NCERT Solutions for Class 8 Maths: Complete Chapter by Chapter Guide (2026)

Difficulty: Moderate. Foundation level: Very High.

This chapter extends what you learned about integers and fractions in Class 7. A rational number is any number that can be written as $\frac{p}{q}$ where $q \neq 0$. You need to understand the properties of rational numbers: closure, commutativity, associativity, the role of zero as the additive identity, and one as the multiplicative identity.

Key concepts to master:

• Finding rational numbers between any two given rational numbers (there are infinitely many!)
• The additive inverse of $\frac{a}{b}$ is $\frac{-a}{b}$ and the multiplicative inverse is $\frac{b}{a}$
• Representing rational numbers on the number line
• Distributive property: $a(b + c) = ab + ac$

Why this matters: Rational numbers are everywhere in higher maths. If you do not understand their properties now, you will struggle with algebraic expressions and equations in Class 9.

Difficulty: Moderate. Foundation level: Very High.

This is your first real taste of solving equations, and it is one of the most important chapters in the entire textbook. You will learn to solve equations where the variable appears on both sides, not just one side like in Class 7.

Key concepts to master:

• Transposing terms: when you move a term to the other side, its sign changes
• Solving equations with variables on both sides: bring all variable terms to one side and constants to the other
• Word problems: converting English sentences into mathematical equations
• Equations involving fractions: find LCM of denominators first, then solve

Example type you must be able to solve: If $\frac{2x+3}{5} = \frac{3x-1}{3}$, find $x$. Cross multiply and simplify step by step.

Why this matters: Linear equations are the backbone of algebra. In Class 9, you will solve pairs of linear equations in two variables. If you cannot confidently handle one variable, two will feel impossible.

Difficulty: Moderate. Foundation level: High.

This chapter covers the properties of different types of quadrilaterals: parallelograms, rectangles, squares, rhombuses, trapeziums, and kites. The angle sum property of a quadrilateral ($360°$) is fundamental.

Key properties to memorise:

• Parallelogram: opposite sides are equal and parallel, opposite angles are equal, diagonals bisect each other
• Rectangle: all angles are $90°$, diagonals are equal and bisect each other
• Rhombus: all sides are equal, diagonals bisect each other at right angles
• Square: all sides equal, all angles $90°$, diagonals are equal and bisect at right angles

The exterior angle sum of any polygon is always $360°$. For a regular polygon with $n$ sides, each exterior angle equals $\frac{360°}{n}$ and each interior angle equals $\frac{(n-2) \times 180°}{n}$.

Common mistake: students confuse the properties of rhombus and rectangle. Remember, a square is both a rectangle AND a rhombus.

Difficulty: Easy to Moderate. Foundation level: Moderate.

This chapter is about constructing quadrilaterals when different combinations of measurements are given. You need to know how to construct a quadrilateral when given:

• Four sides and one diagonal
• Three sides and two diagonals
• Three sides and two included angles
• Two adjacent sides and three angles

The key skill here is splitting the quadrilateral into two triangles using a diagonal, and then constructing each triangle. Always draw a rough sketch first before picking up the compass.

Study tip: This chapter is mostly practical. You need to actually do the constructions with a compass and ruler, not just read about them. Practice at least 5 to 6 constructions from the exercises.

Difficulty: Easy. Foundation level: Moderate.

Data handling in Class 8 introduces you to organising data, drawing histograms, pie charts (or circle graphs), and basic probability.

Key concepts to master:

• Frequency distribution tables with class intervals
• Histograms: bars touch each other (unlike bar graphs where they do not)
• Pie chart: each sector angle = $\frac{\text{value}}{\text{total}} \times 360°$
• Probability of an event = $\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$

This chapter is relatively easy and scoring. Make sure you know how to read and draw histograms, and how to calculate sector angles for pie charts. The probability section here is introductory, but it builds the foundation for a much harder probability chapter in Class 9 and 10.

Difficulty: Moderate. Foundation level: High.

This chapter teaches you about perfect squares, properties of square numbers, and methods to find square roots.

Key formulas and patterns:

• A number ending in 2, 3, 7, or 8 is never a perfect square
• Square of an even number is even, square of an odd number is odd
• Pythagorean triplets: for any $m > 1$, the triplet is $(2m, m^2 - 1, m^2 + 1)$
• Finding square root by prime factorisation: factorise, pair the factors, take one from each pair
• Finding square root by long division method: essential for non perfect squares

Important: The long division method for finding square roots is something many students skip because it seems tedious. Do not skip it. You will need this skill in Class 9 when working with surds and irrational numbers.

Common mistake: forgetting that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ but $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$.

Difficulty: Moderate. Foundation level: Moderate.

This chapter extends the idea of squares to cubes. You learn about perfect cubes, properties of cube numbers, and finding cube roots.

