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# MTH302BusinessMathematicsStatistics.pdf MATH 140B

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This 442 page Class Notes was uploaded by zeee Notetaker on Friday November 20, 2015. The Class Notes belongs to MATH 140B at United States Air Force Academy taught by dr thomas finny in Fall 2015. Since its upload, it has received 13 views. For similar materials see Calculus and Biology I in Applied Mathematics at United States Air Force Academy.

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Business Mathematics & Statistics (MTH 302) Business Mathematics & Statistics (MTH 302) VU TABLE OF CONTENTS : Lesson 1 :COURSE OVERVIEW .......................................................................................................3 Lesson 2 :APPLICATION OF BASIC MATHEMATICS.................................................................... 12 Lesson 3 :APPLICATION OF BASIC MATHEMATICS.................................................................... 22 Lesson 4 :APPLICATION OF BASIC MATHEMATICS.................................................................... 29 Lesson 5 :APPLICATION OF BASIC MATHEMATICS.................................................................. 399 Lesson 6 :APPLICATION OF BASIC MATHEMATICS....................Error! Bookmark not defined.8 Lesson 7 :APPLICATION OF BASIC MATHEMATICS....................Error! Bookmark not defined.9 Lesson 8 :COMPOUND INTEREST............................................................................................... 709 Lesson 9 :COMPOUND INTEREST............................................................................................... 776 Lesson 10:MATRICES.................................................................................................................... 809 Lesson 11: MATRICES................................................................................................................... 854 Lesson 12 :RATIO AND PROPORTION .......................................................................................... 94 Lesson 13 :MATHEMATICS OF MERCHANDISING................................................................... 1009 Lesson 14 :MATHEMATICS OF MERCHANDISING..................................................................... 105 Lesson 15 :MATHEMATICS OF MERCHANDISING................................................................. 11211 Lesson 16 :MATHEMATICS OF MERCHANDISING................................................................. 12120 Lesson 17 :MATHEMATICS FINANCIAL MATHEMATICS........................................................ 12524 Lesson 18 :MATHEMATICS FINANCIAL MATHEMATICS........................................................ 13029 Lesson 19 :PERFORM BREAK-EVEN ANALYSIS.................................................................... 13433 Lesson 20 :PERFORM BREAK-EVEN ANALYSIS.................................................................... 14241 Lesson 21 :PERFORM LINEAR COST-VOLUME PROFIT AND BREAK-EVEN ANALYSIS.... 14746 Lesson 22 :PERFORM LINEAR COST-VOLUME PROFIT AND BREAK-EVEN ANALYSIS.... 15049 Lesson 23 :STATISTICAL DATA REPRESENTATION................................................................ 1587 Lesson 24 :STATISTICAL REPRESENTATION........................................................................16362 Lesson 25 :STATISTICAL REPRESENTATION........................................................................17170 Lesson 26 :STATISTICAL REPRESENTATION........................................................................18079 Lesson 27 :STATISTICAL REPRESENTATION........................................................................18988 Lesson 28 :MEASURES OF DISPERSION................................................................................ 20099 Lesson 29 :MEASURES OF DISPERSION.................................................................................... 207 Lesson 30 :MEASURE OF DISPERASION.................................................................................... 217 Lesson 31 :LINE FITTING.......................................................................................................... 22524 Lesson 32 :TIME SERIES AND.................................................................................................. 24039 Lesson 33 :TIME SERIES AND EXPONENTIAL SMOOTHING................................................ 25352 Lesson 34 :FACTORIALS........................................................................................................... 26059 Lesson 35 :COMBINATIONS ......................................................................................................... 269 Lesson 36 :ELEMENTARY PROBABILITY................................................................................ 