Introduction to Mathematical Modelling Class 9 Solutions and Mind Map (Free PDF Download)

What is Mathematical Modelling?

Mathematical modelling is the process of taking a real-world problem and converting it into a mathematical problem, solving it, and then interpreting the solution back into real-world terms.

Think of it as a bridge between your everyday life and the world of mathematics!

Simple Definition: A mathematical model is a mathematical relation (equation, formula, or graph) that describes a real-life situation.

Real-Life Applications of Mathematical Modelling:

• Launching satellites into space
• Predicting monsoon arrivals
• Controlling vehicle pollution
• Reducing traffic jams in big cities
• Banking and finance calculations
• Weather forecasting
• Medical treatment planning
• Civil engineering projects

The Four Steps of Mathematical Modelling

Step 1: Formulation (Understanding the Problem)

In this step, you:

• State the Problem Clearly: Write down exactly what you need to find
• Identify Relevant Factors: Decide which quantities matter and which you can ignore
• Create Mathematical Description: Write the problem as one or more mathematical equations

Key Question: “What factors really matter for solving this problem?”

Example of Relevant vs Irrelevant Factors:

Step 2: Solution (Solving the Mathematical Problem)

In this step, you:

• Use Mathematical Knowledge: Apply algebra, geometry, or other math concepts
• Solve the Equations: Find the value(s) that satisfy your mathematical equations
• Get Numerical Answer: Obtain the mathematical solution

Step 3: Interpretation (Understanding the Result)

In this step, you:

• Convert Back to Real-World Language: Explain what your mathematical answer means in the original problem
• Check if Answer Makes Sense: Does the answer logically fit the real situation?
• State Your Conclusion: Give a clear, practical answer

Step 4: Validation (Checking Against Reality)

In this step, you:

• Compare with Reality: Check if your mathematical model matches what actually happens
• Test Your Model: If possible, verify with known data or observations
• Modify if Needed: If the model doesn’t match reality well, go back to Step 1 and improve your equations
• Accept or Reject: Decide if the model is good enough to use

This step is the most important difference between solving regular word problems and mathematical modelling!

Real-World Examples (Solved Step-by-Step)

Example 1: Direct Variation – Petrol Consumption

Problem: I travelled 432 km on 48 litres of petrol. How much petrol do I need for a 180 km journey?

Step 1: Formulation

• Problem: Find petrol needed for 180 km
• Relevant Factors: Distance travelled, petrol consumed
• Irrelevant Factors: Weather, road conditions (assumed constant)
• Mathematical Description:
• Let x = distance (in km)
• Let y = petrol needed (in litres)
• Direct variation: y = kx (where k is constant)
• From data: 48 = k × 432
• So: k = 48/432 = 1/9
• Formula: y = (1/9)x

Step 2: Solution

• For x = 180 km:
• y = (1/9) × 180
• y = 20 litres

Step 3: Interpretation

• Answer: 20 litres of petrol are needed for a 180 km journey

Step 4: Validation

• Assumption: The consumption rate remains the same
• If the 180 km route is different (mountains vs plains), the model might not work
• Model is valid only if conditions are similar to the original journey

Example 2: Simple Interest – Investment Problem

Problem: Sudhir invests ₹15,000 at 8% annual interest. He wants to buy a washing machine for ₹19,000. For how many years should he invest?

Step 1: Formulation

• Problem: Find the investment period needed
• Relevant Factors: Principal (₹15,000), Target amount (₹19,000), Interest rate (8%)
• Irrelevant Factors: Bank location, account number
• Mathematical Description:
• Simple Interest Formula: I = (P × n × r) / 100
• Where: P = ₹15,000, r = 8%, I = ₹19,000 – ₹15,000 = ₹4,000
• Formula: 4,000 = (15,000 × n × 8) / 100
• Simplifying: 4,000 = 1,200n

Step 2: Solution

• 1,200n = 4,000
• n = 4,000 / 1,200
• n = 3⅓ years (or 3 years 4 months)

Step 3: Interpretation

• Answer: Sudhir should invest for 3 years and 4 months to buy the washing machine

Step 4: Validation

• Assumption: Interest rate remains constant throughout
• Assumption: Machine price doesn’t increase
• Model works if these conditions hold true

Example 3: Upstream-Downstream – Boat Speed Problem

Problem: A motorboat takes 6 hours upstream and 5 hours downstream to cover the same distance. River speed is 2 km/h. Find the boat’s speed in still water.

Step 1: Formulation

• Problem: Find boat’s speed in still water
• Relevant Factors: Upstream time (6 h), Downstream time (5 h), River speed (2 km/h)
• Irrelevant Factors: Boat length, number of passengers, water temperature
• Mathematical Description:
• Let x = boat’s speed in still water (km/h)
• Let d = distance between towns (km)
• Upstream speed = (x – 2) km/h
• Downstream speed = (x + 2) km/h
• Distance = Speed × Time
• Equations:
  • d = 6(x – 2) … Upstream
  • d = 5(x + 2) … Downstream

Step 2: Solution

• Since both distances are equal:
• 6(x – 2) = 5(x + 2)
• 6x – 12 = 5x + 10
• 6x – 5x = 10 + 12
• x = 22 km/h

Step 3: Interpretation

• Answer: The boat’s speed in still water is 22 km/h

Step 4: Validation

• Check: Upstream speed = 22 – 2 = 20 km/h; Distance = 20 × 6 = 120 km ✓
• Check: Downstream speed = 22 + 2 = 24 km/h; Distance = 24 × 5 = 120 km ✓
• Model is valid!