Key concepts:

• A perfect cube can have any digit (0 to 9) at the units place, unlike perfect squares
• Cube of a negative number is negative: $(-a)^3 = -a^3$
• Hardy Ramanujan numbers: numbers like 1729 that can be expressed as the sum of two cubes in two different ways. $1729 = 12^3 + 1^3 = 10^3 + 9^3$
• Finding cube root by prime factorisation: factorise, group into triples, take one from each triple

Study tip: Do not spend too much time on this chapter. Get comfortable with prime factorisation for cube roots and move on. This chapter is less frequently tested in later classes compared to squares and square roots.

Difficulty: Moderate to Hard. Foundation level: Very High.

This is one of the most practically useful chapters in the entire book. It covers percentages, profit and loss, discount, sales tax/VAT, and compound interest.

Key formulas:

• Profit% = $\frac{\text{Profit}}{\text{Cost Price}} \times 100$
• Discount = Marked Price $-$ Selling Price
• Discount% = $\frac{\text{Discount}}{\text{Marked Price}} \times 100$
• Sales tax on the bill = Tax% $\times$ Bill amount
• Compound Interest: $A = P\left(1 + \frac{r}{100}\right)^n$ where $A$ is the amount, $P$ is the principal, $r$ is the rate per annum, and $n$ is the number of years
• CI = $A - P$

Word problems in this chapter are tricky because you need to figure out what is given and what is being asked. Read each problem at least twice before solving.

Why this matters: Compound interest and percentage problems appear in Class 9 ICSE, Class 10, and even in competitive exams. Get this chapter right and you are set for years.

Difficulty: Hard. Foundation level: Extremely High.

If there is one chapter in this book that you absolutely cannot afford to be weak in, this is it. Algebraic identities are used everywhere in Class 9 and 10 maths.

The four standard identities you must know cold:

Beyond the identities, you also need to master:

• Adding, subtracting, and multiplying polynomials
• Multiplying a monomial by a monomial, a monomial by a polynomial, and a polynomial by a polynomial
• Using identities for quick calculations (for example, $102^2 = (100 + 2)^2 = 10000 + 400 + 4 = 10404$)

Practice tip: Write the four identities on a sticky note and put it on your study table. Use them every single day until they become automatic. When you see $997^2$, your brain should immediately think $(1000 - 3)^2$.

Difficulty: Easy. Foundation level: Moderate.

This chapter is about 3D shapes and their properties. You learn about views of 3D objects, Euler's formula for polyhedra, and mapping.

Key concept:

• Euler's formula: $F + V - E = 2$ where $F$ is faces, $V$ is vertices, and $E$ is edges

You need to know how to identify the top view, front view, and side view of different solid shapes. Also learn to count faces, vertices, and edges for common shapes like cubes, cuboids, pyramids, prisms, and cylinders.

Study tip: This is a relatively easy chapter. Do the exercises carefully and do not overthink. Many students actually enjoy this chapter because it is visual and intuitive.

Difficulty: Hard. Foundation level: Very High.

Mensuration in Class 8 takes a big leap from Class 7. You move from just areas and perimeters to surface areas and volumes of 3D shapes.

Critical formulas:

• Area of a trapezium = $\frac{1}{2}(a + b) \times h$ where $a$ and $b$ are parallel sides
• Area of a general quadrilateral = $\frac{1}{2} \times d \times (h_1 + h_2)$ where $d$ is the diagonal
• Area of a rhombus = $\frac{1}{2} \times d_1 \times d_2$
• Surface area of a cuboid = $2(lb + bh + hl)$
• Surface area of a cube = $6a^2$
• Surface area of a cylinder = $2\pi r(r + h)$
• Volume of a cuboid = $l \times b \times h$
• Volume of a cube = $a^3$
• Volume of a cylinder = $\pi r^2 h$

This chapter has a LOT of formulas. Do not try to memorise them all in one sitting. Learn two or three formulas per day, solve problems using them, and move on to the next batch.

Common mistake: mixing up total surface area and lateral surface area. Lateral surface area of a cuboid is $2h(l + b)$ and does not include the top and bottom faces.

Difficulty: Moderate. Foundation level: High.

This chapter extends exponents to negative exponents and introduces the idea of expressing very large or very small numbers in standard form.

Key laws of exponents:

• $a^m \times a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^{-n} = \frac{1}{a^n}$
• $a^0 = 1$ (for any $a \neq 0$)
• $(ab)^n = a^n b^n$

Standard form: a number written as $k \times 10^n$ where $1 \leq k < 10$.

For example, the speed of light is approximately $3 \times 10^8$ m/s and the size of a red blood cell is about $7 \times 10^{-6}$ m.

Common mistake: thinking $a^{-n}$ means the number is negative. It does not. It means the reciprocal. So $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$, not $-8$.

Difficulty: Easy to Moderate. Foundation level: High.

This chapter teaches you to identify when two quantities are in direct proportion or inverse proportion, and then use that relationship to solve problems.