27675 Lesson 37:PATTERNS OF PROBABILITY: BINOMIAL, POISSON AND NORMAL DISTRIBUTIONS .................................................................................................................................27978 Lesson 38:PATTERNS OF PROBABILITY: BINOMIAL, POISSON AND NORMAL DISTRIBUTIONS .................................................................................................................................28483 Lesson 39:PATTERNS OF PROBABILITY: BINOMIAL, POISSON AND NORMAL DISTRIBUTIONS .................................................................................................................................29796 Lesson 40:PATTERNS OF PROBABILITY: BINOMIAL, POISSON AND NORMAL DISTRIBUTIONS .....................................................................................................................................302 Lesson 41: ESTIMATING FROM SAMPLES: INFERENCE .......................................................... 314 Lesson 42 :ESTIMATING FROM SAMPLE : INFERENCE............................................................ 320 Lesson 43 :HYPOTHESIS TESTING: CHI-SQUARE DISTRIBUTION.......................................... 325 Lesson 44 :HYPOTHESIS TESTING : CHI-SQUARE DISTRIBUTION......................................... 328 Lesson 45 :PLANNING PRODUCTION LEVELS: LINEAR PROGRAMMING............................... 335 2 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU MTH 302 LECTURE 1 COURSE OVERVIEW COURSE TITLE The title of this course is “BUSINESS MATHEMATICS AND STATISTICS”. Instructor’s Resume The instructor of the course is Dr. Zahir Fikri who holds a Ph.D. in Electric Power Systems Engineering from the Royal Institute of Technology, Stockholm, Sweden. The title of Dr. Fikri’s thesis was “Statistical Load Forecasting for Distribution Network Planning”. Objective The purpose of the course is to provide the st udent with a mathematical basis for personal and business financial decisions through eight instructional modules. The course stresses business applications using arithmetic, algebra, and ratio-proportion and graphing. Applications include payroll, cost-volume-profit analysis and merchandising mathematics. The course also includes Statistical Representation of Data, Correlation, Time Series and Exponential Smoothing, Elementary Probability and Probability Distributions. This course stresses logical reasoning and problem solving skills. Access to Microsoft Excel software is required for the course. Course Outcomes Successful completion of this course will enable the student to: 1. Apply arithmetic and algebraic skills to everyday business problems. 2. Use ratio, proportion and percent in the solution of business problems. 3. Solve business problems involving commercial discount, markup and markdown. 4. Solve systems of linear equations graphically and algebraically and apply to cost volume- profit analysis. 5. Apply Statistical Representation of Data, Correlation, Time Series and Exponential Smoothing methods in business decision making 6. Use elementary probability theory and knowledge about probability distributions in developing profitable business strategies. Unit Outcomes Resources/Tests/Assignments Successful completion of the following units will enable the student to apply mathematical methods to business problems solving. Required Student Resources (Including textbooks and workbooks) Text: Selected books on Business Mathematics and Statistics. Optional Resources Handouts supplied by the professor. Instructor’s Slides Online or CD based learning materials. Prerequisites The students are not required to have any mathematical skills. Basic knowledge of Microsoft Excel will be an advantage but not a requirement. Evaluation In order to successfully complete this course, t he student is required to meet the following evaluation criteria: Full participation is expected for this course All assignments must be completed by the closing date. Overall grade will be based on VU existing Grading Rules. All requirements must be met in order to pass the course. 3 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU COURSE MODULES The following are the main modules of this course: Module 1 • Over(Liewture • Perform arithmetic operations in their proper order (Lecture 2) • Convert fractions their percent and decimal equivalents. (Lecture 2) • Solve for any one of percent, portion or base, given the other two quantities. (Lecture 2) • Using Microsoft Excel (Lecture 2) Calculate the gross earnings of employees paid a salary, an hourly wage or commissions. (Lecture 3) • Calculate the simple average or weighted average given a set of values. (Lecture 4) Perform basic calculations of the percentag es, averages, commission, brokerage and discount (Lecture 5) • Simple and compound interest (Lecture 6) • Average due date, interest on drawings and calendar (Lecture 6) Module 2 • Exponents and radicals (Lecture 7) • Solve linear equations in one variable (Lecture 7) • Rearrange formulas to solve for any of its contained variables (Lecture 7) • Solve problems involving a series of compounding percent changes (Lecture 8) • Calculate returns from investments (Lecture 8) • Calculate a single percent change equivalent to a series of percent changes (Lecture 8) • Matrices ( Lecture 9) • Ratios and Proportions ( Lecture10) • Set up and manipulate ratios ( Lecture11) • Allocate an amount on a prorata basis using proportions ( Lecture11) • Assignment Module 1-2 Module 3 • Discounts ( Lectures 12) • MathMem ctian dising ( Lectures 13-16) Module 4 • Applications of Linear Equations ( Lecture 17-18) • Break-even Analysis ( Lecture 19-22) • Assignment Module 3-4 • Mid-Term Examination Module 5 • Statistical data ( Lectures 23) • Measures of central tendency ( Lectures 24-25) • Measures of dispersion and skewness ( Lectures 26-27) Module 6 • Correlation ( Lectures 28-29) • Line Fitting (Lectures 30-31) • Time Series and Exponential Smoothing ( Lectures 31-33) • Assignment Module 5-6 Module 7 • Factorials ( Lecture 34) • Permutations and Combinations ( Lecture 34) • ElemeP ntoyability ( Lectures 35-36) • Patterns of probability: Binomial, Poisson and Normal Distributions ( Lecture 37-40) Module 8 • Estimating from Samples: Inference ( Lectures 41-42) • Hypothesis testing : Chi-Square Distribution ( Lectures 43-44) • Planning Production Levels: Linear Programming (Lecture 45) 4 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU • Assignment Module 7-8 • End-Term Examination Note: The course modules are subject to change. MARKING SCHEME As per VU Rules DESCRIPTION OF TOPICS NO. MAIN TOPIC LECTURE TOPICS RECOMMENDED READING 1 1.0 Module Applications of • Overviewew (Lecture 1) Reference 1 1 Basic Mathematics ( Lectures 1-6) 2 Reference 2, Module • Course Overview Lecture 2 1 • Arithmetic Operations & Tool: Microsoft • Using Microsoft Excel Excel 3 Reference 2, Module • Calculate Gross Earnings Lecture 3 1 Tool: Microsoft • Using Microsoft Excel Excel 4 Reference 2, • Calculating simple or Lecture 4 Module weighted averages Tool: Microsoft 1 • Using Microsoft Excel Excel Reference 6 5 Reference 2, • Basic calculations of Lecture 5 percentages, averages, commission, Module brokerage and discount using Reference 3, Ch 3 1 • Microsoft Excel Tool: Microsoft Excel 6 Reference 2, • Simple and compound Lecture 6 Module interest Reference 3, Ch 3 1 • Average due date, interest on drawings and calendar Tool: Microsoft Excel 7 • Exponents and radicals Reference 2, • Simplify algebraic 2.0 expressions Lecture 7 Module Applications of • Solve linear equations in one Reference 3, Ch 2 2 Basic Algebra variable Tool: Microsoft ( Lectures 7-9) Excel • Rearrange formulas to solve for any of its contained variables 5 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU 8 • Calculate returns from investments Reference 2, • Problems involving a series Lecture 8 of Reference 3, Ch 3 compounding percent changes • Single percent change equivalent Tool: Microsoft to a series of percent changes Excel 9 Reference 2, Lecture 9 Reference 3, Ch 4 • Matrices Tool: Microsoft Excel 10 • Set up and manipulate ratios. Reference 2, 3.0 • Set up and solve proportions. Lecture 10 Applications • Express percent differences Reference 3, Ch 3 Module of Ratio and using proportions. 2 Proportion • Allocate an amount on a ( Lectures 10- prorata basis using 11) proportions. Tool: Microsoft Excel 11 Reference 2, Lecture 11 Module • Set up and manipulate ratios. Reference 3, Ch 3 • Allocate an amount on a 2 prorata basis using proportions Tool: Microsoft Excel 12 4.0 • Calculate the net price of an Reference 2, Merchandising item after single or multiple trade Lecture 12 Module and Financial discounts. Reference 3, Ch 3 3 Mathematics • Calculate an equivalent single ( Lectures 12- discount rate given a series of Tool: Microsoft 16) discounts. Excel 13 Reference 2, Lecture 13 Module • Solve merchandising pricing Reference 3, Ch 3 3 problems involving markup and Tool: Microsoft markdown. Excel 14 Reference 2, Lecture 14 Reference 3, Ch 3 Module 3 • Financial Mathematics Part 1 Reference 5, Ch 16 Tool: Microsoft Excel 15 Module • Financial Mathematics Part 2 Reference 2, 3 Lecture 15 6 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Reference 3, Ch 3 Reference 5, Ch 16 Tool: Microsoft Excel 16 Reference 2, Lecture 16 Module Reference 3, Ch 3 • Financial Mathematics Part 3 Reference 5, Ch 16 3 Tool: Microsoft Excel 17 Reference 2, Lecture 17 5.0 Break-Even Reference 3, Ch 3 Module Analysis • Graph a linear equation in twoReference 5, Ch 16 4 ( Lectures 17- variables. & 18 22) Tool: Microsoft Excel 18 Reference 2, Lecture 18 Module • Solve two linear equations Reference 3, Ch 2 4 with two unknowns Reference 5, Ch 1 Tool: Microsoft Excel 19 • Perform linear cost-volume Reference 2, Module profit and break-even analysis. Lecture 19 4 • Using a break-even chart Tool: Microsoft Excel 20 • Perform linear cost-volume profit and break-even analysis. Reference 2, Module • Using the algebraic approach Lecture 20 4 of solving the cost and revenue Tool: Microsoft Excel functions 21 • Perform linear cost-volume profit and break-even analysis. Reference 2, Module Lecture 21 4 • Using the contribution margin Tool: Microsoft approach Excel 22 • Perform linear cost-volume Reference 2, Module profit and break-even analysis. Lecture 22 4 • Using Microsoft Excel Tool: Microsoft • Assignment Module 3-4 • Mid-Term Examination Excel 6. Statistical 23 Reference 2, Module Representation • Statistical Data Lecture 23 5 of Data Reference 5, Ch 5 ( Lectures 23- Tool: Microsoft 7 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU 27) Excel 24 Reference 2, Lecture 24 Module • Statistical Representation Reference 4, Ch 3 5 Measures of Central Tendency Reference 5, Ch 6 Part 1 Tool: Microsoft Excel 25 Reference 2, • Statistical Representation