, which
number is called "n factorial" and written
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Permutation Formula
A formula for the number of possible permutations of
r objects from a set of n. This is usually written nPr
Formula: nPr = n!/(n-r)
p=number of permutation
n=number of items
r= number of items in each permutation
MATHEMATICAL IMPLEMENTATION
EXAMPLE:
Indicate how many 4 digit numbers greater than 7000
can be from the digits 3, 5,7,8,9
Solution: If the digits are to be greater than 7000
then the first digit can be only one of the 7, 8 and 9.
Now, the first digit can be chosen in 3 ways and theremaining three can be any of the 4 digits left, which
can be chosen in 4p3 ways
Therefore the total number of ways
=3 x 4P3 = 3 x 4 x3 x2 = 72 ways.
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APPLICABILITY IN ECONOMICAL WORKS
The Allocation of Country Telephone Codes
The allocation of country telephone code must be in specific
order. The emphasis on this allocation is in the word order.
For this reason, this implies the principle of permutation in
mathematics.
The Allocation of City/Area Codes
Telephone area codes of all countries in the world are usually
assigned by the countrys Ministry of Telecommunications.
Usually, these digits are 3, selected from the decimal number
system, 0.9.
In applied statistics
The use of permutation methods in applied statistics is
becoming increasingly widespread. Permutation methods
are employed for three reasons:
1. They provide exact significance levels rather than
approximation
2. Their significance levels are distribution free.
3. They yield more powerful statistics.
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COMBINATION
In mathematics combinations is a way of
selecting several things out of a large
group, where order does not matter.
However, repetition in each combination
is not allowed except otherwise stated.
Thus, the combination of 3 objects taken
all at a time.
FORMULA ANALYSIS
The mathematical formula for finding out
different combinations requires a slightanalysis than the formula used for
permutation.
ncr = n!/ {(n-r)! r! }
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MATHEMATICAL IMPLICATION
EXAMPLE:
If a manager of a shop of a readymade
garments wants to display 4 combinations
out of the total 6 colors of ladies suits
received in his store he can display in thefollowing ways:
Solution:
According to the rule,
6c4 = 6!/ {(6-4)!*4!}
=(6*5*4*3*2*1)/ {(2*1) (4*3*2*1)}
=15
Therefore, the manager can display his products in total 15ways.
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Mathematics for Finance
Use of mathematical formulae or models in
business finance or financial decision is
Mathematics for finance.
The financial decision making is based on
the core concepts of interest ,present value,
annuities, creation of sinking funds etc
SIMPLE INTEREST:
The amount earned only on the original initial
principal amount invested is called simple interest.
Formulae: Interest I=Principal/Amount ,
A = P(1+rn)
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Implementation on business:
A company wants to buy a land.If the person borrowed Tk.4000000 at
10% interest for 4 years, find the simple interest after 4 years.
Solution:
Given that, P=4000000
r=10%
n=4
Using the formula of interest, (I)=Pxrxn
=4000000x(10/100)x4
=TK.1600000
Compound interest:
Interest earned on both the initial principal and the
interest re-invested from the prior periods is
compound interest.
Formula: Interest (I) = P{(1+r)^n-1}
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Implementation on business:
A company invested TK. 50000 for 4 years at 12 % rate
of interest. What will be the compound rate of interest
after 4 years if the interest is paid yearly.
Solution:
Given that, P=50000, r=12%, n=4
Interest (I)= 500000{(1+r)^n-1}
=Tk.28676
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SINGKING FUND
Its a fund or account into which a person or company
deposits money on a regular basis in order to repay
some debt or liability of future.
A corporation might establish a sinking fund in order to
accumulate sufficient capital to replace obsolete
equipment.
For long term corporate bodies each years sinking fund
budget needs to allocate funding in accordance with
long term plan.
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Formula modification
Money deposited annually in order to accumulate by its
termination time, the capital amount needs to cover the
anticipated expenditure. The formula is as follows:
A=F*r/[(1+r)n-1]*1
Here,
A= Annual amount.
F=Future value of the sinking fund.
r= Interest rate earned each period.
n= Number of periods over the annual amount A is
deposited.
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Mathematicalimplementation
Example:
Lets consider that a companys major capital expenditure of
TK.100000 is anticipated in 10 years. What amount the company
needs to deposit in the next 10 years to fund such expenditure if
annual interest rate can be earned 5%?
