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BACHELOR OF BUSINESS ADMINISTRATION/ BACHELOR OF COMMERCE BUSINESS MATHEMATICS AND STATISTICS 21BBA102 /21BCM102 A.B. Rao

CHANDIGARH UNIVERSITY Institute of Distance and Online Learning Course Development Committee Chairman Prof. (Dr.) R.S. Bawa Vice Chancellor, Chandigarh University, Punjab Advisors Prof. (Dr.) Bharat Bhushan, Director, IGNOU Prof. (Dr.) Manjulika Srivastava, Director, CIQA, IGNOU Programme Coordinators & Editing Team Master of Business Administration (MBA) Bachelor of Business Administration (BBA) Co-ordinator - Prof. Pragya Sharma Co-ordinator - Dr. Rupali Arora Master of Computer Applications (MCA) Bachelor of Computer Applications (BCA) Co-ordinator - Dr. Deepti Rani Sindhu Co-ordinator - Dr. Raju Kumar Master of Commerce (M.Com.) Bachelor of Commerce (B.Com.) Co-ordinator - Dr. Shashi Singhal Co-ordinator - Dr. Minakshi Garg Master of Arts (Psychology) Bachelor of Science (Travel & Tourism Co-ordinator - Ms. Nitya Mahajan Management) Master of Arts (English) Co-ordinator - Dr. Shikha Sharma Co-ordinator - Dr. Ashita Chadha Bachelor of Arts (General)

Master of Arts (Mass Communication and Co-ordinator - Ms. Neeraj Gohlan Journalism) Bachelor of Arts (Mass Communication and Co-ordinator - Dr. Chanchal Sachdeva Suri Journalism) Co-ordinator - Dr. Kamaljit Kaur Academic and Administrative Management Prof. (Dr.) Pranveer Singh Satvat Prof. (Dr.) S.S. Sehgal Pro VC (Academic) Registrar Prof. (Dr.) H. Nagaraja Udupa Prof. (Dr.) Shiv Kumar Tripathi Director – (IDOL) Executive Director – USB © No part of this publication should be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the author and the publisher. SLM SPECIALLY PREPARED FOR CU IDOL STUDENTS Printed and Published by: Himalaya Publishing House Pvt. Ltd., E-mail: [email protected], Website: www.himpub.com For: CHANDIGARH UNIVERSITY Institute of Distance and Online Learning CU IDOL SELF LEARNING MATERIAL (SLM)

Business Mathematics and Statistics Course Code: BBA102 /BCM10 Credits: 4 Course Objectives:  To identify the basic mathematical tools which are used in business.   To develop insights in mathematical concepts towards understanding business problems.   To analyze managerial situations using mathematical concept to formulate problems for synthesis of information. Syllabus Unit 1: Introduction to Business Mathematics: Basic Concepts and Uses. Unit 2: Simple and Compound Interest: Meaning, Basic Concepts and Applications of Simple and Compound Interest. Unit 3: Annuity: Meaning, Basic Concepts and its Applications. Unit 4: Matrices: Introduction, Standard Definitions and Types of Matrices, Matrix Representation of Data and Operations. Unit 5: Determinant: Properties of Determinants, Determinant of 2 × 2 and 3 × 3 Matrices (Direct and Without Expanding), Cramer’s Rule and Inverse Matrix Method.

Unit 6: Permutation and Combinations: Concept of Factorial, Sum Rule and Product Rule, Properties of Permutations and Combinations, Applications. Unit 7: Linear Programming: Formulation of Equation: Graphical Method of Solution; Problems Relating to Two Variables including the Case of Mixed Constraints; Cases Having No Solution, Multiple Solutions, Unbounded Solution and Redundant Constraints. Unit 8: Introduction of Statistics: Meaning, Scope, Importance and Limitations, Applications of Statistics in Managerial Decision-making. Unit 9: Classification and Tabulation of Data: Meaning, Types and Classifications. Unit 10: Measures of Central Tendency: Arithmetic Mean, Mathematical Properties of Arithmetic Mean, Median, Quartiles, Mode, Geometric Mean. CU IDOL SELF LEARNING MATERIAL (SLM)

