To find the total sales for the two years, we add the matrices: A + B = [30 40 42] [15 40 36] [23 50 15] MBA (Evening) Page 17 Application of Matrix Multiplication Matrix multiplication is useful in many business applications such as: 1. Inventory Management: - Matrix multiplication can be used to calculate the total value of inventory after applying markups or discounts. 2. Financial Modeling: - Matrix multiplication is used to project future values like sales, profits, assets etc. based on estimated growth rates. 3. Production Scheduling: - Matrix multiplication helps determine optimal production quantities by

1. 2011
Application of Mathematics in
Business World
This report indicates the Applications of Mathematical tools
in Business World; for this purpose we have chosen the data
from different organizations; Manufacturing, Trading, Serving
and Non-Profit Organizations. This report gives an idea how
to utilize these Mathematics functions in our real life.
PAF-KIET (City Campus)
MBA (Evening)
01-May-2011

2. Project on
Appllication of M athematicall Toolls in Business Worlld
Ap cati n f M h m ica To s in Bu n Wor
Submitted to
Sir Shahbaz Khan
Advanced Business Mathematics (55035)
Group Participants
 Toufeeq Ahmed (55302)
 Sadia Iftikhar (55502)
 Abdul Waseem (55475)
 Sidra Iqbal (54134)
 Ayesha Khan (54465)
May 01, 2011
MBA (Evening) Page 2

3. ACKNOWLEDGMENT
We are profoundly grateful to Almighty Allah for enabling us to accomplish this Project!
We are sincerely thankful to our supervisor Mr. Shahbaz Khan (Assistant professor at PAF-KIET)
for extending best possible support and cooperation for giving us the idea for making such
report on practical basis. He got a strong command on the subject and the topics he covered,
during the semester, would help us a lot while implementing this knowledge in our practical life.
We are also thankful to all given below organizations which provided us their useful information
for completion of our project.
 Medisure Group of Companies
 Marie Stopes Society, Pakistan
 Interglobe Enterprises
 Meezan Bank Limited
 Trade Polymerz (Pvt.) Limited
With the guidance of our supervisor we really enjoyed working on this project as it was a
learning process with a team’s effort.
MBA (Evening) Page 3

4. Appliicatiion of Mathematiics iin Busiiness World
Appl cat on of Mathemat cs n Bus ness World
You read and you forget!
You see and you remember!
You do and you learn!
MBA (Evening) Page 4

5. Table of Contents
PAF-KIET Overview .............................................................................................................................................. 6
TR, MR, AR (Graphical & Algebraic Methods)
Contributed by : TOUFEEQ AHMED ..........................................................................................
TOUFEEQ AHMED
Medisure Group of Companies............................................................................................... 7
Linear Equation
Contributed by : SADIIA IIFTIIHAR ................................................................................................
SAD A FT HAR
Marie Stopes Society, Pakistan ............................................................................................. 12
Application of Matrix in Business
Contributed by : ABDUL WASEEM ............................................................................................
ABDUL WASEEM
Interglobe Enterprises ........................................................................................................... 16
Logarithmic Function
Contributed by : SIIDRA IIQBAL ...................................................................................................
S DRA QBAL
Meezan Bank Limited ............................................................................................................ 20
Quadratic Function
Contributed by : AYESHA KHAN ................................................................................................
AYESHA KHAN
Trade Polymerz (Pvt) Limited................................................................................................ 25
References/Sources .....................................................................................................................................29
MBA (Evening) Page 5

6. PAF-KIET Overview
Pakistan Air Force Karachi Institute of Economics & Technology was established in 1997 with the aim of
providing quality education economically. The main campus spread over 22 acres is situated at PAF
Korangi Creek. The City Campus is situated at Shahara-e-Faisal, Karachi.
There are over 3000 students at PAF-KIET. Both the Campuses are fully equipped with modern
educational facilities. KIET providing the education services in four different areas:
1. College of Computer Sciences
2. College of Management Sciences
3. College of Engineering & Technology
4. College of Media & Arts
MBA (Evening) Page 6

7. Contributed by
TOUFEEQ AHMED
TOUFEEQ AHMED
Reg # 55302
Reg # 55302
Accounts Manager
MEDISURE GROUP OF COMPANIES
Pharmaceutical Company
MBA (Evening) Page 7

