Maths Project

2011

PAF-KIET (City Campus) MBA (Evening) 01-May-2011

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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.

MBA (Evening) Page 2

Project on

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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


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.


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You read and you forget!

You see and you remember!

You do and you learn!


Table of Contents PAF-KIET Overview .............................................................................................................................................. 6

TR, MR, AR (Graphical & Algebraic Methods)

Contributed by : TTTOOOUUUFFFEEEEEEQQQ AAAHHHMMMEEEDDD ..........................................................................................

Medisure Group of Companies............................................................................................... 7

Linear Equation

Contributed by : SSSAAADDDIIIAAA IIIFFFTTTIIIHHHAAARRR ................................................................................................

Marie Stopes Society, Pakistan ............................................................................................. 12

Application of Matrix in Business

Contributed by : AAABBBDDDUUULLL WWWAAASSSEEEEEEMMM ............................................................................................

Interglobe Enterprises ........................................................................................................... 16

Logarithmic Function

Contributed by : SSSIIIDDDRRRAAA IIIQQQBBBAAALLL ...................................................................................................

Meezan Bank Limited ............................................................................................................ 20

Quadratic Function

Contributed by : AAAYYYEEESSSHHHAAA KKKHHHAAANNN ................................................................................................

Trade Polymerz (Pvt) Limited................................................................................................ 25

References/Sources ..................................................................................................................................... 29


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


Contributed by

TTTOOOUUUFFFEEEEEEQQQ AAAHHHMMMEEEDDD

RRReeeggg ### 555555333000222

Accounts Manager

MEDISURE GROUP OF COMPANIES Pharmaceutical Company


Scenario: Medisure is selling an antibiotic medicine Bredin Tablet; the average daily sales by Karachi Field Force are given below:

The demand function suggested by our Financial Analyst after market research is P = 450 – 3x.

Solution:

Total Revenue Function = P x Q (450 – 3X) X

TR = 450x – 3x2

Marginal Revenue Function = dTR = 450x – 3x2

d X

MR = 450 – 6x

Average Revenue Function = TR = 450x – 3x2

X X

AR = 450 – 3x

Name Units SoldM. Umer Khan 47Jahangir Iqbal 59Raheel Qaiser 73Yasir Hameed 98

Waqas Hussain 65Per Person Average 68


Graphical Method:

Units Sold Total RevenueMarginal Revenue

Average Reveune

0 - 450 4505 2,175.00 420 43510 4,200.00 390 42015 6,075.00 360 40520 7,800.00 330 39025 9,375.00 300 37530 10,800.00 270 36035 12,075.00 240 34540 13,200.00 210 33045 14,175.00 180 31550 15,000.00 150 30055 15,675.00 120 28560 16,200.00 90 27065 16,575.00 60 25570 16,800.00 30 24075 16,875.00 0 22580 16,800.00 -30 21085 16,575.00 -60 19590 16,200.00 -90 18095 15,675.00 -120 165

100 15,000.00 -150 150105 14,175.00 -180 135110 13,200.00 -210 120115 12,075.00 -240 105120 10,800.00 -270 90125 9,375.00 -300 75130 7,800.00 -330 60135 6,075.00 -360 45140 4,200.00 -390 30145 2,175.00 -420 15150 - -450 0

Equations TR = 450x-3x2 MR = 450-6x AR = 450-3x


(500.00)

500.00

1,500.00

2,500.00

3,500.00

4,500.00

5,500.00

6,500.00

7,500.00

8,500.00

9,500.00

10,500.00

11,500.00

12,500.00

13,500.00

14,500.00

15,500.00

16,500.00

17,500.00

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Total Revenue Marginal Revenue Average Reveune

Maximum TR = 16,875at Units Sold = 75.

Equations TR = 450x-3x2 MR = 450-6x AR = 450-3x


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


Contributed by

SSSAAADDDIIIAAA IIIFFFTTTIIIHHHAAARRR

RRReeeggg ### 555555555000222

Coordinator Internal Audit

MARIE STOPES SOCIETY, PAKISTAN NGO - Social Services


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).

City Province Number of Persons

Allocated Amount (12,500 per person)

Karachi Sindh 8 100,000.00Hyderabad Sindh 5 62,500.00Sukkur Sindh 4 50,000.00Larkana Sindh 3 37,500.00Kashmore/Kandhkot Sindh 1 12,500.00Kamber/Shahdadkot Sindh 2 25,000.00Lahore Punjab 7 87,500.00Rawalpindi Punjab 5 62,500.00Sargodha Punjab 3 37,500.00Multan Punjab 4 50,000.00Khushab Punjab 2 25,000.00Chakwal Punjab 3 37,500.00Rahimyar Khan Punjab 4 50,000.00Muzafar Garh Punjab 3 37,500.00Hafizabad Punjab 5 62,500.00TT.Singh Punjab 3 37,500.00Okara Punjab 4 50,000.00Khanewal Punjab 3 37,500.00Attock Punjab 2 25,000.00Mardan KPK 5 62,500.00Noushaehra KPK 4 50,000.00Swabi KPK 3 37,500.00Peshawar KPK 6 75,000.00Turbat 0 0.00

Total 89 1,112,500.00

Field Staff Monthly Budgeted Allowances


SPSS Model Summary and Parameter Estimates

Dependent Variable: Number of Person.