Lecture 25 • Measures of Central Module Tendency Reference 4, Ch 3 5 Part 2 Reference 5, Ch 6 Tool: Microsoft Excel 26 Reference 2, • Measures of Dispersion and Lecture 26 Module Skewness Reference 4, Ch 4 5 Part 1 Reference 5, Ch 6 Tool: Microsoft Excel 27 Reference 2, Lecture 27 Reference 4, Ch 4 • Measures of Dispersion and Reference 5, Ch 6 Module Skewness Tool: Microsoft 5 Part 2 Excel 7. Correlation, 28 Time Series Reference 2, Lecture 28 and Reference 5, Ch Module Exponential • Correlation 13 6 Smoothing Part 1 ( Lectures 28- 33) Tool: Microsoft Excel 29 Reference 2, Lecture 29 Reference 5, Ch • Correlation 13 Part 2 Tool: Microsoft Excel 30 Reference 2, Lecture 30 Reference 5, Ch • Line Fitting 14 Part 1 Tool: Microsoft Excel 8 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU 31 Reference 2, • Line Fitting Lecture 31 Part 2 Tool: Microsoft Excel 32 Reference 2, Lecture 32 • Time Series and Reference 5, Ch • Exponential Smoothing 15 Part 1 Tool: Microsoft Excel 33 Reference 2, • Time Series and Lecture 33 Reference 5, Ch • Exponential Smoothing 15 Part 2 • Assignment Module 5-6 Tool: Microsoft Excel 34 Reference 2, 7. Elementary Lecture 34 Module Probability • Factorials Reference 3, Ch 2 7 ( Lectures 34- • Permutations and 38) Combinations Tool: Microsoft Excel 35 Reference 2, Lecture 35 Module • Elementary Probability Reference 5, Ch 8 Part 1 7 Tool: Microsoft Excel 36 Reference 2, Lecture 36 • Elementary Probability Reference 5, Ch 8 Module Part 2 Tool: Microsoft 7 Excel 37 Reference 2, • Patterns of probability: Binomial, Poisson and Normal Lecture 39 Module Distributions Reference 5, Ch 9 7 Part 1 Tool: Microsoft Excel 38 Reference 2, • Patterns of probability: Lecture 40 Module Binomial, Poisson and Normal Reference 5, Ch 9 7 Distributions Tool: Microsoft Part 2 Excel 9 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU 39 Reference 2, • Patterns of probability: Lecture 41 Module Binomial, Poisson and Normal Reference 5, Ch 9 7 Distributions Part 3 Tool: Microsoft Excel 40 Reference 2, • Patterns of probability: Lecture 41 Module Binomial, Poisson and Normal Reference 5, Ch 9 7 Distributions Part 4 Tool: Microsoft Excel 8. Probability 41 Distributions Reference 2, Lecture 42 ( Lectures 39- • Estimating from Samples: Reference 5, Ch Module 44) Inference 10 8 9. Linear Part 1 Programming (Lecture 45) Tool: Microsoft Excel 42 Reference 2, Lecture 43 • Estimating from Samples: Reference 5, Ch Module Inference 10 8 Part 2 Tool: Microsoft Excel 43 Reference 2, Lecture 44 Module Reference 5, Ch 11 • Hypothesis testing : Chi- 8 Square Distribution Part 1 Tool: Microsoft Excel 44 Reference 2, • Hypothesis testing : Chi- Lecture 45 Module Square Distribution Part 2 Reference 5, Ch 8 11 Tool: Microsoft Excel 45 • Production Planning: Reference 2, Linear Programming Lecture 45 Module Reference 5, Ch 8 • Assignment Module 7-8 18 • End Term Examination Tool: Microsoft Excel Methodology There will be 45 lectures each of 50 minutes du ration as indicated abov e. The lectures will be delivered in a mixture of Urdu and English. Th e lectures will be heavily supported by slide presentations. The slides for a lecture will be made available on the VU website for the course a few days before the actual lecture is televised. will allow students to carry out preparatory reading before the lecture. The course will be provided its own page on the VU’s web site. This will be used to 10 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU provide lecture and other supportin g material from the course to the students. The page will have a link to a web-based discussion and bulletin board for the students. Teaching assistants will be assigned by VU to provide various forms of assist ance such as grading, answering questions posted by students and preparation of slides. Grading There will be a term exam and one final examination. There will also be 4 assignments each covering two modules. The final exam will be comprehensive. These will contribute the following percentages to the final grade: Mid Term Exam 35% Final 50% 4 Assignments 15% Text and Reference Material The course is based on material from different sources. Topics for reading will be indicated on course web site and in professor’s handouts, also to be posted on the course web site. A list of reference books will also be posted and updated on the course web site. The following material will be used by the students as reference: Reference 1: Course Outline 2: Instructor’s Power Point Slides 3: Business Mathematics & Statistics by Prof. Miraj Din Mirza 4: Elements of statistics & Probability by Shahid Jamal 5: Quantitative Approaches in Business studies by Clare Morris 6: Microsoft Excel Help File Schedule of Lectures Given above is the tentative schedule of topics to be covered. Minor changes may occur but these will be announced well in advance. 