8. Scenario: Medisure is selling an antibiotic medicine Bredin Tablet; the average daily sales by Karachi
Field Force are given below:
Name Units Sold
M. Umer Khan 47
Jahangir Iqbal 59
Raheel Qaiser 73
Yasir Hameed 98
Waqas Hussain 65
Per Person Average 68
The demand function suggested by our Financial Analyst after market research is P = 450 – 3x.
Solution:
Total Revenue Function = PxQ
(450 – 3X) X
 TR = 450x – 3x2
Marginal Revenue Function = dTR = 450x – 3x2
dX
 MR = 450 – 6x
Average Revenue Function = TR = 450x – 3x2
X X
 AR = 450 – 3x
MBA (Evening) Page 8

9. Graphical Method:
2
Equations TR = 450x-3x MR = 450-6x AR = 450-3x
Marginal
Units Sold Total Revenue Average Reveune
Revenue
0 - 450 450
5 2,175.00 420 435
10 4,200.00 390 420
15 6,075.00 360 405
20 7,800.00 330 390
25 9,375.00 300 375
30 10,800.00 270 360
35 12,075.00 240 345
40 13,200.00 210 330
45 14,175.00 180 315
50 15,000.00 150 300
55 15,675.00 120 285
60 16,200.00 90 270
65 16,575.00 60 255
70 16,800.00 30 240
75 16,875.00 0 225
80 16,800.00 -30 210
85 16,575.00 -60 195
90 16,200.00 -90 180
95 15,675.00 -120 165
100 15,000.00 -150 150
105 14,175.00 -180 135
110 13,200.00 -210 120
115 12,075.00 -240 105
120 10,800.00 -270 90
125 9,375.00 -300 75
130 7,800.00 -330 60
135 6,075.00 -360 45
140 4,200.00 -390 30
145 2,175.00 -420 15
150 - -450 0
MBA (Evening) Page 9

10. 2
Equations TR = 450x-3x MR = 450-6x AR = 450-3x
17,500.00
16,500.00 Maximum TR = 16,875
at Units Sold = 75.
15,500.00
14,500.00
13,500.00
12,500.00
11,500.00
10,500.00
9,500.00
8,500.00
7,500.00
6,500.00
5,500.00
4,500.00
3,500.00
2,500.00
1,500.00
500.00
(500.00)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Total Revenue Marginal Revenue Average Reveune
MBA (Evening) Page 10

11. Algebraic Method:
If TR = 450x – 3x2
Then 1st derivative of TR is, TR’ = MR = 450 – 6x
As we know that if MR = 0, then TR would be Maximum
When MR = 0
0 = 450 – 6x
– 6x = - 450
X = 450 ÷ 6
X = 75 Units Sold
 When 75 Units are sold, the TR would be Maximum
TR = 450x – 3x2
TR = 450 (75) – 3 (75) 2
TR = 16,875
Therefore, it is proved that if 75 Units are sold, then company will earn Maximum
Revenue i.e. Rs. 16, 875
MBA (Evening) Page 11

12. Contributed by
SADIA IFTIHAR
SADIA IFTIHAR
Reg # 55502
Reg # 55502
Coordinator Internal Audit
MARIE STOPES SOCIETY, PAKISTAN
NGO - Social Services
MBA (Evening) Page 12

13. Field Staff Monthly Budgeted Allowances
Number of Allocated Amount
City Province
Persons (12,500 per person)
Karachi Sindh 8 100,000.00
Hyderabad Sindh 5 62,500.00
Sukkur Sindh 4 50,000.00
Larkana Sindh 3 37,500.00
Kashmore/Kandhkot Sindh 1 12,500.00
Kamber/Shahdadkot Sindh 2 25,000.00
Lahore Punjab 7 87,500.00
Rawalpindi Punjab 5 62,500.00
Sargodha Punjab 3 37,500.00
Multan Punjab 4 50,000.00
Khushab Punjab 2 25,000.00
Chakwal Punjab 3 37,500.00
Rahimyar Khan Punjab 4 50,000.00
Muzafar Garh Punjab 3 37,500.00
Hafizabad Punjab 5 62,500.00
TT.Singh Punjab 3 37,500.00
Okara Punjab 4 50,000.00
Khanewal Punjab 3 37,500.00
Attock Punjab 2 25,000.00
Mardan KPK 5 62,500.00
Noushaehra KPK 4 50,000.00
Swabi KPK 3 37,500.00
Peshawar KPK 6 75,000.00
Turbat 0 0.00
Total 89 1,112,500.00
The monthly budgeted allowance inclusive of the following:
1. Car renting
2. Telephone calls expenses (calculated on the basis of number of calls with durations).
3. Daily allowances.
4. Hotel Stay Expenses (if any).
MBA (Evening) Page 13

