Business Mathematics

Business Mathematics is a comprehensive, application-driven textbook that presents the complete quantitative toolkit a commerce student needs—matrices and determinants, differential and integral calculus, mathematics of finance, and linear programming—through the lens of real business, economic, and financial decision-making.

The book rests on a clear premise: successful business professionals are those who can make quick, informed decisions by applying mathematical reasoning to complex, dynamic situations. Understanding what customers need, deciding what to produce and sell, and determining optimum quantities are all, at bottom, mathematical questions. Accordingly, no technique arrives as abstract theory. The authors organise the entire treatment around three domains of application:

• Business — Investment decisions, return analysis, sales and profit determination, cost allocation, production capacity utilisation, pricing strategy, commission calculation, product requirement planning, transportation logistics, and optimisation of sales mix and production systems
• Economics — Equilibrium pricing, elasticity of demand and supply, income determination, demand and supply analysis, production and cost functions, revenue assessment, and consumer and producer surplus
• Finance — Effective interest rates, equated payment times, annuities, depreciation schedules, loan amortisation, bond valuation, and sinking funds.

What sets the book apart in a crowded field is a structural decision that runs through almost every chapter: each solved example is mirrored by a same-numbered end-of-chapter exercise that poses the identical task with different data. The pattern holds across the book—which is why 11 of the 14 chapters carry exactly as many exercises as examples. The student reads the method worked through in full, then immediately performs it unaided on parallel data, with an answer supplied for comparison.

Behind that architecture sits a substantial body of worked material: 493 fully solved examples and 486 numbered end-of-chapter exercises, every exercise with an answer key and most carrying multiple sub-parts. 294 problems are tagged to actual University of Delhi examination papers, drawn from 42 distinct sittings between 1981 and 2026—and, importantly, 254 of those 294 sit on the solved examples, so the student sees the real exam question worked end to end rather than merely set as practice.

Written in deliberately clear, accessible language, the book is aimed squarely at beginners and at students who have historically found mathematics intimidating. Concepts are developed step by step and consistently anchored to Indian, rupee-denominated business contexts—so students learn not merely to compute, but to read what the computation says about the decision at hand.

The book is structured to align with the B.Com. (Hons.) and B.Com. syllabi under the UGCF–NEP 2020 framework of the University of Delhi; both syllabi are reproduced in full in the front matter. It is equally suited to comparable curricula at other central universities in India and abroad.

This book is intended for the following audience:

• B.Com. (Hons.) Students, University of Delhi—Semester III, DSC 3.1: Business Mathematics. The book's five academic units map one-to-one onto this syllabus, and all 294 tagged past-paper questions are drawn from B.Com. (Hons.) sittings
• B.Com. Students, University of Delhi—Semester V, DSE 5.4: Business Mathematics, referred to in the Preface as B.Com. (Prog.). This syllabus is largely a subset of the Hons. paper, so its topics are covered within the same chapters, with additional material (partial differentiation, annuities, simplex) available for students who want to go further
• Students at Other Central Universities in India and Abroad following comparable Business Mathematics curricula
• Students of Commerce, Economics, Management, and Allied Disciplines needing a working command of quantitative techniques
• Beginners and Mathematics-averse Learners—the authors state explicitly that the book is designed for students who find mathematical concepts challenging, and the matched-pair example/exercise design directly serves that reader
• University of Delhi School of Open Learning (SOL) Students—several questions carry UGCF/NEP SOL tags from 2023–2025 sittings
• Candidates for Professional and Competitive Examinations requiring grounding in business mathematics
• Faculty Seeking a Classroom Text whose sequencing, learning outcomes, and exercise design map onto the prescribed syllabus—the authors credit continued feedback from Delhi University Business Mathematics faculty in shaping the book

The Present Publication is July 2026 Edition, authored by S.K. Sharma & Gurmeet Kaur, with the following noteworthy features:

• [Learn It, Then Do It | A Matched-Pair Learning Design] Each solved example is shadowed by a same-numbered exercise posing the same task on fresh data. Read the worked method, then reproduce it unaided and check your answer. This is the book's defining structural feature and the reason its example and exercise counts align chapter for chapter
• [Worked in Full | 493 Fully Solved Examples] Numbered continuously within each chapter and weighted toward the application-heavy topics
• [Practice with Answers | 486 Graded End-of-Chapter Exercises] Every one carries an answer key, and questions carry multiple sub-parts, so the true problem count runs well beyond the numbered total. Exercise sets are organised under thematic sub-headings
• [45 Years of the Paper | 294 University of Delhi Past-Paper Questions, Tagged by Year] Spanning 42 sittings from 1981 to 2026, with 254 of them worked in full as solved examples
• [NEP-aligned Outcomes | Learning Objectives Open All 14 Chapters] Written as measurable, outcome-based statements using Bloom-style action verbs (Define, Compute, Apply, Analyse, Evaluate, Interpret), consistent with NEP's learning-outcomes orientation
• [Revision-ready | KEY FORMULAS Boxes] Closing the six most formula-dense chapters—Input-Output Analysis, Differentiation, Integration, Applications of Integration, Rates of Interest, and Annuities—each consolidating the chapter's essential formulas after the exercises
• [Business Context Throughout | Practical Applications in a Candid and Lucid Fashion] Examples are drawn from recognisable Indian business situations
• [Visual Intuition | 62 Numbered Figures Across Eight Chapters] Concentrated where seeing it matters most: Annuities (18 timeline diagrams), Application of Differentiation (16 cost/revenue curves), the spreadsheet chapter (10), and Application of Partial Differentiation (7)—supported by uncaptioned graphs (LPP feasible regions) and 56 worked simplex tableaux
• [Nothing Skipped | Step-by-Step Solutions] Full working shown at every stage rather than assumed
• [Beyond the Number | Answers That Interpret, Not Just Compute] In the LPP chapters especially, answer keys carry the full economic reading: optimal product mix, maximum revenue, shadow prices, which centre to expand and why, which resources have unutilised capacity, and whether the solution is degenerate with reasons
• [Practical-ready | A Dedicated Spreadsheet Practicals Unit (Unit VI)] 43 pages documenting 18 Excel functions argument by argument, plus Solver
• [Value-added Coverage | Content Beyond the Prescribed Minimum] Gauss Elimination and Gauss-Jordan Elimination are taught alongside the syllabus-mandated Matrix Inversion and Cramer's Rule, and Chapter 3 covers the closed Leontief model as well as the open model the syllabus requires
• [No Extra Books Needed | Self-Contained Appendices] A how-to on reading logarithmic tables, plus seven numbered tables: common logarithmic, anti-logarithmic, exponential, FVIF, PVIF, FVIFA and PVIFA. Nothing beyond the book is needed to work every exercise

The coverage of the book is as follows:

• Unit I — Matrices and Determinants
• Chapter 1 · Matrices and Matrix Algebra
• Introduction to matrices and order (rows × columns); types—rectangular, square, triangular, diagonal, identity, scalar, null, symmetric, and vectors; algebra of matrices—equality, addition, subtraction, multiplication, integral powers; determinants, minors and cofactors for 2×2 and 3×3; transpose, adjoint and inverse with properties; linear equations in matrix form; Matrix Inversion Method; Cramer's Rule; Gauss Elimination and Gauss-Jordan Elimination
• Chapter 2 · Application of Matrices to Business and Economics
• Applications from simple matrix operations—resource requirement, total cost, payroll, revenue and profit, and depreciation (including a worked sum-of-the-years'-digits vs straight-line comparison). Applications from systems of linear equations—investment and return analysis, sales and profit determination, cost of production, production capacity, selling prices and sales commission, product requirement and usage, transportation capacity and rates, equilibrium prices, resource allocation and product-mix
• Chapter 3 · Input-Output Analysis
• Leontief's Input-Output Model; closed and open models; transaction, technology and input-output matrices; determination of gross output, consumer (final) demand, equilibrium prices in two- and three-sector economies, and primary input requirement; rewriting the transaction matrix for new output; value added; effects of a change in final demand on gross output, primary input and value added
• Unit II — Calculus I
• Chapter 4 · Differentiation
• Interpretation of the derivative (geometric and economic); rules—constant, variable w.r.t. itself, power, sum/difference, product, quotient, chain; derivatives of implicit, inverse, exponential and logarithmic functions; successive (higher-order) differentiation
• Chapter 5 · Application of Differentiation
• Marginal analysis; average vs instantaneous rate of change; increasing and decreasing functions; local maxima and minima; cost functions, average–marginal cost relationship, minimum total cost; revenue functions, TR–AR–MR relationship, maximum total revenue; elasticity of demand and supply, the MR–AR–point-elasticity relationship, income elasticity; monopolist's optimum price and quantity; Economic Order Quantity; optimal trade-in time for replacement
• Unit III — Calculus II
• Chapter 6 · Partial Differentiation
• Bivariate functions; first- and second-order partial derivatives; Young's Theorem; techniques of partial differentiation; homogeneity and Euler's Theorem; total differentials, first and second order; differentiation of implicit functions; bivariate optimisation with a stepwise procedure for local extrema
• Chapter 7 · Application of Partial Differentiation
• Nature of commodities; partial marginal demand and partial elasticities of demand; price elasticity of supply; production functions—marginal product of factors, returns to factors, production elasticities, homogeneous production functions and returns to scale, properties of the linear production function, Cobb-Douglas production function; isoquants, MRTS, elasticity of technical substitution; bivariate optimisation—joint cost minimisation, joint profit maximisation, revenue maximisation; price discrimination
• Chapter 8 · Integration
• Indefinite integration, properties and rules; methods—substitution, by parts, partial fractions; definite integration
• Chapter 9 · Applications of Integration in Business and Economics
• Demand analysis and derivation of the demand function (including from a given elasticity); cost function from marginal cost via indefinite and definite integration; revenue from marginal revenue; profit functions and break-even points; consumer surplus and producer surplus; rate of natural resource consumption; rate of cost savings
• Unit IV — Mathematics of Finance
• Chapter 10 · Rates of Interest: Compounding and Discounting
• Simple and compound interest; discrete, continuous (perpetual) and dynamic-rate compounding; nominal–effective rate relationship; discounting and discount rates; depreciation of assets; equation of value; average due date
• Chapter 11 · Annuities
• The book's most heavily worked chapter. Ordinary annuity, annuity due, deferred annuity, annuity with continuous compounding using integration, perpetual annuity—present and future values throughout; capital expenditure and leasing applications for each type; amortisation of loans (flagged ‘Only for Practicals’); valuation of bonds and debentures; sinking funds
• Unit V — Linear Programming
• Chapter 12 · LPP Formulation and Graphical Solution
• Assumptions of linear programming (including when integer programming is required instead); steps for solving LPPs; formulation—general form, summation notation, matrix notation, standard form; graphical method for maximisation and minimisation; exceptional cases—multiple optimal solutions, unbounded solution, infeasible solution, redundant constraints; applications in business and economics. Every one of its 23 examples has a matched exercise—the tightest pairing in the book
• Chapter 13 · Simplex Method for LPP
• The book's longest chapter. Terminology; basic principles; the simplex procedure; maximisation with all slack variables; ties in incoming and outgoing variables; artificial variables and the Big-M (Penalty) Method; minimisation; recovering the LPP from an initial simplex tableau; economic interpretation of the final tableau—opportunity cost and shadow prices of resources; exceptional cases—multiple optima, unboundedness, infeasibility, degeneracy
• Unit VI — Spreadsheet Practicals
• Chapter 14 · Practicals in Business Mathematics
• Matrices and Determinants — MDETERM, MMULT, MINVERSE, TRANSPOSE, MUNIT, plus Named Arrays (Formulas ▸ Define Name) for reusable matrices. Each documented with syntax, required arguments, accuracy notes and error conditions (when #VALUE! is returned and why). Solving systems by Matrix Inversion and Cramer's Rule in a spreadsheet
• Mathematics of Finance — FV, PV, PMT, PPMT, IPMT, ISPMT, CUMIPMT, CUMPRINC, RATE, EFFECT, NOMINAL, NPER, NPV, each with a full argument breakdown and the sign/units conventions (cash paid out as negative; rate and nper kept consistent). Worked computation of future value, present value, loan amortisation schedules and sinking funds
• Linear Programming with Solver — activating Solver, understanding its parameters, a stepwise solution procedure, and worked maximisation and minimisation cases
• Appendices
• Appendix A
• Finding logarithmic values from common logarithmic tables (a how-to, not just the table)
• Appendix B
• Table 1 Common Logarithmic · Table 2 Anti-Logarithmic · Table 3 Exponential Values · Table 4 FVIF · Table 5 PVIF · Table 6 FVIFA · Table 7 PVIFA