11 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU LECTURE 2 Applications of Basic Mathematics Part 1 OBJECTIVES The objectives of the lecture are to learn about: • Diffeentrsedules • Basic Arithmetic Operations • Starting Microsoft (MS) Excel • Using MS Excel to carry out arithmetic operations COURSE MODULES This course comprises 8 modules as under: • Modules 1-4: Mathematics • Modules 5-8: Statistics Details of modules are given in handout for lecture 01. BASIC ARITHMETIC OPERATIONS Five arithmetic operations provide the foundation for all mathematical operations. These are: • Addition • Subtraction • Multiplication • Division • Exponents Example- Addition 12 + 5 = 17 Example- Subtraction 12 - 5 = 7 Example- Multiplication 12 x 5 = 60 Example- Exponent 16(=)^2 (4)^1/2 = 2 (4)^-1/2 = 1/(4)^1/2 = ½ = 0.5 MICROSOFT EXCEL IN BUSINESS MATHEMATICS & STATISTICS Microsoft Corporation’s Spreadsheet software Excel is widely used in business mathematics and statistical applications. The latest version of this software is EXCEL 2002 XP. This course is based on wide applications of EXCEL 2002. It is recommended that you install EXCEL 2002 XP software on your computer. If your computer has Windows 2000 and EXCEL 2000 even that version of EXCEL can be used as the applications we intend to learn can be done using the earlier version of EXCEL. Those of you who are still working with Windows 98 and have EXCEL 97 installed are encouraged to migrate to newer version of EXCEL software. 12 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Starting EXCEL 2000 XP EXCEL 2000 XP can be started by going through the following steps: Click Start on your computer Click All Programs Click Microsoft Excel The following slides show the operations: The EXCEL window opens and a blank worksheet becomes available as shown below: 13 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU The slide shows a Workbook by the name book1 with three sheets: Sheet1, Sheet2 and Sheet3. The Excel Window has Column numbers starting from A and row numbers starting from 1. the intersection of a row and column is called a Cell. The first cell is A1 which is the intersection of column A and row 1. All cells in a Sheet are referenced by a combination of Column name and row number. 1Example : B15 means cell in column B and row 15. 2Example : A cell in row 12 and column C has reference C12. A Range defines all cells starting from the leftmost corner where the range starts to the rightmost corner in the last row. The Range is specified by the starting cell, a colon and the ending cell. 3:Example A Range which starts from A1 and ends at D15 is referenced by A1:D15 and has all the cells in columns A to D up to and including row 15. A value can be entered into a cell by clicking that cell. The mouse pointer which is a rectangle moves to the selected cell. Simply enter the value followed by the Enter key. The mouse pointer moves to the cell below. If you make a mistake while entering the value select the cell again (by clicking it). Enter the new value. The old value is replaced by the new value. 14 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU If only one or more digits are to be changed then select the cell. Then double click the mouse. The blinking cursor appears. Either move the arrow key to move to the digit to be changed or move the cursor to the desired position. Enter the new value and delete the undesired value by using the Del key. I suggest that you learn the basic operations of entering, deleting and changing data in a worksheet. About calculation operators in Excel In Excel there are four different types of operators: Arithmetic operators Comparison operators Text concatenation operator Reference operators The following descriptions are reproduced from Excel’s Help file for your ready reference. In the present lecture you are directly concerned with arithmetic operators. However, it is important to learn that the comparison operators are used where calculations are made on the basis of comparisons. The text concatenation operator is used to combine two text strings. The reference operators include “:” and “,” or ; as the case maybe. We shall learn the use of these operators in different worksheets. You should look through the Excel Help file to see examples of these functions. Selected material from Excel Help File relating to arithmetic operations is given in in a separate file. The Excel arithmetic operators are as follows: Addition. Symbol: + (Example: =5+4 Result: 9) Subtraction. Symbol: - (Example: =5-4 Result: 1) Multiplication. Symbol: * (Example: =5*4 Result: 20) Division. Symbol: / (Example: =12/4 Result: 3) Percent. Symbol: % (Example: =20% Result: 0.2) Exponentiation: ^ (Example: =5^2 Result: 25) Excel Formulas for Addition All calculations in Excel are made through formulas which are written in cells where result is required. Let us do addition of two numbers 5 and 10. We wish to calculate the addition of two numbers 10 and 5. Let us see how we can add these two numbers in Excel. 1. Open a blank worksheet. 2. Click on a cell where you would like to enter the number 10. Say cell A15. 3. Enter 10 in cell A15. 4. Click cell where you would like to enter the number 5. Say cell B15. 5. Click cell where you would like to get the sum of 10 and 5. Say cell C15. 6. Start the formula. Write equal sign = in cell C15. 7. After =, write “(“ (left bracket) in cell C15. 8. Move mouse and left click on value 10 which is in cell A15. In cell C15, the cell reference A15 is written. 9. Write “+” after “A15” in cell C15. 