14. Field Staff Monthly Allocated Budgeted
100,000 100,000
90,000
87,500
80,000
75,000
70,000
Budgeted Allowances
62,500
60,000
50,000 50,000
40,000
37,500
30,000
25,000
20,000
12,500
10,000
- -
0 1 2 3 4 5 6 7 8
Number of Persons
SPSS Model Summary and Parameter Estimates
Dependent Variable: Number of Person.
Model Summary Parameter Estimates
Equation R Square F df1 df2 Sig. Constant b1 b2
Linear 1.000 . 1 21 . 5.881E-16 8.000E-5
Logarithmic .953 422.171 1 21 .000 -39.265 4.024
Quadratic 1.000 4.504E16 2 20 .000 6.923E-16 8.000E-5 -3.715E-26
Exponential .953 422.171 1 21 .000 1.448 1.894E-5
The independent variable is Total Budgeted Allowance.
MBA (Evening) Page 14

15. Properties of Linear Equation
A linear equation in two variables is an equation which may be written in the standard form
ax + by = c
Where a, b and c are constants. The Linear Equation can also be written in the Y-Intercept form
y = mx + c where m, and c are real numbers.
The graph of a linear equation is a non-vertical line with slope m and y-intercept c.
The x-intercept occurs when y = 0.
Therefore we find the x-intercept by solving mx + b = 0.
The y-intercept occurs when x = 0.
Therefore the y-intercept is c.
Some Application of Linear Equation
1. Temperature Conversion
2. Exchange Rate Calculation
3. Cell Phone Charges Calculation
4. Simple Interest Calculation
5. Salaries & Wages Computation
MBA (Evening) Page 15

16. Contributed by
ABDUL WASEEM
ABDUL WASEEM
Reg # 55475
Reg # 55475
Production Manager
INTERGLOBE ENTERPRISES
Commercial Importer
MBA (Evening) Page 16

17. APPLICATION OF MATRICES TO BUSINESS
What is a matrix?
A matrix is a two dimensional arrangement of numbers in row and columns enclosed by a pair
of square bracket ([ ]), in the form shown blow
11 12 13
21 22 23
31 32 33
Subject of matrix has been researched and expanded by the work of many mathematicians,
who have found numerous applications of matrices in various disciplines such as Economics,
Engineering, Statistics and various other sciences.
In this project the following applications to matrices will be discussed:
 Application of Matrix Addition and Subtraction
 Application of Matrix Multiplication
 Application of System of Linear Equation
Application of Matrix Addition and Subtraction
The applications of addition and subtraction of matrices can be illustrated through the following
examples.
1. The quarterly sales of silicon sealant clear, Black and White for the year 2009 and 2010
are given below.
Year 2009
20 25 22 20
A = 10 20 18 10
15 20 15 15
Year 2010
10 15 20 20
B = 5 20 18 10
8 30 15 10
Find the total quarterly sale of clear, Black and White for the two year.
20 25 22 20 10 15 20 20
A + B = 10 20 18 10 + 5 20 18 10
15 20 15 15 8 30 15 10
MBA (Evening) Page 17

18. 30 40 42 40
A + B = 15 40 33 40
23 50 30 25
 Interglobe Enterprises has the following sale position of its product A and B, at its two centers
Karachi and Lahore at the end of the year.
50 45
A=
60 70
If the sales for the first three months is given as
30 15
B=
20 20
Find the sales position for the last nine months
50 45 30 15
A–B= -
60 70 20 20
20 30
=
40 50
Application of Matrix Multiplication
The application of multiplication of matrices can be illustrated through the following example.
 Interglobe Enterprises produce three products A, B and C requiring the mix of three materials
X, Y and Z. The requirement (per unit) of each product for each material is as follows.
2 3 1
M= 4 2 5
2 4 2
Let per unit cost of material X, Y and Z are represented by 3x1 matrixes as under:
[5]
C = [10]
[5]
With the help of matrix multiplication per unit cost of production of each product would be
calculated as under.
2 3 1 [5]
Cost = 4 2 5 x [10]
2 4 2 [5]
[45]
Cost = [65]
[60]
The total cost of production of the firm produces 200 units of each product would be given as:
MBA (Evening) Page 18

