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Class 9 Ganita ManjariClass 9 Maths Solutions | 2026–27
Chapter 2: Introduction to Linear PolynomialsPage 24 – 26All 5 Questions on This Page

Class 9 Maths Chapter 2 Exercise Set 2.3 Solutions

Complete step-by-step solutions for Class 9 Mathematics Ganita Manjari Chapter 2 Exercise Set 2.3 (Questions 1–5). Covers identifying linear patterns from successive values, constructing tables, establishing linear relationships for cuboid volumes (V = 77h), and modeling book reading linear decay (P = 500 − 20n).

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Key Concepts for Exercise Set 2.3

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1

Recognising Linear Patterns through Successive Differences

When a quantity changes by the same constant amount in each successive step, time interval, or term, the values form a linear pattern. A constant positive difference indicates linear growth, while a constant negative difference indicates linear decay.

Successive difference = constant ⇒ Linear pattern
2

Forming nth-Term Linear Expressions

To find the general expression after n steps or for the nth term, combine the initial starting value with n multiples of the constant change, or use the standard linear model with step index (n − 1).

Linear pattern: Aₙ = A₀ + c · n | Step-wise pattern: aₙ = a₁ + (n − 1)c
3

Translating Real-Life Contexts into Linear Models

Identify the starting fixed quantity (independent of n) and the constant rate of change per unit of time or step. In pocket money savings, initial deposit is ₹500 and rate is +₹150/month; in a rally, initial size is 120 and dropout rate is −9 members/hour.

Aₙ = 500 + 150n and Mₙ = 120 − 9n
4

Linear Variation in Geometric Formulas

In geometric formulas such as Area = length × breadth, if one dimension (length) remains constant while the other (breadth) changes uniformly by a constant difference, the resulting area changes linearly with the changing dimension.

Area = 13 × b ⇒ Constant change in area = 13 × (−2) = −26 cm²
5

Linear Variation in Volume of a Cuboid

When the length and breadth of a rectangular box are fixed, the base area remains constant. The volume varies directly and linearly with the changing height: Volume = (length × breadth) × height = 77h.

Volume = 7 × 11 × h = 77h
Question 1
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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 to represent the amount she will have in the nth month.

Solution

Initially, the student has:

₹500

She receives ₹150 every month.

Let n be the number of months.

Month-by-month pattern
Month n 0 1 2 3 4 5 … n
Amount (₹) 500 500 + 1(150) = 650 500 + 2(150) = 800 500 + 3(150) = 950 500 + 4(150) = 1100 500 + 5(150) = 1250 … 500 + 150n
For example:
Amount after 1 month:
500 + 1(150) = 650
Amount after 2 months:
500 + 2(150) = 800
Amount after 3 months:
500 + 3(150) = 950
Thus, from the second month onwards, the amounts are:
₹800, ₹950, ₹1100, ₹1250, ...

Each month, the amount increases by a constant ₹150.

Therefore, the amount at the end of the nth month is:
Aₙ = 500 + 150n

where Aₙ is the amount in rupees after n months.

Answer:From the second month onwards: ₹800, ₹950, ₹1100, ₹1250, ... | Linear expression: Aₙ = 500 + 150n
Question 2
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A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, ... hours? Find a linear expression to represent the number of members at the end of the nth hour.

Solution

Initially, there are:

120 members

Each hour, 9 members drop out.

Hour-by-hour pattern
Hour n 0 1 2 3 4 5 … n
Members remaining 120 120 − 1(9) = 111 120 − 2(9) = 102 120 − 3(9) = 93 120 − 4(9) = 84 120 − 5(9) = 75 … 120 − 9n
Thus, after 1, 2, 3, ... hours, the numbers of members are:
111, 102, 93, 84, 75, ...

The number of members decreases by a constant 9 every hour.

After n hours, 9n members have dropped out.

Therefore:
Mₙ = 120 − 9n

where Mₙ is the number of members remaining at the end of the nth hour.

Note
Since Mₙ represents an actual count of members, this linear model is valid as long as the remaining number of members is non-negative.
Answer:Members remaining after 1, 2, 3, ... hours: 111, 102, 93, 84, 75, ... | Linear expression: Mₙ = 120 − 9n
Question 3
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Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm, (ii) 10 cm, (iii) 8 cm. Find the linear pattern representing the area of the rectangle.