10. Move mouse and left click on value 5 which is in cell B15. In cell C15, the cell reference B15 is written. 11. Write “) “ (right bracket) in cell C15. 12. PreEsseyr The answer 15 is shown in cell C15. If you click on cell C15, the formula “=A15+B15” is displayed the formula bar to the right of fx in the Toolbar. 15 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU The main steps along with the entries are shown in the slide below. The worksheet MTH302-lec-02 contains the actual entries. The next slide shows addition of 6 numbers 5, 10, 15, 20, 30 and 40. The entries were made in row 34. The values were entered as follows: Cell A34: 5 Cell B34: 10 Cell C34: 15 Cell D34: 20 Cell E34: 30 Cell F34: 40 The formula was written in cell G34. The formula was: =5+10+15+20+30+40 The answer was 120. You can use an Excel function SUM along with the cell range A34:F34 to calculate the sum of the above numbers. The formula in such a case will be: =SUM(A34:F34) You enter “=” followed by SUM, followed by “(“. Click on the cell with value 5(reference: A34). Drag the mouse to cell with value 40(reference: F34) and drop the mouse. Enter “)” and then press the Enter key. 16 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU In the above two examples you learnt how formulas for addition are written in Excel. Excel Formula for Subtraction Excel formulas for subtraction are similar to those of addition but with the minus sign. Let us go through the steps for subtracting 15 from 25. Enter values in row 50 as follows: Cell A50: 25 Cell B50: 15 Write the formula in cell C50 as follows: =A50-B50 To write this formula, click cell C50, where you want the result. Enter “=”. Click on cell with value 25 (reference:A50). Enter “-“(minus sign). Click on cell with value 15 (reference B50). Press enter key. If you enter 15 first and 25 later, then the question will be to find result of subtraction 15- 25. 17 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Excel Formula for Multiplication Excel formula for multiplication is also similar to the formula for addition. Only the sign of multiplication will be used. The Excel multiplication operator is *. Let us look at the multiplication of two numbers 25 and 15. The entries will be made in row 60. Enter values as under: Cell A50: 25 Cell B50: 15 The formula for multiplication is: =A50*B50 18 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Click on cell C50 to write the formula in that cell. Enter “=”. Click on cell with number 25 (reference: A50). Enter “*”. Click on cell with number 15 (reference: B50). Press Enter key. The answer is 375 in cell C50. Excel Formula for Division The formula for division is similar to that of multiplication with the difference that the division sign “/” will be used. Let us divide 240 by 15using Excel formula for division. Let us enter numbers in row 75 as follows: Cell A75: 240 Cell B75: 15 The formula for division will be written in cell C75 as under: =A75/B75 The steps are as follows: Click the cell A75. Enter 240 in cell A75. Click cell B75. Enter 15. Click cell C75. Enter “=”. Click on cell with value 240 (reference: A75). Enter “/”. Click cell with number 15 (reference: B75). Press enter key. The answer 16 will be displayed in cell C75. Excel Formula for Percent The formula for converting percent to fraction uses the symbol %. To convert 20% to fraction the formula is as under: =20% If you enter 20 in cell A99, you can write formula for conversion to fraction by doing the following: Enter 2o in cell A99. In cell B99 enter “=”. Click on cell A99. Enter”%”. Press Enter key. The answer 0.2 is given in cell B99. 19 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Excel Formula for Exponentiation The symbol for exponentiation is ^. The formula for calculating exponents is similar to multiplication with the difference that the carat symbol ^ will be used. Let us calculate 16 raised to the power 2 by Excel formula for exponentiation. The values will be entered in row 85. The steps are: Select Cell A85. Enter 16 in this cell. Select cell B85 Enter 2 in this cell. Select cell C85. Enter”=”. Select cell with value 16 (reference:A85). Enter “^”. Select number 2 (reference: B85) Press Enter key. The result 256 is displayed in cell C85. 20 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Recommended Homework Download worksheet MTH302-lec-02.xls from the course web site. Change values to see change in results. Set up new worksheets for each Excel operator with different values. Set up worksheets with combinations of operations. 