Solution:
We estimate this by using the formula where
F=100000,
r=5% and
n=10
Putting the values according to the formula:
A=100000*0.05/[(1+0.05)10-1]
=TK.7950
Hence, the amount needed to be deposited annually by the
com an is TK.7950.
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Applicability of sinking fund
The opportunity costs of the sinking fund requirement
may be the inability to secure long term debt.
The accumulation of funds in a special account also
provides the issuer with security against future
business conditions that may be detrimental to its
ability.
The insurance provide by a sinking fund decreases
the interest rates, therefore the expenses.
The issuer can book capital gains on debt
retirement if it purchases bonds in the openmarket below book value.
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LINEAR EQUATION
The practical use of linear equation is in evolving certain
relations and finding out the value of the unknown.
The concept was first invented by the Russian
Mathematician L.V. Kantorovich and developed later by
George B. Dantzig
A system of simultaneous equations is helpful for
finding unique values for unknowns, where the
equations should be equal to the number of unknowns.
IMPLEMENTATION OF LINEAR EQUATION
IN BUSINESS DECISION
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The main theme of theory is to optimize the use of
scarce resources which include machine, man-
power, ware-house, raw- materials etc.
There are several theoretical tools to accomplish
this purpose but those are not adequate for
treating a complex economic problem.
For tackling such problems that are used for linear
programming has been found to be most useful.
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SYSTEM OF LINEAR EQUATION
IN-CONSISTENT CONSISTENT
LINEAR
EQUATION
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GRAPHICAL SOLUTION
There can be graphical solutions as well as
algebraic solutions of equations.
The former courses of graphical solution were
not prcised but easy to use in some cases.
The equations are consistent but dependent.
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QUADRATIC EQUATION
An equation having the highest index as 2 is called the
quadratic equation. For example:
x2+5x+6=0
The general form of quadratic equation is ax2+bx=0.But
another form of this equation is {-b(b2-4ac)}/2a
It is an equation which when reduced to the rational
integral form contains the square of the unknown
quantity.
MATHEMATICAL ANALYSIS
Demand for goods of an industry is given by the
equation pq=100, where p is the price and q is its
quantity, supply is given by the equation 20+3p.We
need to count the equilibrium price and quantity
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Solution:
The demand equation is pq=100 and----(1)
supply equation is 20+3p=q------------(2)
Substituting the value of q from (2) in (1), we get
P(20+3p)=100
3p2+20p-100=0
P = -20(400+1200)
Hence, the equilibrium price (p) =10/3 ,
Quantity exchanged (q) =30 (ans)
QUADRATIC EQUATION IN BUSINESS PLANNING AND
DECISION MAKING
It indicates the relationship between quantity of
goods and costs.
Indicates the relationship between order and
spending
Indicates relation between investment and income.
Sometimes complicated verbal statement when
translated in equation or inequalities can be solved
with great ease.
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Set theory
Applicability of set theory in making surveys
regarding decision making
A SET IS A COLLECTION OF WELL-DEFINED AND
WELL DISTINGUISHED OBJECTS.THE SETS IS OF MUCH WIDER APPLICATION,
ESPECIALLY THEY HELP IN PREPARING THE
PROGRAMME FOR FEEDING IN TO THE MACHINE.
Applying set theory
In sets we deal with a group of objects which can
be defined in terms of their distinctive feature,
magnitudes etc.
It is possible to tell whether a given object
belongs to a set or not.
The basic feature of a set is that it is well defined
and well-distinguished for easy recognition.
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In the view of the previous mentioned topics, each of the
respective parts discussed played a vital role in dealing with
business decision. Their impacts are mentioned viz:-
The compound interest, rate of interest, salary,
commission, depreciation etc are being calculated easily
with the help of log.
Permutation helps in allocating utility expenses as well
calculating all relates expenses.
Combination is the process of maintaining a selectiveorder while a company has a choice among a huge
amount of choices.
Decisional works are incomplete without the effect of
financial mathematics. It mainly decides the best type of
investment field.
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Another part of the financial math is the sinking fund
which declares itself the safest way of investment and
as a decision.
The role of equations in decisional works is
indescribable. The linear equation commits to
evolving certain relations and finding out the value ofthe unknown.
The quadratic equation plays a vital role in calculating
the relation between the demand, supply and
quantity of product of a company.
In sets we deal with a group of objects and find out
relation between elements and within the givenconditions.
In the perspective of the above mentioned topics
decisions regarding the success of the business we can
consider them and their prescribed rules needs to be
followed thoroughly.