The document provides an overview of topics in business mathematics including linear equations, cost-output functions, matrices, logarithmic and exponential functions, and their applications in business and real life. Examples of applications given include using linear equations to model costs, revenues, and profits; break-even analysis; modeling population growth with exponential functions; and using derivatives and integrals to solve problems in physics, engineering, and economics. The document aims to demonstrate how mathematics is relevant across many domains including business, science, and daily life.

1. BUSINESS MATHS
BUSINESS AND REAL LIFE EXAMPLES
By
Ahmad Badar

2. CONTENT
 •Linear Equation and Function, short review of the lines
 •Systems of linear equations and inequalities, Linear function applications
 •Cost-output function, Break even analysis and Linear demand function
 •Matrices and determinate
 •Quick review of logarithmic and exponential functions
 •The need for the Logarithms in business applications
 •Simple interest and future value
 •Simple discount and present value
 •Ordinary annuities (Future value, Sinking funds and Amortization)
 •Average and Instantaneous rates of change: The Derivative Formulae
 •Application of Differential Calculus
 •Maxima and minima of functions
 •Anti-Derivatives: the indefinite integral
 •Area and definite integral
 •Consumer's and Producer's surplus

3. Linear Equation and Function, Short
Review Of Line
In Real Life:
 Variable Costs
 Rates
 Budgeting
 Making Predictions
 In real life, the two variables of the equations are husband and wife in a
family. The relationship between the two variables gives a straight line in
linear equations. Likewise, both husband and wife in the family must shed
their differences and should go in the same direction to run the family
smoothly.

4. Linear Equation and Function, Short
Review Of Line
In Business:
 Model cost,
 Revenue,
 Profit,
 Supply
 Demand using linear functions.

5. In Real Life:
Speed limit
Legal speed on the highway ≤ 65 miles per hour
Credit card
Monthly payment ≥ 10% of your balance in that billing
cycle
Text messaging
Allowable number of text messages per month ≤ 250
Travel time
Time needed to walk from home to school ≥ 18 minutes
Time Zone
Canada’s time is 9:00 ≤ PST

6. linear equations and inequalities,
linear functions
In Business:
 Business Graphs
 Cost and Revenue
 Profit Makeup
 Economics
 Investment

7. Cost Output Function, Breakeven
Analysis and Linear Demand Function
In Real Life and in Business:
 Price Decreasing
 Price Increasing
 Variable Costs
 Profit
 Safety margin
 Price
 Production

8. Matrices and Determinants
 In Real Life:
 Computer Based Applications
 Physics
 Medical
 Automation
 Information Technology

9. In Business
 Economics
 Linear Programing
 Commerce
 Business Decisions
 Forecasts

10. Applications of Exponential Functions
in Real Life And Business Life.
 Population
 Exponential Decay
 Compound Interest

11. APPLICATION OF SIMPLE INTEREST
IN REAL LIFE AND BUSINESS LIFE.
 Car Loans
 Certificates of Deposits
 Consumer (and Other) Loans
 Discounts on Early Payments

12. Application of Logarithm in Real Life
And Business Life.
 Safety Index
 pH value
 Binary Search
 Order of magnitude
 Growth rates

13. Real Life And Business Applications
Of Simple Interest.
 Trade Discount
 The amount of discount that the wholesaler or retailer receives off the list
price or the difference between the list price and the net price.
 Cash Discount
 An incentive that a seller offers to a buyer in return for paying bill owed
before the scheduled due date.

14. Methods of calculating Cash
Discount
 Ordinary Dating Method
Credit terms 2/10, n/30. This means you will get a 2 percent if you pay
within 10 days of receiving the invoice. You must pay the bill within 30
days or you start to incur interest charges.
 End Of The Month Method
Credit terms 2/10, n/30E.O.M .This means you will get a 2 percent if you pay
within 10 days of receiving the invoice. You must pay the bill within the
first 30 days of the next month or you start to incur interest charges.
 Receipt Of Good Dating Method
Credits terms is 2/10 R.O.G . This means you get a 2% discount if you pay
within 10 days after the goods are received.

