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Short Intro

Exercise 2.3 introduces students to the concept of linear patterns and how mathematical expressions can represent real-life situations involving growth and reduction. These detailed step-by-step solutions help students understand sequences, linear expressions, and pattern formation easily.

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Quick Information Box

Topic	Details
Chapter	Introduction to Linear Polynomials
Exercise	Exercise 2.3
Subject	Mathematics
Class	Grade 9
Main Concepts	Linear Patterns & Linear Expressions
Difficulty Level	Moderate
Useful For	School Exams & Olympiad Preparation

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Concepts Used (Topics Covered)

• Linear Patterns
• Sequences
• Linear Expressions
• Arithmetic Growth
• Linear Decay
• Area of Rectangle
• Volume of Cuboid
• Daily Reduction Problems
• Algebraic Representation

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Important Formulas

Linear Pattern Formula

$an+b$

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Area of Rectangle

$A=l\times b$

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Volume of Cuboid

$V=l\times b\times h$

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Savings Formula

$$\text{Total Amount} = \text{Initial Amount} + (\text{Monthly Increase})$$

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Questions & Step-by-Step Solutions

Question 1

A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression for the nth month.

Solution

Initial amount = ₹500

Monthly increase = ₹150

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After 2nd month

$$500 + 2(150)$$$$= 500 + 300$$$$= ₹800$$

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After 3rd month

$$500 + 3(150)$$$$= ₹950$$

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Linear Expression

If n = number of months,$$500 + 150n$$

Final Answer

✅ Amount after nth month = ₹(500 + 150n)

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Question 2

A rally starts with 120 members. Each hour, 9 members drop out. Find the number of members after 1, 2, 3 … hours and write a linear expression.

Solution

Initial members = 120

Decrease every hour = 9

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After 1 hour

$$120 – 9 = 111$$

After 2 hours

$$120 – 18 = 102$$

After 3 hours

$$120 – 27 = 93$$

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Linear Expression

$$120 – 9n$$

Final Answer

✅ Members after nth hour = 120 − 9n

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Question 3

Suppose the length of a rectangle is 13 cm. Find the area if breadth is 12 cm, 10 cm and 8 cm. Find the linear pattern representing the area.

Solution

Area formula:$$A=l\times b$$

Length = 13 cm

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(i) Breadth = 12 cm

$$13 \times 12 = 156$$

(ii) Breadth = 10 cm

$$13 \times 10 = 130$$

(iii) Breadth = 8 cm

$$13 \times 8 = 104$$

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Linear Pattern

$$A=13b$$

Final Answer

✅ Linear expression = 13b

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Question 4

Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if height is 5 cm, 9 cm and 13 cm. Find the linear pattern representing the volume.

Solution

Volume formula:$$V=l\times b\times h$$

Length = 7 cm

Breadth = 11 cm$$7 \times 11 = 77$$

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(i) Height = 5 cm

$$77 \times 5 = 385$$

(ii) Height = 9 cm

$$77 \times 9 = 693$$

(iii) Height = 13 cm

$$77 \times 13 = 1001$$

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Linear Pattern

$$V=77h$$

Final Answer

✅ Linear expression = 77h

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Question 5

Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.

Solution

Total pages = 500

Pages read daily = 20

Pages read in 15 days:$$20 \times 15 = 300$$

Pages left:$$500 – 300 = 200$$

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Linear Pattern

$$500 – 20n$$

Final Answer

✅ Pages left after 15 days = 200
✅ Linear expression = 500 − 20n

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Common Mistakes

❌ Forgetting increase/decrease signs
❌ Using wrong formula for area or volume
❌ Multiplication errors in large numbers
❌ Confusing pattern variable with actual values

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Exam Tips

✔ Write formulas before solving
✔ Identify whether quantity increases or decreases
✔ Use variables carefully
✔ Check calculations step-by-step

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Practice MCQs

1. Which expression shows linear growth?

A) $5 – 2n$
B) $3n + 7$
C) $n^2 + 1$
D) $2^n$

Answer

✅ B) $3n + 7$

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2. What is the area of rectangle with length 13 cm and breadth 9 cm?

A) 117
B) 127
C) 107
D) 137

Answer

✅ A) 117

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3. Volume of cuboid formula is:

A) $l+b+h$
B) $l\times b$
C) $l\times b\times h$
D) $2(l+b)$

Answer

✅ C) $l\times b\times h$

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4. If 5 decreases every day from 100, expression becomes:

A) $100+5n$
B) $100-5n$
C) $5n-100$
D) $100n$

Answer

✅ B) $100-5n$

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FAQ Section

Q1. What is a linear pattern?

A linear pattern is a sequence where the difference between consecutive terms remains constant.

Q2. What is linear growth?

When a quantity increases by a fixed amount regularly, it is called linear growth.

Q3. What is linear decay?

When a quantity decreases by a fixed amount regularly, it is called linear decay.

Q4. Why are patterns important in Maths?

Patterns help us predict values and create mathematical formulas.

Q5. Can linear patterns be represented algebraically?

Yes, linear patterns can be represented using algebraic expressions.

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