Important Formulas and Concepts

Direct Variation Formula

y = kx (where k is a constant)

• “y varies directly with x”
• When x increases, y increases proportionally

Simple Interest Formula

I = (P × n × r) / 100

• I = Interest earned
• P = Principal (amount invested)
• n = Number of years
• r = Rate of interest per annum (%)

Distance-Speed-Time Formula

Distance = Speed × Time or d = st

Upstream-Downstream Relations

• Upstream speed = (boat speed – river speed)
• Downstream speed = (boat speed + river speed)

Identifying Relevant vs Irrelevant Factors

Strategy to Identify:

Relevant Factors Ask Yourself:

1. Does this factor directly affect the answer?
2. If I change this, will the answer change?
3. Is this mentioned in the problem or necessary for calculation?

Irrelevant Factors:

1. Would the answer be the same if I ignore this?
2. Is this just additional descriptive information?
3. Can solving the problem proceed without knowing this?

Quick Examples:

Important Points

Point 1: Mathematical Models in Daily Life

Every formula you’ve learned in previous classes is a mathematical model:

• Simple Interest: I = Pnr/100
• Area of rectangle: A = length × width
• Perimeter formulas
• Distance formulas

Point 2: The Validation Step is Crucial

This step separates mathematical modelling from regular word problem solving. You must check if your mathematical answer makes sense in the real world.

Point 3: Assumptions Are Important

Every mathematical model requires assumptions. You must state these clearly:

• “We assume the speed remains constant”
• “We assume no external factors change”
• “We assume the given data is accurate”

Point 4: Models Can Be Improved

Your first model might not perfectly match reality. You can:

• Go back to Step 1 (Formulation)
• Identify what went wrong
• Modify your mathematical equations
• Test again in Step 4 (Validation)

Point 5: Balancing Accuracy and Usability

A good model:

• ✓ Is simple enough to use
• ✓ Is accurate enough for the purpose
• ✓ Doesn’t include unnecessary complexity
• ✓ Works within specified limitations

Advantages of Mathematical Modelling

1. Cost-Effective: No need for expensive experiments

• Example: Testing pollution effects on Taj Mahal using models instead of real experiments

1. Time-Saving: Get answers without waiting for real events

• Example: Predicting school infrastructure needs 5 years from now

1. Safety: Avoid dangerous real-world testing

• Example: Testing earthquake resistance of buildings mathematically

1. Predictions: Estimate future outcomes

• Example: Weather forecasting, population growth projections

1. Understanding: Gain insights into how systems work

• Example: Understanding disease spread, traffic flow

1. Decision Making: Help make better choices

• Example: Government policy planning, business strategy

Limitations of Mathematical Modelling

1. Oversimplification: Real world is too complex to model perfectly

• A model is like a map – it shows some features but not all

1. Assumption Errors: If assumptions are wrong, results are wrong

• Example: Assuming constant speed when speed actually varies

1. Limited Tools: Some real situations are too complex to solve mathematically

• Example: Weather models are very complex, exact solutions are impossible

1. Accuracy vs Simplicity Trade-off:

• Adding more details makes models more accurate but harder to use
• Adding fewer details makes models simpler but less accurate

1. Limited Application Range: A model works only within certain limits

• A model for small distances might fail for very large distances
• A model for low speeds might fail at high speeds

1. Data Limitations: Results depend on quality of input data

• Wrong or incomplete data leads to wrong conclusions

Quick Summary Table

Tips for Success in Mathematical Modelling

1. Read the Problem Carefully

• Underline what you need to find
• Circle all given information
• Ask: “What exactly is being asked?”

2. Identify Relevant Information

• Ask: “Does this fact affect my answer?”
• Make a list of relevant factors
• Make a list of irrelevant factors

3. Write Mathematical Equations

• Use variables (x, y, etc.) clearly
• Show the relationship between variables
• Check: “Are my equations correct?”

4. Solve Step-by-Step

• Write all steps clearly
• Check calculations twice
• Use correct mathematical operations

5. Interpret Your Answer

• Convert back to real-world language
• Use appropriate units
• State complete answer: “The answer is _ because _“

6. Validate Before Concluding

• Check if answer is reasonable
• Test with given data if possible
• Ask: “Does this answer make sense?”

7. State Assumptions Clearly

• List all assumptions you made
• Explain why you made them
• Note limitations of your model

Important Questions & Answers

Q1: Why is the validation step important?

Answer: The validation step ensures your mathematical model actually represents the real situation accurately. Without it, you might get an answer that is mathematically correct but wrong in reality.

