Mathematics for Class 8 under the West Bengal Board of Secondary Education (WBBSE) includes a variety of topics designed to build a solid foundation for higher studies.

**All following are class 8 math solution for wbbse**

**Chapter 1: Rational Numbers**

**Understanding Rational Numbers**

Rational numbers are numbers that can be written as a fraction, where both the top number (numerator) and the bottom number (denominator) are whole numbers. The denominator should not be zero. For example, 3/4 is a rational number.

**Example Problem and Solution**

**Problem**: Simplify the rational number 48/64.

**Solution**:

**Find the Greatest Common Divisor (GCD)**: The first step is to find the largest number that can divide both 48 and 64 without leaving a remainder. For this, we find the common factors of both numbers. The common factors of 48 and 64 include 1, 2, 4, 8, and 16. The largest common factor is 16.**Divide the Numerator and Denominator by the GCD**: Next, we divide both the numerator and the denominator by 16.- 48 divided by 16 is 3.
- 64 divided by 16 is 4.

Therefore, 48/64 simplifies to 3/4.

**Chapter 2: Linear Equations in One Variable**

**Understanding Linear Equations**

A linear equation in one variable is an equation that involves only one variable and can be written in a simple form. For example, x + 5 = 10 is a linear equation where x is the variable.

**Example Problem and Solution**

**Problem**: Solve the equation x + 7 = 12.

**Solution**:

**Isolate the Variable**: To solve for x, we need to get x alone on one side of the equation. We do this by performing the same operation on both sides of the equation. Here, we subtract 7 from both sides.- x + 7 – 7 = 12 – 7

**Simplify**: Simplifying both sides, we get:- x = 5

So, the solution to the equation x + 7 = 12 is x = 5.

**Chapter 3: Understanding Quadrilaterals**

**Types of Quadrilaterals**

Quadrilaterals are four-sided figures. There are various types of quadrilaterals, including squares, rectangles, parallelograms, rhombuses, and trapezoids. Each type has its own properties.

**Example Problem and Solution**

**Problem**: Find the perimeter of a rectangle with length 8 cm and width 5 cm.

**Solution**:

**Understand the Perimeter**: The perimeter of a rectangle is the total distance around the edge of the rectangle. It is calculated by adding the lengths of all four sides.- A rectangle has two lengths and two widths.
- Perimeter = 2 * (Length + Width)

**Calculate**: Here, the length is 8 cm and the width is 5 cm.- Perimeter = 2 * (8 + 5)
- Perimeter = 2 * 13
- Perimeter = 26 cm

So, the perimeter of the rectangle is 26 cm.

**Chapter 4: Practical Geometry**

**Drawing Geometric Shapes**

Practical geometry involves drawing shapes accurately using tools like a ruler and compass. It includes constructing angles, triangles, and other polygons.

**Example Problem and Solution**

**Problem**: Draw a triangle with sides 5 cm, 6 cm, and 7 cm.

**Solution**:

**Draw the Base**: Using a ruler, draw a straight line 7 cm long. This will be the base of the triangle.**Construct the First Side**: From one end of the base, use a compass to draw an arc with a radius of 5 cm.**Construct the Second Side**: From the other end of the base, use the compass to draw another arc with a radius of 6 cm. The point where these two arcs intersect is the third vertex of the triangle.**Complete the Triangle**: Connect the vertex to both ends of the base using a ruler.

Now you have a triangle with sides 5 cm, 6 cm, and 7 cm.

**Chapter 5: Data Handling**

**Organizing Data**

Data handling involves collecting, organizing, and interpreting data. This can include creating bar graphs, pie charts, and line graphs.

**Example Problem and Solution**

**Problem**: Represent the following data in a bar graph: Number of students in different classes – Class 6: 40, Class 7: 35, Class 8: 50, Class 9: 45.

**Solution**:

**Draw the Axes**: Draw a horizontal axis (x-axis) and a vertical axis (y-axis). Label the x-axis with the classes and the y-axis with the number of students.**Mark the Data**: On the x-axis, mark the classes (Class 6, Class 7, Class 8, Class 9). On the y-axis, mark a scale that includes the highest number of students (50 in this case).**Draw the Bars**: For each class, draw a bar that reaches up to the corresponding number of students:- Class 6: Draw a bar up to 40.
- Class 7: Draw a bar up to 35.
- Class 8: Draw a bar up to 50.
- Class 9: Draw a bar up to 45.

**Chapter 6: Squares and Square Roots**

**Understanding Squares and Square Roots**

A square of a number is that number multiplied by itself. The square root of a number is a value that, when multiplied by itself, gives the original number.

**Example Problem and Solution**

**Problem**: Find the square root of 81.

**Solution**:

**Identify the Square Root**: Think of a number that, when multiplied by itself, equals 81.- The number 9 multiplied by itself (9 x 9) equals 81.

So, the square root of 81 is 9.

**Chapter 7: Cubes and Cube Roots**

**Understanding Cubes and Cube Roots**

A cube of a number is that number multiplied by itself twice. The cube root of a number is a value that, when used in three multiplications, gives the original number.

**Example Problem and Solution**

**Problem**: Find the cube root of 27.

**Solution**:

**Identify the Cube Root**: Think of a number that, when multiplied by itself twice, equals 27.- The number 3 multiplied by itself twice (3 x 3 x 3) equals 27.

So, the cube root of 27 is 3.

**Chapter 8: Comparing Quantities**

**Understanding Ratios and Percentages**

Comparing quantities often involves using ratios and percentages. A ratio compares two quantities, while a percentage represents a part of a whole as parts per hundred.

**Example Problem and Solution**

**Problem**: Find the ratio of 20 to 50 and express it as a percentage.

**Solution**:

**Find the Ratio**: To find the ratio, divide both numbers by their greatest common divisor. The greatest common divisor of 20 and 50 is 10.- Ratio: 20 divided by 10 is 2, and 50 divided by 10 is 5.
- So, the ratio is 2:5.

**Convert to Percentage**: To convert the ratio to a percentage, think of it as a fraction and multiply by 100.- Ratio as a fraction: 2 out of 5 is 2/5.
- Convert to percentage: (2/5) multiplied by 100 is 40%.

So, the ratio 2:5 is equivalent to 40%.

**Chapter 9: Algebraic Expressions and Identities**

**Understanding Algebraic Expressions**

Algebraic expressions are combinations of numbers, variables, and operations. For example, 3x + 2 is an algebraic expression.

**Example Problem and Solution**

**Problem**: Simplify the expression 2x + 3x – x.

**Solution**:

**Combine Like Terms**: To simplify, combine all terms that have the same variable (x in this case).- 2x + 3x – x: Combine 2x and 3x to get 5x, then subtract x.

So, the simplified expression is 4x.

**Chapter 10: Visualizing Solid Shapes**

**Understanding Solid Shapes**

Solid shapes include 3D objects like cubes, cylinders, spheres, and cones. These shapes have length, width, and height.

**Example Problem and Solution**

**Problem**: Identify the shape with 6 equal faces.

**Solution**:

**Identify the Shape**: A shape with 6 equal faces is known as a cube. Each face of the cube is a square, and all faces are the same size.

So, the shape with 6 equal faces is a cube.

**Chapter 11: Mensuration**

**Understanding Mensuration**

Mensuration involves calculating the area, perimeter, and volume of different shapes. Area is the amount of space inside a shape, perimeter is the distance around a shape, and volume is the space a 3D shape occupies.

**Example Problem and Solution**

**Problem**: Find the area of a rectangle with length 10 cm and width 5 cm.

**Solution**:

**Calculate the Area**: To find the area, multiply the length by the width.- Area: 10 cm multiplied by 5 cm equals 50 square cm.

So, the area of the rectangle is 50 square cm.

**Chapter 12: Exponents and Powers**

**Understanding Exponents**

Exponents are a way to express repeated multiplication of the same number. For example, 2 to the power of 3 (written as 2^3) means 2 multiplied by itself 3 times.

**Example Problem and Solution**

**Problem**: Simplify the expression 2^3.

**Solution**:

**Calculate the Power**: 2 to the power of 3 means 2 multiplied by itself 3 times.- 2 x 2 x 2 equals 8.

So, 2^3 simplifies to 8.

**Chapter 13: Direct and Inverse Proportions**

**Understanding Proportions**

Proportions compare two ratios. In direct proportion, as one quantity increases, the other also increases. In inverse proportion, as one quantity increases, the other decreases.