15. Average Rate of Change
 The easiest example for the average rate of change is speed. Speed is
simply distance covered by a body in a particular amount of time. The
formula for speed is:
 Speed = Distance Covered/Total Time Taken

16. Let’s say a body movies in a straight line from a point A
to a point B. Point A is 100 Kilometers away and Point B
is 150 Kilometers away, then the distance that the body
has covered comes out to be 150 – 100 = 50 km. Suppose,
the body was at Point A at 04:30 PM and reached Point B
at 08:30 PM. The time taken by the body to cover the
distance comes out to be 4 hours. Hence, if we want to
calculate the average rate of change of distance with
respect to time, then it simply comes out to be 50/4 i.e.
12.5 km per hour.
Another very good example of average rate of change is
when you find the slope of a line. The slope of a line is
nothing but the change in Y coordinates with respect to
the change is the X coordinates. If a line crosses two
points with coordinates (x1, y1) = (2, 2) and (x2, y2) = (3,
0), then the slope of the line is (y2-y1)/(x2-x1) = (0-2) /
(3-2) = -2/1. Hence, the average rate of change in the Y
coordinates with respect to the X coordinates a.k.a. the
slope of the line is -2.

17. Practical Applications of the Average
Rate of Change
 As described before, average rate of change is used to measure the speed of
an object undergoing motion. An advanced level of the formula is used in
various astronomical equations pertaining to rocket science and space travel.
 The equation is used to measure the rate of a chemical reaction in chemistry.
This application of average rate of change is very useful in Chemical
Engineering Calculations.
 The formula is used in various behavioral calculations relating to
management and the human psyche.
 A good application in real life can be found in predicting your electricity bill.
If you have an idea of the average rate in which electricity is consumed in
your household, then you can predict your electricity bill for a month. These
fundamentals are used in various devices that help you monitor your
electricity bill.

18. Instantaneous Velocity
 Suppose we drop a tomato from the top of a 100 foot building and time
its fall.
 How long did it take for the tomato to drop 100 feet? (2.5 seconds)
 How far did the tomato fall during the first second? (100 – 84 = 16 feet)
 How far did the tomato fall during the last second? (64 – 0 = 64 feet)

19. Maxima and Minima of functions
 A real-valued function f defined on a domain X has
a global (or absolute) maximum point at x∗ if f(x∗) ≥ f(x) for all x in X.
Similarly, the function has a global (or absolute) minimum
point at x∗ if f(x∗) ≤ f(x) for all x in X. The value of the function at a
maximum point is called the maximum value of the function and the
value of the function at a minimum point is called the minimum value of
the function.

20. Applications in real life and business
of Maxima and Minima of functions

 Maxima and minima pop up all over the place in our daily lives. They can be
found anywhere we are interested in the highest and/or lowest value of
a given system; if you look hard enough, you can probably find them just
about anywhere! Here are just a few examples of where you
 A meteorologist creates a model that predicts temperature variance with
respect to time. The absolute maximum and minimum of this function over
any 24-hour period are the forecasted high and low temperatures, as
later reported on The Weather Channel or the evening news.
 The director of a theme park works with a model of total revenue as a
function of admission price. The location of the absolute maximum of this
function represents the ideal admission price.

21. An actuary works with functions that represent the probability of various negative events occurring. The
local minima of these functions correspond to lucrative markets for his/her insurance company – low-risk,
high-reward ventures.
It is used in Chemistry. We have used the maxima of wave function and radial probability distribution
functions to determine where an electron is most likely to be found in any given orbital.
A NASA engineer working on the next generation space shuttle studies a function that computes the
pressure acting on the shuttle at a given altitude. The absolute maximum of this function represents the
pressure that the shuttle must be designed to sustain.

22. Anti-derivatives: the Indefinite Integral
 Antiderivatives are related to definite integrals through the fundamental
theorem of calculus: the definite integral of a function over an interval is
equal to the difference between the values of an antiderivative evaluated
at the endpoints of the interval.

