NCERT Solutions for Class 9 Maths (Ganita Manjari Part I) – All Chapters

This page gives you free NCERT Solutions for all 8 chapters. Every exercise set is solved step by step, in simple language, with the reason for each step. You can also find the topic list, the main formulas and answers to common doubts.

NCERT means National Council of Educational Research and Training. It is the body that writes school textbooks in India.

NCERT Solutions for Class 9 Maths: All Chapters

Click any chapter to open its full solutions. Exercise-wise links are given under each chapter below.

The book also has graph paper at the end, on page 197.

Chapter 1: Orienting Yourself: The Use of Coordinates

This chapter teaches you how to describe a position using two numbers. You learn the Cartesian plane, the origin, the four quadrants and ordered pairs. You then learn to find the distance between two points and the midpoint of a line.

Topics you will study

• Axes, origin, quadrants and ordered pairs
• Plotting points and reading coordinates
• Distance between two points
• Midpoint of a line segment
• Using graph paper and grids

Formulas and key ideas

• Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
• Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Common doubt: Students often mix up the order of x and y. Always write the x-value first, then the y-value.

Exercise-wise solutions

Chapter 2: Introduction to Linear Polynomials

This chapter builds algebra step by step. You start with expressions, then move to linear polynomials, and then see them as graphs. The main idea is that one rule can be shown as a table, as algebra and as a straight line.

Topics you will study

• Variables, constants and terms
• Linear polynomials and their value
• Writing rules from number patterns and picture patterns
• Drawing and reading graphs of linear relations
• Slope and y-intercept of y = ax + b

Formulas and key ideas

• Linear polynomial: p(x) = ax + b
• Value at x = k: p(k) = ak + b
• Linear equation form: ax + b = c

Common doubt: A polynomial is not the same as an equation. p(x) = 2x + 3 is a rule. 2x + 3 = 9 is an equation you solve.

Exercise-wise solutions

Chapter 3: The World of Numbers

This chapter takes you from fractions to real numbers. You learn why some numbers can never be written as a fraction, and how to show them on the number line.

Topics you will study

• Rational numbers and how to place them on the number line
• Density: infinitely many rational numbers lie between any two rational numbers
• Decimal form of rational numbers
• Irrational numbers
• Proof that √2 and √3 are irrational
• The square root spiral

Formulas and key ideas

• A rational number can be written as p/q, where q ≠ 0
• An irrational number cannot be written as p/q
• Every point on the number line stands for one real number

Common doubt: A long decimal is not always irrational. 0.3333… repeats, so it is rational. It equals 1/3.

Exercise-wise solutions

Chapter 4: Exploring Algebraic Identities

Here you do not just learn identities by heart. You see where they come from, using squares and rectangles drawn on paper. After that, factorisation becomes much easier.

Topics you will study

• Standard algebraic identities
• Showing identities with geometric models and algebra tiles
• Expanding expressions
• Factorising expressions, with and without tiles
• Making new identities
• Simplifying rational expressions

Formulas and key ideas

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• (a + b)(a − b) = a² − b²
• (x + a)(x + b) = x² + (a + b)x + ab

Common doubt: (a + b)² is not a² + b². The middle term 2ab is always there.

Exercise-wise solutions

Chapter 5: I’m Up and Down, and Round and Round

The name sounds unusual, but this is the circle geometry chapter. You learn the parts of a circle and then prove results about chords, arcs and angles.

Topics you will study

• Centre, radius, chord, arc and circumference
• Circumcircle and circumcentre of a triangle
• Equal chords and the angles they make
• Perpendicular from the centre to a chord
• Distance of a chord from the centre
• Angle made by an arc at the centre and on the circle
• Cyclic points and cyclic quadrilaterals

Formulas and key ideas

• A circle is the set of all points at a fixed distance from one fixed point
• That fixed point is the centre. That fixed distance is the radius
• The angle made by an arc at the centre is double the angle it makes at any point on the rest of the circle

Common doubt: In a proof, always mark equal parts on the figure first. Most marks are lost for missing reasons, not wrong answers.

Exercise-wise solutions

Chapter 6: Measuring Space: Perimeter and Area

This is the mensuration chapter. You measure the boundary and the space inside flat shapes. The chapter also shows how these formulas were found, including work by Indian mathematicians.

Topics you will study

• Perimeter of common shapes
• π (pi) and why it is irrational
• Length of an arc and area of a sector
• Area of rectangles, parallelograms and triangles
• Heron’s formula
• Squaring a rectangle, from Baudhayana’s Sulbasutras
• How the area of a circle is derived
• Brahmagupta’s formula for a cyclic quadrilateral

Formulas and key ideas

• Area of rectangle = l × b
• Perimeter of rectangle = 2(l + b)
• Area of triangle = ½ × base × height
• Circumference of circle = 2πr
• Area of circle = πr²
• Heron’s formula: s = (a + b + c)/2, Area = √[s(s − a)(s − b)(s − c)]

Common doubt: Keep your units right. Perimeter is in cm, area is in cm².

Exercise-wise solutions

Chapter 7: The Mathematics of Maybe: Introduction to Probability

Probability is the maths of chance. This chapter starts with simple experiments, such as tossing a coin, and then moves to counting outcomes.

Topics you will study

• Randomness and chance events
• The probability scale
• Empirical probability, found from experiments and data
• Theoretical probability, found from equally likely outcomes
• Showing outcomes in tables and tree diagrams

Formulas and key ideas

• P(E) = number of favourable outcomes ÷ total number of outcomes
• 0 ≤ P(E) ≤ 1

Common doubt: Empirical probability comes from what actually happened. Theoretical probability comes from what should happen. They are usually close, not equal.

Exercise-wise solutions

Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions

This chapter is new for Class 9. You look at number patterns and learn to write a rule for them. This is a big help later, in Class 10.

Topics you will study

• Sequences and how to list terms
• Explicit rule and recursive rule
• Arithmetic Progression (AP): the nth term
• Sum of the first n natural numbers
• Geometric Progression (GP): the nth term
• Patterns in fractals and the Tower of Hanoi puzzle

Formulas and key ideas

• AP: a, a + d, a + 2d, a + 3d, …
• nth term of an AP: aₙ = a + (n − 1)d
• GP: a, ar, ar², ar³, …
• nth term of a GP: aₙ = arⁿ⁻¹
• Sum of first n natural numbers = n(n + 1)/2

Common doubt: In an AP you add the same number each time. In a GP you multiply by the same number each time.

Exercise-wise solutions

What Changed in Class 9 Maths for 2026-27

The old book had 15 chapters and started with Number Systems. The new book has 8 chapters in Part I and starts with coordinates. Here is the simple comparison.