Review & Problem-Solving: Mastering Business Math Techniques

TAKEAWAYS:

• Problem-solving frameworks: Learn to classify problems (LP, calculus, financial math) and select the right method (graphic/simplex, integration/differentiation, TVM formulas).
• Visual thinking: Master graphical methods (feasible regions, indifference curves) and interpret business data through charts and diagrams.
• Real-world connections: See how optimization, interest calculations, and dynamic systems apply to eSewa transactions, Daraz logistics, and NEPSE stock trends.
• Exam strategies: Avoid common pitfalls in simplex tables, constraint handling, and unit conversions through structured templates and error-checklists.
• Integration of concepts: Combine differentiation, integration, and linear algebra to solve complex business scenarios (e.g., profit maximization with constraints).
• Time management: Prioritize problem types by difficulty and marks, using the "3-minute rule" for quick solvable questions.

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1. Problem Classification and Method Selection

Business math problems can be grouped into 5 core categories. Identify the type first to choose the right tool:

Worked Example 1: Classifying a Problem

Problem: A factory produces two products, X and Y. Profit per unit is ₹500 for X and ₹300 for Y. Constraints:

• Labor: 4 hours for X, 2 hours for Y; total 160 hours/week.
• Material: 3 kg for X, 1 kg for Y; total 90 kg/week. Formulate the LP problem and identify the method to solve it.

Solution:

1. Objective: Maximize $Z=500x+300y$.
2. Constraints:
   • Labor: $4x+2y\le 160$ → $2x+y\le 80$
   • Material: $3x+y\le 90$
   • Non-negativity: $x,y\ge 0$.
3. Method: Graphic method (2 variables) or Simplex (if scaled up).
4. Real-world tie: This mirrors Daraz’s warehouse optimization, where products (X=phones, Y=accessories) compete for shelf space (labor/material constraints).

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2. Linear Programming: Graphic vs. Simplex Methods

A. Graphic Method (2 Variables)

When to use: Problems with 2 decision variables and linear constraints. Steps:

1. Plot constraints as lines (equality form).
2. Shade feasible regions.
3. Identify corner points (optimal solutions lie here).
4. Evaluate objective function at corners.

Worked Example 2: Graphic Solution Problem: Maximize $Z=5x_1+3x_2$ subject to:

• $2x_1+x_2\le 5$
• $x_1+x_2\le 4$
• $x_1,x_2\ge 0$

Solution:

1. Plot constraints:
   • For $2x_1+x_2=5$: Intercepts at $(2.5,0)$ and $(0,5)$.
   • For $x_1+x_2=4$: Intercepts at $(4,0)$ and $(0,4)$.
2. Feasible region: Shaded area satisfying all constraints (see figure below).
3. Corner points: $(0,0)$, $(0,4)$, $(1.67,0.67)$, $(2.5,0)$.
4. Evaluate Z:
   • At $(1.67,0.67)$: $Z=5(1.67)+3(0.67)=10.33$ (optimal).

Real-world tie: Pathao’s driver allocation uses similar constraints to maximize rides (objective) given driver availability (constraints).

B. Simplex Method (3+ Variables)

When to use: Problems with 3+ variables or non-linear constraints. Steps:

1. Convert inequalities to equalities (slack/surplus variables).
2. Write initial tableau.
3. Pivot until no negative coefficients in $Z$-row.

Worked Example 3: Simplex Tableau Problem: Minimize $Z=5x+3y$ subject to:

• $2x+3y\ge 6$
• $x+y\ge 2$
• $x,y\ge 0$.

Solution:

1. Convert to standard form (maximize $-Z$):
   • $-Z=-5x-3y$
   • $2x+3y-s_1=6$
   • $x+y-s_2=2$.
2. Initial tableau:

   | Basis | x   | y   | s1  | s2  | RHS |
   |-------|-----|-----|-----|-----|-----|
   | s1    | 2   | 3   | 1   | 0   | 6   |
   | s2    | 1   | 1   | 0   | 1   | 2   |
   | -Z    | -5  | -3  | 0   | 0   | 0   |
   

3. Pivot: Enter $x$ (most negative in $Z$-row), leave $s_2$ (smallest ratio).
4. Final tableau (after pivoting):

   | Basis | x   | y   | s1  | s2  | RHS |
   |-------|-----|-----|-----|-----|-----|
   | s1    | 0   | 1.5 | 1   | -2  | 4   |
   | x     | 1   | 1   | 0   | 1   | 2   |
   | -Z    | 0   | 2   | 0   | 5   | 10  |
   

5. Optimal solution: $x=2$, $y=0$, $Z=10$.

Comparison Table:

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3. Calculus Applications: Profit Maximization

Key Idea: Use derivatives to find critical points (maxima/minima) for cost, revenue, or profit functions.

Worked Example 4: Profit Maximization Problem: A company’s profit $P(x)=-0.5x^2+100x-200$ (in ₹ lakhs), where $x$ is units sold. Find the output level for maximum profit.

Solution:

1. Find derivative: $P^{\prime }(x)=-x+100$.
2. Set $P^{\prime }(x)=0$: $-x+100=0$ → $x=100$.
3. Second derivative test: $P^{\prime \prime }(x)=-1<0$ → maximum.
4. Maximum profit: $P(100)=-0.5(100{)}^2+100(100)-200=4800$ ₹ lakhs.