Each chapter runs to a fixed five-part pattern, so students always know where to look:

• Learning Objectives — Outcome-based statements of what the student will be able to do
• Concept Development — Plain-language exposition in short numbered sub-sections
• Solved Examples — Worked in full, tagged where drawn from a Delhi University paper
• Exercises — One set per chapter, thematically sub-headed, matched to the examples, every question answered
• Key Formulas — Included in the six formula-dense chapters, a consolidated revision sheet

About the Author

S.K. Sharma

Dr S.K. Sharma is a former senior faculty member of S.G.T.B. Khalsa College, University of Delhi. He completed his B.Com. (Hons.) at Shri Ram College of Commerce and his M.Com., M.Phil. and PhD at the Department of Commerce, Delhi School of Economics, University of Delhi.

Across a long teaching career, he has taught Accounting, Finance, Computers and Quantitative Techniques—including statistics and mathematics—to B.Com. (Hons.) and B.B.E. students. At the postgraduate level, he has taught computer and finance subjects on the M.Com. and MHROD programmes at both the North and South Campuses of the Department of Commerce, and served for several years as visiting faculty to the Department of Computer Science, University of Delhi. Under the Colombo Plan, he was deputed to Sherubtse College, Kanglung, and to Gedu College of Business Studies, Royal University of Bhutan.

He has co-authored books on Business Analytics, Business Statistics, Business Mathematics and Statistics, Fundamentals of Financial Management, Fundamentals of Investment, and Applications of Computers in Business, and is the author of Computerised Accounting System. He has also contributed chapters to a volume on Database Design for Accounting published by the NCERT.

Gurmeet Kaur

Prof. (Dr) Gurmeet Kaur began her teaching career in 1998 at S.G.T.B. Khalsa College, University of Delhi, and earned her doctorate from the Department of Commerce, Delhi School of Economics, University of Delhi. She is currently a senior faculty member at Daulat Ram College, University of Delhi.

Across nearly three decades, she has taught Business Mathematics, Business Statistics, Business Analytics, Finance, Investment and Marketing. As a Fellow at the Institute of Lifelong Learning, University of Delhi, she contributed to developing more than 50 e-lessons and conducted capacity-building workshops for MOOCs. She was selected for the ICSSR Fellowship, has completed research projects for leading institutions, and has supervised doctoral students to completion.

Her research appears in Scopus-indexed, Web of Science, ABDC, ESCI, SSCI and UGC-CARE journals and has attracted some 1,100 citations, particularly for her work on green marketing. She has co-authored widely used textbooks on Business Statistics, Business Mathematics, Fundamentals of Investment, and Business Mathematics and Statistics.