21 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU LECTURE 3 Applications of Basic Mathematics Part 2 OBJECTIVES The objectives of the lecture are to learn about: • Evaluations • Calculate Gross Earnings • Using Microsoft Excel Evaluation In order to successfully complete this course, the student is required to meet the evaluation criteria: • Evaluation Criterion 1 • Full participation is expected for this course • Evaluation Criterion 2 • All assignments must be completed by the closing date • Evaluation Criterion 3 • Overall grade will be based on VU existing Grading Rules • Evaluation Criterion 4 • All requirements must be met in order to pass the course Grading There will be a term exam and one final exam; there will also be 4 assignments. The final exam will be comprehensive. These will contribute the following percentages to the final grade: Mid Term Exam 35% Final 50% 4 Assignments 15% Collaboration The students are encouraged to develop collaboration in studying this course. You are advised to carry out discussions with other students on different topics. It will be in your own interest to prepare your own solutions to Assignments. You are advised to make your original original submissions as copying other students’ assignments will have negative impact on your studies. ETHICS Be advised that as good students your motto should be: • No copying • No cheating • No short cuts 22 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Methodology There will be 45 lectures each of 50 minutes duration. The lectures will be delivered in a mixture of Urdu and Englis. The lectures will be heavily supported by slide presentations. The slides available on the VU website before the actual lecture is televised. Students are encouraged to carry out preparatory reading before the lecture. This course has its own page on the VU’s web site. There are lecture slides as well as other supporting material available on the web site. Links to a web-based discussion and bulletin board will also been provided. Teaching assistants will be assigned by VU to provide various forms of assistance such as grading, answering questions posted by students and preparation of slides Text and Reference Material This course is based on material from different sources. Topics for reading will be indicated on course web site and in professor’s handouts. A list of reference books to be posted and updated on course web site. You are encouraged to regularly visit the course web site for latest guidelines for text and reference material. PROBLEMS If you have any problems with understanding of the course please contact: mth302@vu.edu.pk Types of Employees There may be three types of employees in a company: • Regular employees drawing a monthly salary • Part time employees paid on hourly basis • Payments on per piece basis To be able to understand how calculations of gross earnings are done, it is important to understand what gross earnings include. GROSS EARNINGS/SALARY Gross salary includes the following: • Basic salary • Allowances Gross salary may include: • Basic salary • House Rent • Conveyance allowance • Utilities allowance Accordance to the taxation rules if allowances are 50% of basic salary, the amount is treated as tax free. Any allowances that exceed this amount are considered taxable both for the employee as well as the company. Example 1 The salary of an employee is as follows: Basic salary = 10,000 Rs. Allowances = 5,000 Rs. 23 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU What is the taxable income of employee? Is any add back to the income of the company? % Allowances = (5000/10000) x 100 =50% Hence allowances are not taxable. Total taxable income = 10,000 Rs. Add back to the income of the company = 0 Example 2 The salary of an employee is as follows: Basic salary = 10,000 Rs. Allowances = 7,000 Rs. What is the taxable income of employee? Is any add back to the income of the company? % Allowances = (7000/10000) x 100 =70% Allowed non-taxable allowances = 50% = 0.5 x 10000 = 5,000 Rs. Taxable allowances = 70% – 50% = 7000 - 5000 = 2,000 Rs. Hence 2000 Rs. of allowances are taxable. Total taxable income = 10,000 + 2000 = 12,000 Rs. Add back to the income of the company = 20% allowances = 2,000 Rs. Structure of Allowances The common structure of allowances is as under: • House Rent = 45 % • Conveyance allowance = 2.5 % • Utilities allowance = 2.5 % Example 3 The salary of an employee is as follows: Basic salary = 10,000 Rs. What is the amount of allowances if House Rent = 45 %, Conveyance allowance = 2.5 % and Utilities allowance = 2.5 %? House rent allowances = 0.45 x 10000 = 4,500 Rs. Conveyance allowance = 0.025 x 10000 = 250 Rs. Utilities allowance = 0.025 x 10000 = 250 Rs. Thus total allowances are 4500+250+250 = 5000Rs Provident Fund According to local laws, a comthny can establish a Provident Trust Fund for the benefit of the employees. By law, 1/11 of Basic Salary per month is deducted by the company from the gross earnings of the employee. An equal amount, i.e 1/11thof basic salary per month, is contributed by the company to the Provident Fund to the account of the employee. The company can invest the savings in Provident Fund in Government Approved securities such as defense saving Certificates. Interest earned on investments in Provident Fund is credited to the account of the employees in proportion to their share in the Provident Fund. Example 4 The salary of an employee is as follows: Basic salary = 10,000 Rs. Allowances = 5,000 Rs. 24 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU What is the amount of deduction on account of contribution to the Provident Trust Fund? What is the contribution of the company? What is the total saving of the employee per month on account of Provident Trust Fund? Employee contribution to Provident Fund = 1/11 x 10000 = 909.1 Rs. Company contribution to Provident