Equation

Model Summary Parameter Estimates

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.

100,000

62,500

50,000

37,500

12,500

25,000

87,500

62,500

37,500

50,000

25,000

37,500

50,000

37,500

62,500

37,500

50,000

37,500

25,000

62,500

50,000

37,500

75,000

--

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

100,000

0 1 2 3 4 5 6 7 8

Budg

eted

Allo

wan

ces

Number of Persons

Field Staff Monthly Allocated Budgeted


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


Contributed by

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RRReeeggg ### 555555444777555

Production Manager

INTERGLOBE ENTERPRISES

Commercial Importer


APPLICATION OF MATRICES TO BUSINESS

WWhhaatt iiss aa mmaattrriixx??

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

푎11 푎12 푎13푎21 푎22 푎23푎31 푎32 푎33

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

A= 20 25 222010 20 181015 20 1515

Year 2010

B= 10 15 20205 20 18108 30 1510

Find the total quarterly sale of clear, Black and White for the two year.

A + B = 20 25 222010 20 181015 20 1515

+ 10 15 20205 20 18108 30 1510


A + B = 30 40 424015 40 334023 50 3025

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.

A = 50 4560 70

If the sales for the first three months is given as

B = 30 1520 20

Find the sales position for the last nine months

A – B = 50 4560 70

- 30 1520 20

= 20 3040 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.

M = 2 3 14 2 52 4 2

Let per unit cost of material X, Y and Z are represented by 3x1 matrixes as under:

C = [5]

[10][5]

With the help of matrix multiplication per unit cost of production of each product would be calculated as under.

Cost = 2 3 14 2 52 4 2

x [5]

[10][5]

Cost = [45][65][60]

The total cost of production of the firm produces 200 units of each product would be given as:


Total cost = [200 200 200] x [45][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.

Material

Chemicals C1 C2 C3 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 41 1 23 2 1

푥푦푧

= 29= 13= 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


Contributed by

SSSIIIDDDRRRAAA IIIQQQBBBAAALLL

RRReeeggg ### 555444111333444

Phone Banking Officer

MEEZAN BANK LIMITED Banking Service Industry


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 Amount00001 130 Rs. 16,900.0000002 2,140 Rs. 19,600.0000003 3,140 Rs. 19,600.0000004 4,140 Rs. 19,600.0000005 5,144 Rs. 20,736.0000006 6,147 Rs. 21,609.0000007 7,147 Rs. 21,609.0000008 8,159 Rs. 25,281.0000009 9,180 Rs. 32,400.0000010 10,180 Rs. 32,400.0000011 11,200 Rs. 40,000.0000012 12,200 Rs. 40,000.0000013 13,205 Rs. 42,025.0000014 14,243 Rs. 59,049.0000015 15,243 Rs. 59,049.0000016 16,255 Rs. 65,025.0000017 17,290 Rs. 84,100.0000018 18,310 Rs. 96,100.0000019 19,310 Rs. 96,100.0000020 20,310 Rs. 113,569.0000021 21,337 Rs. 122,500.0000022 22,350 Rs. 128,881.0000023 23,359 Rs. 160,000.0000024 24,400 Rs. 179,776.0000025 25,424 Rs. 179,776.00


Rs. 0.00

Rs. 20,000.00

Rs. 40,000.00

Rs. 60,000.00

Rs. 80,000.00

Rs. 100,000.00

Rs. 120,000.00

Rs. 140,000.00

Rs. 160,000.00

Rs. 180,000.00

- 3,000 6,000 9,000 12,000 15,000 18,000 21,000 24,000 27,000

Valu

e in

PKR

Quantity Imported

Denim Fabric from China



Dependent Variable: Quantity Purchased

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.


Rules for Logarithmic Function

1) logb(mn) = logb(m) + logb(n)

2) logb(m/n) = logb(m) – logb(n)

3) logb(mn) = 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


Contributed by

AAAYYYEEESSSHHHAAA KKKHHHAAANNN

RRReeeggg ### 555444444666555

Business Development Officer

TRADE POLYMERZ (PVT) LIMITED Trading Commercial Organization


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 Machinery Sales Summary

3,402,000

3,920,000

4,466,000

5,200,000

6,330,000

7,344,000

6,510,000

6,017,500

5,319,600

4,425,200

3,885,000

3,480,800

3,000,000.00

3,500,000.00

4,000,000.00

4,500,000.00

5,000,000.00

5,500,000.00

6,000,000.00

6,500,000.00

7,000,000.00

7,500,000.00

Jan/10 Feb/10 Mar/10 Apr/10 May/10 Jun/10 Jul/10 Aug/10 Sep/10 Oct/10 Nov/10 Dec/10

Sale

s Val

ue

Month

Circular Weaving Machinary Sales Summary



Dependent Variable: Quantity.

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


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 b² − 4ac > 0

Two Real Solutions If the discriminant is a perfect square the roots are rational. Otherwise, they are irrational.

Discriminant is Zero b² − 4ac = 0

One Real Solution

Negative Discriminant b² − 4ac < 0

No Real Solutions Two Imaginary Solutions


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.

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Maths Project