23. Applications in real life and business
in Anti-derivative: Indefinite Integral
 To determine the rate of a chemical reaction and to determine some
necessary information of Radioactive decay reaction.
 Electric Charges have a force between them that varies depending on the
amount of charge and the distance between the charges. We use
integration to calculate the work done when charges are separated.

24. To calculate the center of mass, center of gravity and mass moment of inertia of vehicles, satellites, a
tower and basically every other building or structure which you can imagine.
To calculate the velocity and trajectory of a satellite while placing it an orbit like at exactly which point you
need to give how much thrust to get the desired trajectory.
To study of the spread of infectious disease — relies heavily on calculus. It can be used to determine how
far and fast a disease is spreading, where it may have originated from and how to best treat it.

25. Area and Definite Integral
 The area of a flat, or plane figure is the number of unit squares that can
be contained within it. The unit square is usually some standard unit, like
a square meter, a square foot, or a square inch.

26. Applications in real life and business in
Area and Definite
 Parking lots are sized depending on the area of the associated building.
 Building lobbies are designed to provide enough area for the people who
will be passing through, and lingering.
 Airports are designed to provide enough area for the planes to get to the
gates without bumping in to each other.
 Ports and docks are designed so ships can be tied up without bumping
into each other.

27. Consumers and Producers Surplus
 When supply and demand are equal the economy is said to be at
equilibrium. At this point (point (P⋆,Q⋆)), the allocation of goods is most
efficient because the amount of goods being supplied is exactly the same
as the amount of goods being demanded. At the equilibrium price
suppliers are selling all the goods that they have produced and
consumers are getting all the goods that they are demanding.

28. Applications in real life and business
in Consumers and Producers Surplus
 Coffee is a good example of a product because it is essentially the same across all
producers. However, depending on where it is sold, the price of a cup of coffee can
vary widely. Starbucks can charge more than McDonald’s for a cup of coffee because
coffee drinkers have strong preferences regarding where they buy their coffee drinks
and what they believe is a reasonable price for a cup of coffee. The difference
between the lowest available price for a cup of coffee and the highest price is the
producer surplus.
 The sky is dumping freezing rain, it’s 11:30 at night, and you’re standing on a street
corner 15 miles from home. You open your Uber app and see that the trip is going to
cost you roughly Rs.50, as prices are surging 200 percent above normal. But you
summon the car anyways. In fact, you might not have hesitated to go as high as
Rs.100, just to get dry in that comprising moment. Economists would call that Rs.50
difference—between what you pay and what you’re willing to pay—a “consumer
surplus.” They see it as a hard representation of Uber’s value and utility to you, as if
you’ve pocketed Rs.50 in benefits on an otherwise wet and miserable day.

29. Many people appreciate Mercedes automobiles for their styling, mechanical performance and status
appeal. Those who favor owning a Mercedes will pay what they must to have the car of their choice.
The manufacturer of Mercedes automobiles tries to produce the exact number of high-end, mid-price
and low-end autos to meet consumer demand for cars at those price levels. Consumers who can't
afford a new Mercedes at the lowest price level or who want the top of the line but can't pay the price
can always buy their favored Mercedes in the used car market. So, the price of a Mercedes, whether
new or used, is set by the price the buyer will pay relative to the price the seller will charge. This sets
up a situation where buyers and sellers often negotiate final transaction prices, which are based on
the seller's supply of cars in relation to the buyer's desire to buy or demand for the product.

30. CONCLUSION
 Whether you aspire to study sociology, psychology, physics, biology or
even economics, math is held in high regard, and you will be called
on solve various maths problems, as part of your work.
 Many students and adults never think to use the maths they have learnt, or
are still learning in their everyday lives. In this, they are wrong for many
reasons!
 First of all, as we will see, mathematics is present in many aspects of your
daily life, from a trip to the bank, to cooking and even doing DIY.

31. HENCE PROVED
Math is everywhere

32. THANKS FOR LISTENING