19. [45]
Total cost = [200 200 200] x [65]
[60]
The total cost of production will be Rs. 34,000
Application of System of Linear Equation:
The following examples can be used to illustrate the common method of solving system of linear
equation the result from applied business problems.
 Interglobe Enterprises uses three types of material M1, M2 and M3, for producing
three types of chemicals product C1, C2 and C3. The chemical requirement (in
Kgs) for each type of chemicals is given blow.
Chemicals
C1 C2 C3
Material 2 3 4
1 1 2
3 2 1
Determine the number of chemical product of each type which can be produced using 29, 13
and 16 kgs of material of the three types respectively.
Solution:
Let X, Y and Z denote the number of chemical product that can be produced of each type. Then
we have
The above information can be represented using the matrix method, as under.
3 2 4 = 29
1 1 2 = 13
3 2 1 = 16
The above equation can be solved using Gauss Jordan Elimination method.
X + y + 2z=13 eq # 1
Y=3 eq # 2
-5z=-20 eq # 3
Hence the solution is:
X=2, y=3 and z=4
Verification
2(2) +3 (3) +4(4) = 29
2+3+2 (4) = 13
3(2) +2(3) +4 = 16
MBA (Evening) Page 19

20. Contributed by
SIDRA IQBAL
SIDRA IQBAL
Reg # 54134
Reg # 54134
Phone Banking Officer
MEEZAN BANK LIMITED
Banking Service Industry
MBA (Evening) Page 20

21. M/s XYZ Denim Limited imports Boys Pants from Yong Denim Fabric China the
details of LCs opend at ABC Bank Limited are given below
Bill # Quantity Amount
00001 130 Rs. 16,900.00
00002 2,140 Rs. 19,600.00
00003 3,140 Rs. 19,600.00
00004 4,140 Rs. 19,600.00
00005 5,144 Rs. 20,736.00
00006 6,147 Rs. 21,609.00
00007 7,147 Rs. 21,609.00
00008 8,159 Rs. 25,281.00
00009 9,180 Rs. 32,400.00
00010 10,180 Rs. 32,400.00
00011 11,200 Rs. 40,000.00
00012 12,200 Rs. 40,000.00
00013 13,205 Rs. 42,025.00
00014 14,243 Rs. 59,049.00
00015 15,243 Rs. 59,049.00
00016 16,255 Rs. 65,025.00
00017 17,290 Rs. 84,100.00
00018 18,310 Rs. 96,100.00
00019 19,310 Rs. 96,100.00
00020 20,310 Rs. 113,569.00
00021 21,337 Rs. 122,500.00
00022 22,350 Rs. 128,881.00
00023 23,359 Rs. 160,000.00
00024 24,400 Rs. 179,776.00
00025 25,424 Rs. 179,776.00
MBA (Evening) Page 21

22. Denim Fabric from China
Rs. 180,000.00
Rs. 160,000.00
Rs. 140,000.00
Rs. 120,000.00
Value in PKR
Rs. 100,000.00
Rs. 80,000.00
Rs. 60,000.00
Rs. 40,000.00
Rs. 20,000.00
Rs. 0.00
- 3,000 6,000 9,000 12,000 15,000 18,000 21,000 24,000 27,000
Quantity Imported
MBA (Evening) Page 22

23. SPSS Model Summary and Parameter Estimates
Dependent Variable: Quantity Purchased
Model Summary Parameter Estimates
Equation R Square F df1 df2 Sig. Constant b1 b2 b3
Linear .869 153.037 1 23 .000 4.198E3 .133
Logarithmic .978 1.011E3 1 23 .000 -8.792E4 9.339E3
Quadratic .950 210.443 2 22 .000 -400.894 .298 -9.040E-7
Cubic .973 253.645 3 21 .000 -5.345E3 .572 -4.426E-6 1.220E-11
Compound .403 15.525 1 23 .001 3.849E3 1.000
Exponential .403 15.525 1 23 .001 3.849E3 1.344E-5
The independent variable is Values in PKR.
MBA (Evening) Page 23