Fund = 1/11 x 10000 = 909.1 Rs. Total savings of employee in Provident Fund = 909.1 + 909.1 = 1,818.2 Rs. Gratuity Fund According to local laws, a thmpany can establish a Gratuity Trust Fund for the benefit of the employees. By law, 1/11 of Basic Salary per month is contributed by the cothany to the Gratuity Fund to the account of the employee. Thus there is a saving of 1/of basic salary on behalf of the employee in Gratuity Fund. The company can invest the savings in Gratuity Fund in Government Approved securities such as defence saving Certificates. Interest earned on investments in Gratuity Fund is credited to the account of the employees in proportion to their share in the Gratuity Fund. Example 5 The salary of an employee is as follows: Basic salary = 10,000 Rs. Allowances = 5,000 Rs. What is the contribution of the company on account of gratuity to the Gratuity Trust Fund? Company contribution to Gratuity Fund = Total savings of employee in Gratuity Fund = 1/11 x 10000 = 909.1 Rs. Leaves All companies have a clear leaves policy. The number of leaves allowed varies from company to company. Typical leaves allowed may be as under: • Casual Leave = 18 Days per year • Earned Leave = 18 Days per year • Sick Leave = 12 Days per year Example 6 The salary of an employee is as follows: Basic salary = 10,000 Rs. Allowances = 5,000 Rs. What is the cost on account of casual, earned and sick leaves per year if normal working days per month is 22? What is the total cost of leaves as percent of gross salary? Gross salary = 10000 + 5000 = 15,000 Rs. Cost of casual leaves per year = {18 / (22 x 12)} x 15000 x 12 = 12,272.7 Rs. Cost of earned leaves per year = {18 / (22 x 12)} x 15000 x 12= 12,272.7 Rs Cost of Sick leaves per year = {12 / (22 x 12) x 15000 x 12 = 8,181.8 Rs Total cost of leaves per year = 12272.7 + 12272.7 + 8181.8 = 32,727.3 Rs. Total cost of leaves as percent of gross salary = (32727.3/(12 x 15000))x 100 = 18.2% Social Charges 25 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Social charges comprise leaves, group insurance and medical. Typical medical/group insurance is about 5% of gross salary. Other social benefits may include contribution to employee’s children’s education, club membership, leave fare assistance etc. Such benefits may be about 5.8%. Leaves are 18.2% of gross salary (as calculated in above example) Total social charges therefore may be = 18.2 + 5 + 5.8 = 29% of gross salary. Other companies may have more social benefits. The 29% social charges are quite common. Example 7 The salary of an employee is as follows: Basic salary = 10,000 Rs. Allowances = 5,000 Rs. What is the cost of the company on account of leaves (18.2%), group insurance/medical (5%) and other social benefits (5.8%)? Leaves cost = 0.182 x 15000 = 2,730 Rs. Group insurance/medical = 0.05 x 15000 = 750 Rs. Other social benefits = 0.058 x 15000 = 870 Rs. Total social charges = 2730 + 750 + 870 = 4,350 Rs. SUMMARY Summary of different components of salary is as follows: Basic salary Allowances 50 % of basic salary Gratuity 9.09 % of basic salary Provident Fund 9.09 % of basic salary Social Charges 29 % of gross salary Gross remuneration is pay or salary, typically monetary payment for services rendered, as in an employment. It includes. 1. Basic Salary 2. House rent allowance 3. Conveyance allowance 4. Utilities 5. Provident fund 6. Gratuity fund 7. Leaves 8. Group insurance (medical etc) 9. Miscellaneous social charges. Benefits can also include more factors and are not limited to the above list. The purpose of the benefits is to increase the economic security of employees Example 8 The salary of an employee is as follows: Basic salary = 6,000 Rs. 26 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU The calculations are shown in the slide below. There is mistake in calculating gross remuneration in example given below. Total amount of leaves are 19636 which is the amount for 1 year not 1 month. So divide the amount of leaves by 12 and then calculate gross remuneration. Converting fraction to percent Calculate percent by multiplying fraction by 100. and put the percent sign (%) Percent = Fraction X 100 Example 9 Convert 0.1 to percent. 0.1 X 100 = 10% Common Fraction Common fraction is a fraction having an integer as a numerator and an integer as a denominator. For example ½, 10/100 are common fractions. Converting percent into Common Fraction Example 11 20%= 20/100= 0.2 Decimal fraction . Any number written in the form: an integer followed by a decimal point followed by a (possibly infinite) string of digits. For example 2.5, 3.9 etc. Converting percent into decimal fraction 27 © Copyright Virtual University of Pakistan Business Mathematics & Statistics (MTH 302) VU Example 20% = 0.2 Percent 20% or 20/100=0.2 Percentage Percentage is formed by multiplying a number called the base by a percent, called the rate. Thus Percentage = Base x Rate Example 13 What percentage is 20% x of 120? Here, rate = 20% = 20/100 = 0.2 Base = 120 Percentage = 20/100 x 120 Or 0.2 X 120 = 24 Example 14 What Percentage is 6 % of 40? Percentage = Rate X Base = 0.06 X 40 = 2.4 Base Base = Percentage/Rate Example 15 Fin

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