Geometric Model of the Rectangle

The length remains constant at 13 cm, so the area changes linearly with the breadth b.

Solution

The length of the rectangle is fixed:

Length = 13 cm

Let the breadth be b cm.

We know:
Area = length × breadth
Therefore:
A = 13 × b
A = 13b

where A is the area of the rectangle.

Finding the areas
(i) Breadth = 12 cm
A = 13 × 12
= 156 cm²
Therefore:

156 cm²

(ii) Breadth = 10 cm
A = 13 × 10
= 130 cm²
Therefore:

130 cm²

(iii) Breadth = 8 cm
A = 13 × 8
= 104 cm²
Therefore:

104 cm²

Linear pattern

Show the relationship using a horizontal table.

Breadth b (cm) 12 10 8 6 …
Area A (cm²) 156 130 104 78 …
Note
The breadth 6 cm and area 78 cm² are the next values obtained by continuing the observed pattern.

The area values decrease as:

156, 130, 104, 78, ...

For the first two changes:

130 − 156 = −26
104 − 130 = −26
So, as the breadth decreases by 2 cm, the area decreases by 26 cm².

The direct relationship between breadth and area is:

A = 13b

This is the main linear expression requested by the question.

Step-by-step pattern:

If the steps are numbered n = 1, 2, 3, ..., each step decreases the area by 26 cm²:

Aₙ = 156 − 26(n − 1) = 182 − 26n

where Aₙ is the area at step n.

Answer:(i) 156 cm² | (ii) 130 cm² | (iii) 104 cm² | Linear pattern: 156, 130, 104, 78, ... | Linear expression: A = 13b (or Aₙ = 182 − 26n)
Question 4
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Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.

Geometric Model of the Rectangular Box

Length (7 cm) and breadth (11 cm) are fixed, so the volume changes linearly with height h.

Solution

The length and breadth of the rectangular box are fixed:

Length = 7 cm
Breadth = 11 cm

Let the height be h cm.

We know:
Volume = length × breadth × height
Therefore:
V = 7 × 11 × h
V = 77h

where V is the volume of the rectangular box.

Finding the volumes
(i) Height = 5 cm
V = 7 × 11 × 5
= 77 × 5
= 385 cm³
Therefore:

385 cm³

(ii) Height = 9 cm
V = 7 × 11 × 9
= 77 × 9
= 693 cm³
Therefore:

693 cm³

(iii) Height = 13 cm
V = 7 × 11 × 13
= 77 × 13
= 1001 cm³
Therefore:

1001 cm³

Linear pattern

Show the relationship using a horizontal table.

Height h (cm) 5 9 13 17 …
Volume V (cm³) 385 693 1001 1309 …
Note
The height 17 cm and volume 1309 cm³ are the next values obtained by continuing the observed pattern.

The volume values increase as:

385, 693, 1001, 1309, ...

For the first two changes:

693 − 385 = 308
1001 − 693 = 308
So, as the height increases by 4 cm, the volume increases by 308 cm³.

The direct relationship between height and volume is:

V = 77h

This is the main linear expression requested by the question.

Answer:(i) 385 cm³ | (ii) 693 cm³ | (iii) 1001 cm³ | Linear pattern: 385, 693, 1001, 1309, ... | Linear expression: V = 77h
Question 5
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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

Initially, the book has:

500 pages

Sarita reads 20 pages every day.

Day-by-day pattern
Day n 0 1 2 3 4 … n
Pages left 500 480 460 440 420 … 500 − 20n

After 1 day:

500 − 20 = 480
After 2 days:
500 − 40 = 460
After 3 days:
500 − 60 = 440
Thus, the pages left form the linear pattern:
500, 480, 460, 440, ...

The number of pages decreases by a constant 20 pages each day.

Therefore, after n days:
P = 500 − 20n

where P is the number of pages left after n days.

Pages left after 15 days
P = 500 − 20(15)
= 500 − 300
= 200
Therefore:

200 pages are left after 15 days.

Answer:After 15 days: 200 pages | Linear expression: P = 500 − 20n | Linear pattern: 500, 480, 460, 440, ...

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