Unit 11: Dispersion: Range, Quartile Deviation, Standard Deviation, Coefficient of Variation. Unit 12: Sampling: Introduction of Sampling, Difference between Census and Sampling. Unit 13: Probability: Types of Probability and Non-probability Sample. Unit 14: Correlation Analysis: Significance, Types, Methods of Correlation Analysis: Karl Pearson’s Correlation Coefficient, Rank Correlation Coefficient, Properties of Correlation. Unit 15: Regression Analysis: Regression Lines; Probable Error, Relationship between Correlation and Regression Coefficients. Unit 16: Time Series Analysis: Introduction, Trend Analysis (Using Moving Averages Method and Least Square Method only). Text Books: 1. Veerarajan, T. (2016), Discrete Mathematics, New Delhi: Tata McGraw-Hill. 2. Singaravelu (2013), Allied Mathematics, Chennai: Meenakshi Agency. Reference Books: 1. Vittal, P.R. (2017), Allied Mathematics, Chennai: Margham Publications. 2. Venkatachalapathy, S.G. (2007), Allied Mathematics, Chennai: Margham Publications.

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CONTENTS 1–8 9–34 Unit 1: Introduction to Business Mathematics 35–65 Unit 2: Simple and Compound Interest 66–93 Unit 3: Annuities 94 – 116 Unit 4: Matrices 117 – 132 Unit 5: Determinants 133 – 162 Unit 6: Permutations and Combinations 163 – 175 Unit 7: Linear Programming 176 – 198 Unit 8: Introduction to Statistics 199 – 240 Unit 9: Classification and Tabulation of Data 241 – 279 Unit 10: Measures of Central Tendency 280 – 283 Unit 11: Dispersion 284 – 294 Unit 12: Sampling 295 – 330 Unit 13: Types of Probability 331 – 353 Unit 14: Correlation Analysis 354 – 383 Unit 15: Regression Analysis Unit 16: Time Series Analysis 384 References

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Introduction to Business Mathematics 1 UNIT 1 INTRODUCTION TO BUSINESS MATHEMATICS Structure 1.0 Learning Objectives 1.1 Introduction 1.2 Need and Uses of Mathematics 1.3 Application of Mathematics 1.4 Summary 1.5 Key Words/Abbreviations 1.6 Learning Activity 1.7 Unit End Questions (MCQ and Descriptive)

1.8 References 1.0 Learning Objectives After studying this unit, you will be able to:  Explain the importance of mathematics for business data.   Elaborate the various mathematical aspects and in business data.   Study the references of mathematical approach to decision-making in business and also to learn some topics in operations research. 1.1 Introduction Nowadays, many a business concern maintains its own department of statistics, for evaluating its business policies on profits, sales, consumer preferences, manufacturing costs, quality of products, CU IDOL SELF LEARNING MATERIAL (SLM)

2 Business Mathematics and Statistics productivity and consumers good will. Economic planning without accurate and reliable statistics is unthinkable. Statistical charts, graphs and diagrams pertaining to quantitative data, are visual aids and immediately appeal to the eye and render business figures and facts, more significant and easily intelligible. Success in trade and commerce at present, is largely determined by correct estimates and probabilities and favorable forecasts based on statistics collected systematically, presented accurately and interpreted wisely. Whether a person is a banker, an investor, a manufacturer or a stock exchange broker, he would be benefited immensely; if he possesses statistical knowledge relating to monetary savings, rates of interest, diverse investment markets, costs of different raw materials and their markets, stocks and share markets etc. One of the several reasons for the Insurance Companies to make profits is that, with the knowledge of the operation of the Law of Statistical Regularity, of statistics of births and deaths due to various causes — their actuaries calculate insurance premiums. Statistics is also known as the Arithmetic of Human Welfare. Human Welfare is greatly enhanced by agricultural and industrial progress. Statistical techniques are golden keys to industrial progress for modern statistical techniques such as Quality Control, Market Research etc. when properly used by industrial establishments can lead to increased output, improved quality, reduced costs, encouraging profits, saving in wastage of materials and labour and wise decisions in purchase policy and launching of sales and thereby to the promotion of industrial prosperity. In other words the tools of statistical analysis may not lead us to peaks of precision or to depths of illusion, but they do certainly and invariably show us the practical levels of human expectation in quantitative terms. Large scale and as well as small scale industrial establishments can profit much by maintaining statistical departments and employing expert statisticians. Such departments under wise and expert statistical supervision collect, record and analyse data relating to industry, prepare statistical charts, diagrams and maps, design and execute market research surveys, furnish periodical progress reports and tender the most sensible statistical advice to the business executives. Undoubtedly statistics, and statisticians are indispensable aids in trade, commerce and industry.