Key ideas:

• Direct proportion: when one quantity increases, the other also increases at the same rate. If $x$ and $y$ are in direct proportion, then $\frac{x_1}{y_1} = \frac{x_2}{y_2}$
• Inverse proportion: when one quantity increases, the other decreases. If $x$ and $y$ are inversely proportional, then $x_1 \times y_1 = x_2 \times y_2$

Real life examples: cost of items (direct proportion with number of items), time taken to finish work (inverse proportion with number of workers), speed and time for a journey (inverse proportion).

Study tip: The best way to master this chapter is to practice identifying whether a situation is direct or inverse proportion before you start calculating. Once you get the identification right, the calculation is straightforward.

Difficulty: Moderate to Hard. Foundation level: Extremely High.

Factorisation is essentially the reverse of what you did in Chapter 9. Instead of expanding expressions, you are breaking them down into factors. This is one of the most heavily tested skills in Class 9 and 10.

Methods of factorisation:

• Taking out common factors: $6xy + 9y^2 = 3y(2x + 3y)$
• Regrouping: when there is no single common factor, rearrange terms and group them
• Using identities: recognise the patterns from Chapter 9 and work backwards

For example:
• $x^2 + 6x + 9 = (x + 3)^2$ (using $(a + b)^2$ identity)
• $4x^2 - 9 = (2x + 3)(2x - 3)$ (using $a^2 - b^2$ identity)
• $x^2 + 5x + 6 = (x + 2)(x + 3)$ (using $(x + a)(x + b)$ identity)

Division of algebraic expressions: dividing a polynomial by a monomial and dividing a polynomial by another polynomial.

Practice tip: Do at least 20 factorisation problems beyond what the textbook gives. This is a skill that needs repetition. The more you practice, the faster you get at spotting the patterns.

Difficulty: Easy. Foundation level: High.

This chapter introduces you to reading and plotting graphs on the Cartesian plane. You will work with line graphs, linear graphs, and learn to plot points using coordinates.

Key concepts:

• The Cartesian plane has two axes: the horizontal $x$ axis and the vertical $y$ axis
• A point is written as $(x, y)$ where $x$ is the distance along the horizontal axis and $y$ is the distance along the vertical axis
• A linear graph is a straight line. If the equation is $y = mx + c$, the graph is always a straight line
• Reading information from graphs: finding values, identifying trends, and comparing data

This chapter is the foundation for coordinate geometry in Class 9 and 10. It may seem simple now, but make sure you are comfortable with plotting points and reading graphs accurately.

Study tip: Practice plotting at least 10 points on graph paper. It sounds basic but many students make careless mistakes by swapping the $x$ and $y$ coordinates.

Difficulty: Easy to Moderate. Foundation level: Moderate.

This chapter is about number puzzles, divisibility rules, and writing numbers in generalised form.

Key concepts:

• General form of a 2 digit number: $10a + b$ (where $a$ is the tens digit and $b$ is the units digit)
• General form of a 3 digit number: $100a + 10b + c$
• Reversing a 2 digit number: $10b + a$
• Divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11

Divisibility by 11: a number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or divisible by 11.

This chapter is interesting because it helps you understand why certain number tricks work. For example, why the sum of a 2 digit number and its reverse is always divisible by 11: $(10a + b) + (10b + a) = 11a + 11b = 11(a + b)$.

Study tip: This chapter is fun and not too heavy. Enjoy the puzzles but make sure you understand the algebra behind them.

Start with Chapter 1 (Rational Numbers) and Chapter 2 (Linear Equations in One Variable). These two chapters set the tone for everything else. Spend a full week on each. In week 3, cover Chapter 12 (Exponents and Powers) since it is closely related to number concepts.

By the end of week 3, you should be comfortable with properties of rational numbers, solving linear equations with variables on both sides, and working with negative exponents.

This is the most important stretch. Spend a full week on Chapter 9 (Algebraic Expressions and Identities), then a full week on Chapter 14 (Factorisation). These two chapters are connected and must be done together. Use week 6 for Chapter 8 (Comparing Quantities), which has the compound interest formula you need to practice thoroughly.

Do extra problems from these chapters if you can. They are the ones that matter most for Class 9 preparation.

Cover Chapter 3 (Understanding Quadrilaterals) in week 7. Spend week 8 on Chapter 11 (Mensuration), which is formula heavy and needs extra time. In week 9, do Chapter 4 (Practical Geometry) and Chapter 10 (Visualising Solid Shapes) together since they are both visual and less calculation intensive.

Make sure you memorise the mensuration formulas during this stretch. Flashcards or a formula sheet on your wall works well.

Cover Chapter 6 (Squares and Square Roots), Chapter 7 (Cubes and Cube Roots), and Chapter 13 (Direct and Inverse Proportions) in weeks 10 and 11. These chapters are moderate in difficulty and you should be able to handle them faster by now.

Week 12 is for Chapter 5 (Data Handling), Chapter 15 (Introduction to Graphs), and Chapter 16 (Playing with Numbers). These are the lighter chapters.

Use any remaining time for revision. Go back to Chapters 2, 8, 9, 11, and 14 and redo the exercises you found hardest.