24. Rules for Logarithmic Function
1) logb(mn) = logb(m) + logb(n)
m
2) logb( /n) = logb(m) – logb(n)
n
3) logb(m ) = n · logb(m)
Some Application of Logarithmic Function
1. Interest Rate Calculations
2. Mortgage Calculations
3. Population growth function
4. Radioactive Decay Problems
5. Earthquake Problems
MBA (Evening) Page 24

25. Contributed by
AYESHA KHAN
AYESHA KHAN
Reg # 54465
Reg # 54465
Business Development Officer
TRADE POLYMERZ (PVT) LIMITED
Trading Commercial Organization
MBA (Evening) Page 25

26. Circular Weaving Machinery Sales Summary
Month Qty Rate Amount (PKR)
Jan-2010 18 189,000 3,402,000
Feb-2010 20 196,000 3,920,000
Mar-2010 22 203,000 4,466,000
Apr-2010 25 208,000 5,200,000
May-2010 30 211,000 6,330,000
Jun-2010 34 216,000 7,344,000
Jul-2010 31 210,000 6,510,000
Aug-2010 29 207,500 6,017,500
Sep-2010 26 204,600 5,319,600
Oct-2010 23 192,400 4,425,200
Nov-2010 21 185,000 3,885,000
Dec-2010 19 183,200 3,480,800
Circular Weaving Machinary Sales Summary
7,344,000
7,500,000.00
7,000,000.00
6,510,000
6,500,000.00 6,330,000
6,017,500
6,000,000.00
Sales Value
5,500,000.00 5,319,600
5,200,000
5,000,000.00
4,466,000 4,425,200
4,500,000.00
3,920,000 3,885,000
4,000,000.00
3,402,000 3,480,800
3,500,000.00
3,000,000.00
Jan/10 Feb/10 Mar/10 Apr/10 May/10 Jun/10 Jul/10 Aug/10 Sep/10 Oct/10 Nov/10 Dec/10
Month
MBA (Evening) Page 26

27. SPSS Model Summary and Parameter Estimates
Dependent Variable: Quantity.
Model Summary Parameter Estimates
Equation R Square F df1 df2 Sig. Constant b1 b2
Linear .790 37.616 1 10 .000 -59.568 .000
Logarithmic .778 35.101 1 10 .000 -987.635 82.941
Quadratic .886 34.821 2 9 .000 633.721 -.007 1.759E-8
Growth .802 40.544 1 10 .000 -.208 1.696E-5
Exponential .802 40.544 1 10 .000 .812 1.696E-5
The independent variable is Rate.
Some Application of Quadratic Function
1. Maximum Profit & Maximum Revenue
2. Minimum Cost Calculation
3. Break-even Point for Optimum Production
4. Salaries with other Benefits Computation
5. Electricity Consumption Model
MBA (Evening) Page 27

28. Properties of Quadratic Function
A quadratic function, in mathematics, is a polynomial function of the form
Properties of Graphs of Quadratic Functions
1. The graph of a quadratic function f(x) = ax + bx + c is called a parabola.
2. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward.
3. The lowest point of a parabola (when a > 0) or the highest point (when a < 0) is called
the vertex.
Properties of Quadratic Formula
Value of the discriminant Type and number of Solutions Example of graph
Positive Discriminant
Two Real Solutions
If the discriminant is a perfect
b² − 4ac > 0 square the roots are rational.
Otherwise, they are irrational.
Discriminant is Zero
One Real Solution
b² − 4ac = 0
Negative Discriminant
No Real Solutions
b² − 4ac < 0 Two Imaginary Solutions
MBA (Evening) Page 28

29. References/Sources
 Essential Mathematics for Economics & Business by Teresa Bradly.
 Applied Mathematics for Business, Economics and the Social
Sciences by Frank S. Budnick.
 All the above mentioned Organizations’ databases.
 Internet Search Engine www.google.com.
 PAF-KIET site www.pafkiet.edu.pk.
 SPSS Software.
 Microsoft Excel Tools.
The End
------
MBA (Evening) Page 29