In the modern times high-speed technology has made it possible to analyse data pertaining to Business, Economics and financial facts and figures systematically, scientifically with accuracy and reliability. CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Business Mathematics 3 Information technology based on high-speed computerisation has led to progressive development in the appropriate interpretation of numerical facts, thereby ensuring the effective usefulness. Business figures relating to capital investments, sales, profits, losses, rates, financial ratios, work, time and distance, percentages, appreciation, depreciation, balance sheets, stocks, shares etc. form the substance of Business Mathematics. The statement ‘Number rules the universe’ can be justified by the impact of computer technology on facts and figures in the modern world of Business, science and operational processes. Mathematical approach to decision making in Business becomes more reliable, and meaningfully significant. Linear Programming, transportation, Models, Assignment Models, PERT, CPM, Queuing Models are effective tools in operations-research and quantitative data. Business Mathematics is an integral part of Commerce and Management courses. Commercial Arithmetic which deals with Rates, Brokerage, Commission, Discount, Dividend, Mensuration, Permutations, Combinations, etc. are all useful to Business Management. 1.2 Need and Uses of Mathematics Mathematics is used in almost every aspect of life. It also plays an important role in business decision making. Business mathematics is used to record and manage business operations. Many Business activities like accounting, financial analysis, sales etc. are done by using business mathematics. When we talk about decision making for business, mathematics plays an important role in it. A businessman has to take thousands of decisions every day for his business, therefore, mathematics helps in rationalizing the information and it gives the accurate results inherent for any decision making.

Mathematics has numerous concepts like statistics, probability, algebra, calculus etc. that are used in business mathematics. These concepts provide accurate statistics solutions for various business activities. Mathematics is an important subject which enhances the knowledge of the person and it also improves problem-solving skills. CU IDOL SELF LEARNING MATERIAL (SLM)

4 Business Mathematics and Statistics 1.3 Application of Mathematics Mathematics is used in almost every field of daily life. Business involves the buying and selling of goods in order to earn profit, it uses mathematics to record, classify, summarize and analyse the business transactions. So mathematics is used by commercial enterprises to record and manage the business operations such as, elementary arithmetic involving fractions, decimals, percentage, elementary algebra, statistics and probability. Now a days business management is using advanced mathematics such as calculus matrix algebra and liner programming. Practical applications include checking accounts, forecasting the sales, price discounts, mark-ups, mark- downs, payroll calculations, simple and compound interest, reducing wastage of resources. Some applications of mathematics in business and commerce are listed below: 1. Algebra 2. Operation Research 3. Statistics and Probability 4. Calculus 5. Matrix and Linear Algebra 2.1 Algebra Mathematical principles are needed to study accounting. It incorporates successful exploration of numerical, geometrical and logical relationships. Mathematics benefits accountant in comparison — mathematical formulas help business and commerce to compare income, cost, expenses and profits. The various formulas are derived using various percentage, ratios and equations. The various ratios are derived such as: inventory turnover ratios, profitability ratios, debtor turnover ratio, debt-equity ratio etc. Mathematics is helpful in deriving accounting equation. The basic idea in accounting is that

