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Short Intro

Exercise 2.4 explains the concepts of linear growth and linear decay using real-life situations like plant growth, mobile depreciation, population increase, and prepaid balance reduction. These detailed step-by-step solutions help students understand linear functions and algebraic modelling easily.

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Quick Information Box

Topic	Details
Chapter	Introduction to Linear Polynomials
Exercise	Exercise 2.4
Subject	Mathematics
Class	Grade 9
Main Concepts	Linear Growth & Linear Decay
Difficulty Level	Moderate
Useful For	School Exams & Olympiad Preparation

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Concepts Used (Topics Covered)

• Linear Growth
• Linear Decay
• Linear Functions
• Algebraic Expressions
• Table of Values
• Depreciation
• Population Growth
• Balance Reduction
• Mathematical Modelling

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Important Formulas

Linear Growth Formula

$y=a+bx$

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Linear Decay Formula

$y=a-bx$

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Depreciation Formula

$$\text{Value} = \text{Initial Value} – (\text{Decrease per year} \times \text{Years})$$

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Population Growth Formula

$$P = P_0 + rt$$

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Questions & Step-by-Step Solutions

Question 1

Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month.

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(i) Find the height after 7 months

Solution

Initial height = 1.75 feet

Monthly growth = 0.5 feet

After 7 months:$$1.75 + 7(0.5)$$$$= 1.75 + 3.5$$$$= 5.25$$

Final Answer

✅ Height after 7 months = 5.25 feet

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(ii) Table of Values

Month (t)	Height h (feet)
0	1.75
1	2.25
2	2.75
3	3.25
4	3.75
5	4.25
6	4.75
7	5.25
8	5.75
9	6.25
10	6.75

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(iii) Expression relating h and t

$$h = 1.75 + 0.5t$$

This represents linear growth because the height increases by a constant amount every month.

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Question 2

A mobile phone is bought for ₹10,000. Its value decreases by ₹800 every year.

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(i) Find value after 3 years

Solution

Initial value = ₹10,000

Decrease every year = ₹800

After 3 years:$$10000 – 3(800)$$$$= 10000 – 2400$$$$= 7600$$

Final Answer

✅ Value after 3 years = ₹7600

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(ii) Table of Values

Year (t)	Value v (₹)
0	10000
1	9200
2	8400
3	7600
4	6800
5	6000
6	5200
7	4400
8	3600

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(iii) Expression relating v and t

$$v = 10000 – 800t$$

This represents linear decay because the value decreases by a fixed amount every year.

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Question 3

The initial population of a village is 750. Every year, 50 people move from a nearby city to the village.

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(i) Population after 6 years

Solution

Initial population = 750

Increase per year = 50

After 6 years:$$750 + 6(50)$$$$= 750 + 300$$$$= 1050$$

Final Answer

✅ Population after 6 years = 1050

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(ii) Table of Values

Year (t)	Population P
0	750
1	800
2	850
3	900
4	950
5	1000
6	1050
7	1100
8	1150
9	1200
10	1250

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(iii) Expression relating P and t

$$P = 750 + 50t$$

This represents linear growth because population increases by a constant number every year.

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Question 4

A telecom company charges ₹600 for a recharge scheme. This prepaid balance reduces by ₹15 every day after recharge.

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(i) Equation for remaining balance

Solution

Initial balance = ₹600

Reduction every day = ₹15$$b(x)=600-15x$$

This represents linear decay because balance decreases by a fixed amount daily.

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(ii) After how many days will balance become zero?

Solution

$$600 – 15x = 0$$$$15x = 600$$$$x = 40$$

Final Answer

✅ Balance will run out after 40 days

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(iii) Table of Values

Days (x)	Balance b(x)
1	585
2	570
3	555
4	540
5	525
6	510
7	495
8	480
9	465
10	450

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Common Mistakes

❌ Using wrong sign in growth and decay problems
❌ Forgetting initial value in expressions
❌ Calculation mistakes in tables
❌ Confusing increase with decrease

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Exam Tips

✔ Growth means addition (+)
✔ Decay means subtraction (−)
✔ Write formula before solving
✔ Make tables carefully and systematically

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Practice MCQs

1. Which expression represents linear growth?

A) $500-20x$
B) $1000+50x$1000+50x
C) $x^2+5$
D) $2^x$

Answer

✅ B) $1000+50x$

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2. Which expression represents linear decay?

A) $200+15x$
B) $500-10x$
C) $x^2-1$
D) $3x^2$

Answer

✅ B) $500-10x$

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3. A value decreases by ₹100 every year. Which expression is correct?

A) $500+100x$
B) $500-100x$
C) $100x-500$
D) $500x$

Answer

✅ B) $500-100x$

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4. Linear growth means:

A) Variable increase
B) Constant increase
C) Random increase
D) Quadratic increase

Answer

✅ B) Constant increase

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FAQ Section

Q1. What is linear growth?

Linear growth means a quantity increases by a fixed amount over equal intervals.

Q2. What is linear decay?

Linear decay means a quantity decreases by a fixed amount over equal intervals.

Q3. What is a linear function?

A function represented by a straight-line equation is called a linear function.

Q4. Why are tables important in linear functions?

Tables help us understand patterns and relationships clearly.

Q5. Can real-life situations be represented using linear equations?

Yes, many daily-life situations can be represented using linear equations.

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