Q2: Can a mathematical model be perfect?

Answer: No, a mathematical model is always a simplification of reality. It can never perfectly represent the real world because real situations are too complex. But a good model is accurate enough for its intended purpose.

Q3: What should I do if my model doesn’t match reality?

Answer: Go back to Step 1 (Formulation) and:

• Check your assumptions
• Identify what factors you neglected
• Modify your mathematical equations
• Test again using validation

Q4: Is every formula in my textbook a mathematical model?

Answer: Yes! Every formula represents a mathematical relationship that describes a real-world situation. For example, the simple interest formula I = Pnr/100 is a mathematical model for how bank interest works.

Q5: Can I ignore some factors in the problem?

Answer: Yes, but only if they are truly irrelevant to the answer. A factor is irrelevant if:

• It doesn’t affect the final answer
• Its effect is so small that it can be neglected
• The problem would become impossible to solve if you included it

Q6: Why do scientists use mathematical models?

Answer: Scientists use mathematical models because:

• Real experiments are often expensive or impossible
• Mathematical models save time and money
• They help predict future events
• They allow testing dangerous scenarios safely

Q7: What’s the difference between a word problem and mathematical modelling?

Answer:

Q8: How do I know if my assumptions are reasonable?

Answer: Your assumptions are reasonable if:

• They don’t change the problem significantly
• The real situation stays relatively constant
• The effect of ignored factors is very small
• Similar successful models use the same assumptions

Q9: What if the model takes too long to solve?

Answer: If your model is too complicated:

• You can simplify it (but lose accuracy)
• You can use approximate methods
• You can use computers/calculators
• Remember: Balance accuracy with practicality

Q10: Can two different models give different answers?

Answer: Yes, and this is normal! Different models may:

• Make different assumptions
• Consider different relevant factors
• Use different mathematical approaches
• Both could be “correct” within their own limitations

Real-World Application: Gender Equality in Education

This is a real example from the NCERT textbook about predicting girl student enrollment in primary schools.

The Situation:

The UN’s goal is to achieve 50% enrollment of girls in schools by 2015. India collects data year by year.

The Mathematical Model:

Using 10 years of data, scientists created an equation to predict future enrollment:
Enrollment % = 41.9 + 0.22n

Where n = number of years after 1991

Prediction:

By solving 50 = 41.9 + 0.22n, they predicted enrollment would reach 50% in the year 2027.

Validation:

They checked their model against actual data and found small differences of 0.3-0.5%. This was acceptable, so they used this model for predictions.

How to Score Well in Mathematical Modelling

In Examinations:

1. ✓ Clearly state relevant and irrelevant factors
2. ✓ Write all mathematical equations properly
3. ✓ Show all steps in solution
4. ✓ Interpret answer in real-world terms
5. ✓ If possible, validate your answer
6. ✓ State all assumptions you made

Common Mistakes to Avoid:

• ✗ Not identifying relevant factors
• ✗ Missing the validation step
• ✗ Not stating assumptions
• ✗ Wrong mathematical equations
• ✗ Calculation errors
• ✗ Not converting answer back to real-world terms

Practice Strategy:

1. Solve 3-4 examples completely (all 4 steps)
2. Identify patterns in similar problems
3. Create your own real-world problems
4. Exchange with classmates and solve theirs
5. Explain your solution to someone else

Connection with Other Chapters

Mathematical modelling uses concepts from:

• Linear Equations: Writing and solving mathematical relationships
• Algebra: Using variables and formulas
• Geometry: Calculating areas, volumes, distances
• Trigonometry: Finding heights, angles in real situations
• Statistics: Using data to create models

Practice Problems

Exercise A2.1

Problem 1: A company can hire a computer for ₹2,000/month or buy for ₹25,000. After how many months is buying cheaper?

Solution Outline:

• Relevant: Monthly cost, purchase price
• Irrelevant: Computer brand, color
• Equation: 2,000n = 25,000
• Answer: n = 12.5 months (buy after 13 months)

Problem 2: Two cars travel toward each other at 40 km/h and 30 km/h. Distance between them is 100 km. When will they meet?

Solution Outline:

• Relevant: Speed of each car, distance between them
• Irrelevant: Car models, driver names
• Equation: 40t + 30t = 100
• Answer: t = 10/7 ≈ 1.43 hours

Problem 3: Moon is 384,000 km from Earth and orbits in 24 hours. Find orbital speed.

Solution Outline:

• Distance = 2πr = 2 × 3.14 × 384,000
• Distance = 2,411,520 km
• Speed = Distance / Time = 2,411,520 / 24 ≈ 100,480 km/h

Problem 4: Water heater uses ₹8/hour. Base electricity bill ₹1,000 (without heater), ₹1,240 (with heater). Average hours per day?

Solution Outline:

• Extra bill with heater = ₹1,240 – ₹1,000 = ₹240
• Hours of use per month = 240 / 8 = 30 hours
• Hours per day = 30 / 30 ≈ 1 hour

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