total wealth of business is called Assets. There are two possible claims on assets (A) called liabilities (L) and capital (C). By using mathematical relation A = L + C, accountants use mathematics in order to arrive the total cost and taking decision regarding manufacturing or buying the product. The total cost formula for business is T = a + bx; where ‘T’ is total cost ’a’ is fixed cost, ‘b’ is cost per unit produced and ‘x’ is no. of units produced. Also profits are determined by subtracting total cost from total revenue and helps in analysing the financial health of business and prices are CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Business Mathematics 5 determined by adding some markup to cost. So accountant used addition and percentage to determine the prices of product. 2.2 Operational Research (OR) OR is concerned with determining the maximum (profit, performance, yields) or minimum (loss, cost, risk) of some real world objectives. OR includes game theory, linear programming formulation techniques, PERT, CPM, transportation problems. Linear programming also called linear optimization is a method to achieve the best outcome (as maximum profit or lowest cost or ensuring best use of available resources) in a mathematical model whose requirements are represented by linear relationship. Some of industries that use LP model include transportation, energy, telecommunication and manufacturing. Linear function to be maximized by mathematical function. 2.3 Statistics and Probability Statistics is very indispensable for the businessman. It formulates various plans and policies and forecasts trends of future such as change in demand, market fluctuations using statistical techniques. On the other hand, future events are uncertain and to predict these uncertainties, probability is an effective tool to forecast sales, scenario, future returns and risk evaluation in the business world. Before introducing the product, team of market research analyse data relating to population, income of consumer, tastes, preferences, habits, pricing policy of competitors by using various statistical techniques. We can collect and analysis the data in the field of economy by using statistical methods. Probability theory serves as a useful tool for decision making, estimating number of defective units, sales expected and also in business policies. Through the use of statistical (regression) techniques Levine and Zervos (1998) attempted to find the empirical relationship between various measures of stock market development, banking development, and long-run economic growth. They concluded that even after controlling for many factors associated with growth, stock market liquidity and banking development are both positively and robustly correlated with contemporaneous and future rates of economic growth, capital accumulation, and productivity growth. The small business firms especially those in the fashion industry should learn and apply probability theory since there line of business was

more prone to chance occurrences (Orga, &Ogbo, 2012). In their study in Nigeria they observed that the small business firms fail despite the programmes of government directed at their survival. The application of probability theory in small business was examined to find the implications and in restoring the gap between the rich and the CU IDOL SELF LEARNING MATERIAL (SLM)

6 Business Mathematics and Statistics poor through better and informed decisions. The findings indicated that probability theory has wide application in small business firms; probability shows specificity in business situations and is inevitable in this era of information overload caused by ICT. In nutshell, statistics and probability are very useful in taking various decisions relating to material, production, finance, personnel and marketing in an Industry. 2.4 Calculus Calculus is another branch of mathematics made up of two fields — differential calculus and integral calculus. Differential calculus plays valuable role in management and business for decision making in production (e.g., supply of raw material, wage rates and taxes). In calculus, the case when ‘y’ is a function of ‘x’ or we can say one variable (y) is dependent on other variable (x) and the derivative of ‘y’ w.r.t. ‘x’, i.e., dy/dx measures the change of variable ‘y’ w.r.t. change in variable ‘x’. Derivative enables a firm to make important production decisions. It is also called marginal function. Demand can be assumed as a function of price. This operator is also helpful in calculating minimum cost and maximum profit. Also total cost of production and marketing depends on no. of units in mathematical relations, which can be described as c(x) = F + v(x), where c(x) is cost function v(x) is variable cost and F is fixed cost. Revenue function R(x) can be represented as R(x) = xp(x), where ‘x’ is no. of units and ‘p’ is rate per unit. Hence, knowledge of derivative is essential for understanding the economic relations. Another integral operator is used to calculate the total revenue in case of marginal revenue is given. So calculus plays a vital role in taxes, profit and revenue calculations which are very important for any business. 2.5 Matrix and Linear Algebra Matrices play prominent role in developing a solution required for commercial organizations. It has knowledge to deal with unique needs of various sectors of Industry. It gives opportunities to finance and logistics management and customer relationship by providing them a variety of solutions. Also product price matrices are helpful to set bulk purchase discount. Determinants and Cramer rule

are helpful in problem solving related to business and economy. It enables oneself in obtaining and optimal solution to maximize profit or minimize cost problems. Linear algebra serves a purpose of powerful tool for its application in business. As total cost, revenue, supply, demand and population are all related with a system of linear equations. Leontief (1987) derived a production equation in input- output analysis and got Noble prize for his contribution. The model given by him was X = CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Business Mathematics 7 CX + d, where x is the production factor, c is consumption matrix and d is demand vector. If matrix I – C is invertible then appropriate production for a given final demand can be computed directly via X = (I – C)–1 d. This basic input-output analysis however is a very powerful tool (Miller & Blair, 2009). It can Predict what happens to an economy when final demand changes. By changing the consumption matrix this can represent what happens to an economy when the relative cost in terms of other goods (a change in one or more entries in internal demand) of producing one good can change both internal and final demand economy. Dyck & Sumaila (2010), has applied the Leontief technological coefficients at total current impact of the fisheries sector at current production and then estimate total output supported throughout the economy at the current level of production. They recognized that the non-linearity of fisheries production could cause problems when doing predictions at various levels of production. 1.4 Summary Mathematics and statistics are of much relevance in the study of business data relating to manufacturing costs, consumer preferences, sales, profits, quality of products progress in trade and commerce can be known through estimates and probabilities. Rates, brokerage, commission, dividend permutation, combinatin are significant in business mathematics. 1.5 Key Words/Abbreviations Business policies, consumer preferences, estimates, probabilities, discounts, manufacturing costs, stock exchange, brokerage, decision-making, linear programming, quality control, PERT, CPM.

1.6 Learning Activity Study the progressive aspects of a business concern, by using percentages, rates, ratios in the interpretation of sales, profits, costs, consumer preferences during the last 3 years. ............................................................................................................................................................ ............................................................................................................................................................. CU IDOL SELF LEARNING MATERIAL (SLM)

8 Business Mathematics and Statistics 1.7 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions 1. Describe briefly the scope and usefulness of Business Mathematics in trade commerce and industry. 2. Explain the significance of Business Mathematics with reference to financial management. 3. Explain the usage of mathematical methods with reference to ‘decision-making’ in Business. 4. Describe briefly the use of certain mathematical techniques in certain spheres of business activity. 5. Discuss the statement ‘Business Mathematics provides accuracy to statements of facts and figures whereas statistical calculations are supportive estimates based on probability.’ 6. “Mathematical methods if used properly, ensure reliability to business transactions.” Discuss. 7. Write an explanatory note on; Use of Mathematics tools in ‘Decision-making’. B. Multiple Choice/Objective Type Questions 1. Business mathematics is concerned with the following except ________. (a) Sales and profits (b) Financial ratios (c) Capital investments (d) State capital (e) Commission and brokerage 2. In which one of these environments the usage of business mathematics is most significant.

(a) Sound environment (b) Political environment (c) Economic environment (d) Business environment (e) Culural environment Answers: (1) (d); (2) (d). 1.8 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 9 UNIT 2 SIMPLE AND COMPOUND INTEREST Structure 2.0 Learning Objectives 2.1 Introduction 2.2 Simple Interest 2.3 Present Worth and True Discount 2.4 Banker’s Discount 2.5 Average Due Date 2.6 Compound Interest 2.7 Summary

2.8 Key Words/Abbreviations 2.9 Learning Activity 2.10 Unit End Questions (MCQ and Descriptive) 2.11 References 2.0 Learning Objectives After studying this unit, you will be able to:  Elaborate the meanings and the methods of calculation of S.I. and C.I.   Analyse the use and the procedure of calculation of the present worth, true discount, banker’s discount.  Work out various types of examples.   Explain the significance of these in practical business. CU IDOL SELF LEARNING MATERIAL (SLM)

10 Business Mathematics and Statistics 2.1 Introduction The money that is paid for the use of money that is taken as a loan is known as ‘Interest’. The person giving the loan is the money lender and the person taking it is the borrower. The money lender lends money in consideration of which the borrower makes a payment. The sum of money that is borrowed is called the ‘Principal’ and the money that is paid for its use is called ‘interest’. The interest that is charged for a loan of ` 100, for one year, is called the ‘Rate percent per annum’. The total amount that is due from the borrower at any time is: = Principal + Interest due till that time i.e., Amount = Principal + Interest Notations: P = Principal I = Interest accrued r = Rate of interest per annum n = Time in years A = Amount due i = r/100 = rate of interest per annum per unit of money (1 rupee)

2.2 Simple Interest Interest is said to be ‘Simple’ if it is not added to the Principal at the end of each year, but it accumulates in proportion to the time or period of the loan. In other words the ‘Principal’ is fixed but the interest accrues in proportion to time. Calculations Explained: (i) Given P, n, r to find I Pnr Procedure: I = 100 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 11 (ii) Given I, n, r to find P Procedure: P = 100 uI nr (iii) Given I, P, N; to find r Procedure: r = 100 uI Pn (iv) Given P, n, r; to find A § Pnr · § nr · Procedure A = P ¨ ¸ P¨1 + ¸ ©100 ¹ © 100¹ ? A = P (1 + ni) where i = r/100 = interest on ` 1 for 1 year. Remark: If the time period = D days Illustrative Examples: then n =

365 D 1. Find the S.I. on ` 2,800 for 5 years at 8% per annum. Pnr 2800  5  8 I = 100 = 100 = ` 1,120 2. Find the rate per cent at which Rs.4600, would give ` 1,150 as interest, in 4 years. 100×I 100  1,150 25 r = Pn = 4,600  4 = 4 = 6¼% 3. Find the principal that is necessary to give ` 560 as interest in 3½ years at 5% S.I. per annum. 100×I 100 q 560 P = nr = 3,5 q 5 = ` 3,200 4. A sum of money, lent out at simple interest, amounts to ` 2,520 in 2 years and to ` 2,700 in 5 years. Find the sum of money and the rate of interest. [N.D.A.] CU IDOL SELF LEARNING MATERIAL (SLM)

12 Business Mathematics and Statistics Solution: = ` 2,700 In 5 years the amount In 2 years the amount = ` 2,520 ? In 3 years S.I. = ` 180 2 In 2 years S.I. = ` 180 × 3 = ` 120 ? Principal = Amount in 2 years – S.I. in 2 years = ` (2,520 – 120) = ` 2,400 Rate of Interest = 120  100 2400  2 = ` 2½ Rate of S.I. is 2½%. 2.3 Present Worth and True Discount Suppose a person ‘A’ owes a definite sum of money to another ‘B’ and the payment of that money would be due on some future date. In the mean time, if ‘A’ desires to make an immediate payment of the debt in full then he would be entitled to make a deduction from the amount due, because he makes an immediate payment. The money so deducted is known as ‘discount’ and the

remaining amount that is paid after such a deduction is known as the ‘Present Value’ or ‘Present Worth’ of the debt. Usually, in the case of ‘Bills’ such discounts calculated on percentage basis are allowed in consideration of an immediate cash payment. This discount is called ‘true discount’. Thus if ‘A’ is the amount of a bill that would be due for payment after a period of n years and the rate of interest for one rupee for the purpose of discount is ‘i’ then CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 13 A P.W. = 1 ni where P.W. = Present Wroth A  r = i  1 nr /100   100 A  100 = 100 + nr 100 ? P.W. = Debt × 100 + nr We note the following: (1) Debt P.W. (100 + nr) = 100 nr (2) Discount = P.W. × 100 or Discount = Debt – P.W. 100 = Debt – Debt × 100 + nr § 100 · ¨1 ¸ = Debt 100 + nr ¹ ©

§ 100 + nr100 · ¨ ¸ = Debt © 100 + nr ¹  nr  = Debt    100 + nr  nr  =A   100 + nr Ani = 1 + ni  A =   ×n×i  1 + ni  CU IDOL SELF LEARNING MATERIAL (SLM)

14 Business Mathematics and Statistics = (P.W.) × n × i = interest on Present Worth The following facts should be noted well:  ‘Debt’ corresponds to ‘Amount’.   ‘True Discount’ corresponds to ‘Interest’.   ‘Present Value’ or ‘Present Worth’ corresponds to ‘Principal’.   ‘Rate of Discount’ corresponds to ‘Rate of Simple Interest’. Now-a-days in practical business calculations ‘true discount’ is not of much use. 2.4 Banker’s Discount The interest that a Banker charges for discounting a Bill before the date on which it falls due is known as ‘Banker’s Discount’. It is obtained by calculating the interest on the face value of the bill for the remaining tenure of the bill. Banker’s Discount = Ani where A = face value of the bill n = number of years after which it is due i = rate of interest The present Value or the Present Worth of a bill is: = (face value of the bill) – (Banker’s Discount) = A – Ani

= A (1 – ni) The gain of the Banker in discounting a bill as per this method, is equal to the excess of this ‘Discount’ over the ‘True Discount’. That is, Banker’s Gain = B.D. – T.D. Ani = Ani – 1 + Ani A(ni)2 = CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 15 Ani = 1 ni × ni = Interest on True Discount 2.5 Average Due Date If a number of bills for amounts a1, a2 , a3, ........ are due after n1, n2, n3, ........ days but it is felt necessary to cash these bills on one and the same day for the total a1 + a2 + a3 + ......... of the different amounts of the bills, then the date on which all these bills could be cashed is called the ‘Average Due Date’. If N denotes the number of days after which the full value of the bills would be due, then the following equation enables us to determine the ‘Average Due Date’. a n +a n + a n + ........ N= 1 1 22 33 a1 + a2 + a3 + .............. Explanation: The above equation is derived as follows: The P.V. of the total of amounts a1 + a2 + a3 + …… = (a1 + a2 + a3 + ........) – (a1 + a2 + a3 + .........)ni where ‘I’ is the interest per rupee per day. But the present values of the different bills are respectively. a1 – a1n1i

a2 – a2n2i a3 – a3n3i .............. .............. ? (a1 + a2 + a3 + ……) – (a1 + a2 + a3 + ……)ni = (a1 + a2 + a3 + ……) – (a1n1 + a2n2 + a3n3 + ……)i ? (a1 + a2 + a3 + ............) NI = (a1n1 + a2n2 + a3n3 + ……)i a1n1 + a2 n2 + a3 n3 + … ?n= a2 + a2 + a3 + … CU IDOL SELF LEARNING MATERIAL (SLM)

16 Business Mathematics and Statistics Illustrative Examples Example1: A radio-set is offered for ` 950 cash or ` 1,000 on instalments on the following terms: ` 100 cash down and the balance in 9 equal monthly instalments. What is the average rate of simple interest charged? Solution: Since the first payment is ` 100, the payment of the balance, i.e., ` 1,000 – ` 100 = ` 900 may be considered as the repayment of a loan of ` 900 in 9 monthly instalments of ` 100 each. The equated time of payment is obtained from: 900 × t = 100 (1 + 2 + 3 +........ + 9) 100  9 (10) ? t = 900 2 =5 The interest is ` 50, therefore nr I = p × 12 × 100 5r 50 = 900 ×12 × 100 50  12  100 ?r= 900  5

= 120/9 = 13.33% Example 2: A person desires to create an endownment fund to provide for a prize of ` 300 every year. If the fund can be invested at 4% per annum, find the value of the endownment. Solution: The amount of endownment fund = ` 300/0.04 = 100 × 300/4 = 7,500 Ans.: ` 7,500 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 17 Example 3: A person undertakes to pay back a loan of ` 8,000 in monthly instalments of ` 200 plus interest at 12% on the outstanding balance. What is the average rate of interest earned by the money lender? 8,000 Solution: The number of instalments = 200 = 40 The equated time t = 200 (1 + 2 + 3 + ........ + 40) 200  40 = 41/2 = 20½ months. ? The interest on ` 8,000 for 20½ months at 12%. 8000  0.12  41 = 12  2 8000  12  41 = 100  12  2 = 40×41 = ` 1,640 If the interest on ` 8,000 for 40 months is ` 1,640, then the rate of interest is 1640  100  12 123 = = 6.15%

800  40 20 Note: In this case, the amount of each instalment would be: 8000 1640 = 9640 40 40 =`241 Example 4: The cash price of a Pressure Cooker is ` 240. A customer desires to take it on the basis of 12 equal monthly instalments. The seller agrees provided the customer pays 14% simple interest. What is the value of each instalment? Solution: Let an instalment be denoted by p, then the toal amount of 12 instalments would be 12p. An instalment of p for 12 months (including the 1st payment) is equivalent to a single payment of 12p at the end of: p (1 + 2 + 3 + ......... + 12) = 6½ months 12p CU IDOL SELF LEARNING MATERIAL (SLM)

18 Business Mathematics and Statistics Interest on ` 240 @ 14% p.a. for 6½ months is:  240  14  13  = `   12  100  2  182 = ` 10 = ` 18.20 ? The total amount payable in 12 instalments = ` 240 + ` 18.20 = ` 258.20 ? The value of each instalment = ` 258.20 12 = ` 21.52 Example 5:10% Debenture Stock purchased at 108 in 1990 is due to be paid off in 2000 at ` 105 for every 100 stock. What is the return to the investor on the basis of simple interest? Solution: Premium = ` 5 Interest = ` 100 Total return = ` 105

? The purchaser gets ` 105 in 10 years on an investment of ` 108 ? The rate of interest received from the investment is 105  100 = 108  10 = 9.44% p.a. Example 6: A money lender advances ` 8,400 on the condition that a sum of ` 9,000 should be paid back at the end of a year. Find the return on the lender’s investment. Solution: Interest earned = ` 9,000 – ` 8,400 = ` 600 Interest on ` 8,400 for 1 year is ` 600, therefore rate of interest is 600  100 = = 7.14% p.a. 8400 CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 19 Example 7: A certain sum of money is deposited with a Banker at a specified rate of interest. An amount of ` 480 was withdrawn at the end of the first year. Again at the end of the second year ` 260 was withdrawn and then the sum remaining at credit was ` 1,000. If no withdrawals had been made the sum deposited would have earned an annual simple interest of ` 80. What is the original sum deposited and the rate of interest? Solution: Let the original amount deposited be ‘p’ and ‘r’ the rate of interest, then … (1) pr = 80 … (2) and [ p (1 + r) – 480] (1+ r) – 260 = 1000 80 Putting r = p in equation (2), we get · ºª 80 º ª § 80 p¨1 ¸480 1 1260 « »« » ¬ © p¹ ¼¬ p ¼ ? (p – 400) (p + 80) = 1260 p p2 – 1580p – 32000 = 0 (p – 1600) (p + 20) = 0 ? p = 1600 The sum deposited is ` 1,600 80 Rate of interest = 1600

= 0.05 The rate of interest is ` 5% p.a. Example 8: A loan of ` 8,500 has to be repaid in monthly instalments of ` 200 each. What is the rate of interest charged? Solution: Equating these instalments of ` 200 each for 50 months, with a single payment of ` 8,500 at the end of the period t, where t is the period of the average due date, we get 200 (1 + 2 + 3 + ........ + 50) = 10000t 200  50 (50 + 1) 10000  2 =t CU IDOL SELF LEARNING MATERIAL (SLM)

20 Business Mathematics and Statistics 51 ?t= 2 Interest charged = ` 1,500 t ? 1,500 = Pnr where n = 12 1500 r = Pn 1500 1500 = =  8500  51 p (t/12)   2  12  2400 = 289 Rate of interest = 8.35% Example 9: A building society advances ` 35,000 to be repaid in 15 equal annual instalments 1 along with Simple Interest 6 4 % p.a. Find the amount of each instalment. Solution: Let p = the annual instalment

The ‘average due date’ is px1 + px2 + ........ + px15 = 15p 1 + 2 + ........ + 15 = 15 15  16 = 2  15 = 8 years Interest due on ` 35,000 in 8 years @ 6¼% per annum is: 35000 25 = 100 × 4 × 8 = 17,500 Total sum to be paid = 15p CU IDOL SELF LEARNING MATERIAL (SLM)

Simple and Compound Interest 21 = 35,000 + 17,500 = ` 52,500 Each annual instalment = p 52,000 = 15 = ` 3,500 Example 10: A bill for ` 5,460 was drawn on 12th June 2007 at 6 months date and discounted on 17th July 2007 at the rate of 5%. Find the Banker’s Discount in this bill. Solution: Bill drawn on = 12-06-2007 = 12-12-2007 Adding 6 months = 15-12-2007 = 15-12-2007 Adding 3 days of grace = 17-07-2007 = 151 days Bill matures on Days Bill discounted on 14 31 Remainder of the tenure 30 31 Working Month 30 July Aug. Sept. Oct. Nov.