**Example Problem and Solution**

**Problem**: If 5 pens cost 25 rupees, how much will 8 pens cost?

**Solution**:

**Find the Cost of One Pen**: First, find the cost of one pen by dividing the total cost by the number of pens.- Cost of one pen: 25 rupees divided by 5 pens equals 5 rupees per pen.

**Calculate the Cost of 8 Pens**: Multiply the cost of one pen by 8.- 8 pens: 5 rupees per pen multiplied by 8 equals 40 rupees.

So, 8 pens will cost 40 rupees.

**Chapter 14: Factorization**

**Understanding Factorization**

Factorization involves breaking down a number or an expression into its factors. Factors are numbers or expressions that multiply together to get the original number or expression.

**Example Problem and Solution**

**Problem**: Factorize the number 24.

**Solution**:

**Find the Factors**: To factorize 24, find the numbers that multiply together to give 24.- Factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24.

So, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

**Chapter 15: Introduction to Graphs**

**Understanding Graphs**

Graphs are visual representations of data. Common types include bar graphs, line graphs, and pie charts.

**Example Problem and Solution**

**Problem**: Plot a line graph for the following data: Day 1: 5 books sold, Day 2: 7 books sold, Day 3: 9 books sold, Day 4: 6 books sold.

**Solution**:

**Draw the Axes**: Draw a horizontal axis (x-axis) for the days and a vertical axis (y-axis) for the number of books sold.**Mark the Data Points**: On Day 1, mark a point at 5 books sold on the y-axis. On Day 2, mark a point at 7 books sold, and so on.**Connect the Points**: Draw a line connecting all the points from Day 1 to Day 4.

Now you have a line graph representing the number of books sold over 4 days.

**Chapter 16: Playing with Numbers**

**Understanding Patterns**

Playing with numbers involves recognizing patterns and using them to solve problems. This includes identifying number sequences and understanding their rules.

**Example Problem and Solution**

**Problem**: What is the next number in the sequence 2, 4, 6, 8, 10?

**Solution**:

**Identify the Pattern**: The sequence increases by 2 each time.**Find the Next Number**: Add 2 to the last number in the sequence (10).- 10 + 2 equals 12.

So, the next number in the sequence is 12.

**Chapter 17: Probability**

**Understanding Probability**

Probability is the chance of an event happening. It is expressed as a number between 0 and 1, where 0 means the event will not happen, and 1 means it will definitely happen.

**Example Problem and Solution**

**Problem**: What is the probability of flipping a coin and getting heads?

**Solution**:

**Identify Possible Outcomes**: When flipping a coin, there are two possible outcomes: heads or tails.**Determine the Probability**: Since there are two outcomes and only one is heads, the probability of getting heads is 1 out of 2.

So, the probability of flipping a coin and getting heads is 1/2 or 0.5.

**Chapter 18: Understanding Quadratic Equations**

**Understanding Quadratic Equations**

A quadratic equation is an equation that involves the square of a variable. For example, x^2 + 5x + 6 = 0 is a quadratic equation.

**Example Problem and Solution**

**Problem**: Solve the quadratic equation x^2 – 4 = 0.

**Solution**:

**Isolate the Square Term**: Add 4 to both sides to isolate the square term.- x^2 = 4

**Find the Square Roots**: The values of x that satisfy this equation are the numbers whose square is 4.- The numbers are 2 and -2.

So, the solutions to the quadratic equation x^2 – 4 = 0 are x = 2 and x = -2.

**Chapter 19: Introduction to Trigonometry**

**Understanding Trigonometry**

Trigonometry deals with the relationships between the sides and angles of triangles. It is especially useful for right-angled triangles.

**Example Problem and Solution**

**Problem**: In a right-angled triangle, if one angle is 30 degrees and the hypotenuse is 10 cm, find the length of the side opposite the 30-degree angle.

**Solution**:

**Use Trigonometric Ratios**: In a right-angled triangle, the side opposite a 30-degree angle is half the hypotenuse.- Half of 10 cm is 5 cm.

So, the length of the side opposite the 30-degree angle is 5 cm.

By understanding and practicing these example problems and solutions, Class 8 students can build a strong foundation in mathematics. Regular practice and review of these concepts will help students excel in their exams and prepare for higher-level math in the future.