Real-world tie: NEPSE’s stock price trends are modeled using similar calculus techniques to predict peaks and troughs.

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4. Financial Mathematics: Time Value of Money (TVM)

Key Formulas:

1. Future Value (FV): $FV=PV(1+r{)}^n$.
2. Present Value (PV): $PV=\frac{FV}{(1+r{)}^n}$.
3. Annuity PV/FV: $P{V}_{annuity}=PMT\left(\frac{1-(1+r{)}^{-n}}{r}\right)$.

Worked Example 5: Loan Repayment Problem: You take a ₹50,000 loan at 10% annual interest, repayable in 3 equal yearly installments. Calculate the annual payment.

Solution:

1. Given:
   • $PV=50,000$, $r=0.10$, $n=3$.
2. Annuity formula: $PV=PMT\left(\frac{1-(1+r{)}^{-n}}{r}\right)$.
3. Plug in values: $50,000=PMT\left(\frac{1-(1.10{)}^{-3}}{0.10}\right)$.
4. Solve for PMT: $PMT=\frac{50,000}{2.4869}\approx ₹20,104$.

Real-world tie: Khalti’s buy-now-pay-later service uses this exact formula to calculate installments for customers.

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5. Optimization in Dynamic Systems (Difference Equations)

Key Idea: Model recursive relationships (e.g., population growth, inventory) using difference equations.

Worked Example 6: Inventory Management Problem: A shop’s inventory $I_{t+1}=0.8I_t+100$, where $I_t$ is stock at time $t$. Find the equilibrium stock level.

Solution:

1. Equilibrium condition: $I_{t+1}=I_t=I^{\ast }$.
2. Set equation: $I^{\ast }=0.8I^{\ast }+100$.
3. Solve: $0.2I^{\ast }=100$ → $I^{\ast }=500$.

Real-world tie: Daraz’s warehouse stock levels are optimized using similar recursive models to avoid overstocking/shortages.

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6. Problem-Solving Strategies for Exams

A. Step-by-Step Template

Use this 5-step framework for any problem:

1. Understand: Identify problem type (LP, calculus, etc.).
2. Formulate: Write equations/objective function.
3. Solve: Apply the correct method (graphic/simplex/derivatives).
4. Verify: Check constraints, units, and reasonableness.
5. Interpret: State the answer in business context (e.g., "Maximum profit is ₹X at Y units").

B. Common Pitfalls and Fixes

C. Time Management Tips

• 3-minute rule: If a problem takes >3 minutes to classify, move on and return later.
• Marks allocation: Spend 60% time on 70% marks (e.g., LP problems).
• Partial credit: Always show steps—even incorrect work may earn points.

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In the Real World

1. eSewa’s Transaction Fees:

   • Concept: Profit maximization (calculus).
   • How: eSewa uses derivative analysis to set transaction fees (e.g., 2.5% for bills, 3% for transfers) that maximize revenue without driving users to competitors like Khalti.

2. Daraz’s Delivery Route Optimization:

   • Concept: Linear programming.
   • How: Daraz’s logistics team solves LP problems to assign delivery routes (constraints: driver capacity, time windows) to minimize fuel costs and delays. A similar problem was solved in Worked Example 1.

3. NEPSE’s Stock Price Predictions:

   • Concept: Differential equations.
   • How: Analysts use dynamic models (e.g., $\frac{dP}{dt}=kP(1-\frac{P}{M})$) to predict stock trends, where $P$ = price, $M$ = market cap. This mirrors Worked Example 6 but with continuous time.

4. Ncell’s Data Plan Pricing:

   • Concept: Time value of money (TVM).
   • How: Ncell calculates the present value of future revenue from prepaid plans (e.g., ₹1000 for 30 days) using annuity formulas to ensure profitability. See Worked Example 5.

5. Kathmandu Traffic Light Timing:

   • Concept: Optimization (dynamic systems).
   • How: The Kathmandu Metropolitan City uses recursive models to adjust traffic light cycles, balancing pedestrian flow and vehicle throughput. The equilibrium solution (like Worked Example 6) minimizes congestion.

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Exam Tip

1. For LP Problems:

   • Always draw the feasible region (even if not asked) to visualize constraints.
   • In simplex, label each pivot step clearly (e.g., "Pivot on $x_1$").
   • Dual problems (like in past exams) can be solved by swapping objective and constraints. For the primal: $\text{Min }Z=5x+3y$ $2x+3y\ge 6$ $x+y\ge 2$ The dual is: $\text{Max }W=6u+2v$ $2u+v\le 5$ $3u+v\le 3$ $u,v\ge 0$.

2. For Calculus Problems:

   • Label critical points as maxima/minima in your answer (examiners love this).
   • If asked for economic interpretation, relate to marginal cost/revenue (e.g., "MC = MR at $x=100$").

3. For Financial Math:

   • Show the formula you’re using before plugging in numbers.
   • For annuities, state whether it’s PV or FV clearly.

4. General:

   • Units matter: Always include units (₹, units, years) in final answers.
   • Assumptions: If a problem lacks data (e.g., interest rate), state a reasonable assumption (e.g., "Assume 8% annual interest").
   • Graphs: For calculus problems, sketch the curve with labeled axes and critical points.

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Final Visual Summary: