Description:

The chapter “Variation” in the Class 10 Mathematics syllabus of the West Bengal Board introduces students to the mathematical relationships between two or more quantities that change with respect to one another. This chapter focuses on three main types of variations: Direct Variation, Inverse Variation, and Joint Variation. Each type is explained using simple definitions, formulas, and real-life examples to help students identify and apply them in various mathematical and practical contexts.

1. Concept of Variation

Variation refers to the relationship between variables where the change in one causes a systematic change in the other. The purpose of this chapter is to help students understand how quantities are related proportionally and how to represent them algebraically.

There are three main types of variation:

• Direct Variation 
• Inverse Variation 
• Joint Variation 

2. Direct Variation

2.1 Definition:

When two quantities increase or decrease together in the same ratio, they are said to be in Direct Variation.
Mathematical expression:
If y ∝ x, then y = kx, where k is the constant of variation.

2.2 Characteristics:

• Both variables increase or decrease together. 
• The ratio y/x remains constant. 
• The graph is a straight line passing through the origin. 

2.3 Example:

Suppose a car travels at a constant speed of 50 km/h.

• In 2 hours, it covers 100 km (50 × 2) 
• In 3 hours, it covers 150 km (50 × 3)
  Here, time and distance vary directly. 

3. Inverse Variation

3.1 Definition:

When one quantity increases and the other decreases in such a way that their product remains constant, it is called Inverse Variation.
Mathematical expression:
If y ∝ 1/x, then y = k/x, where k is the constant of variation.

3.2 Characteristics:

• One variable increases while the other decreases. 
• The product of the two variables (x × y) remains constant. 
• The graph is a curve (hyperbola). 

3.3 Example:

A task is completed by 4 workers in 6 days.
If the number of workers increases to 8, the task will be done in 3 days.
(4 × 6 = 24 = 8 × 3)
This is an inverse variation between the number of workers and the time required.

4. Joint Variation

4.1 Definition:

When a variable depends on two or more other variables simultaneously (either directly or inversely), it is called Joint Variation.
Mathematical expression:
If y ∝ xz, then y = kxz

4.2 Characteristics:

• Combines more than one variation. 
• Involves multiple variables. 
• Frequently used in complex real-world problems. 

4.3 Example:

Let’s consider a situation where the time taken to complete a job depends on both the number of workers and their efficiency.
Time ∝ 1 / (number of workers × efficiency)
This is a case of joint variation.

5. Steps to Solve Variation Problems

5.1 Finding the Constant (k):

Identify the relationship and use given values to determine k.

Example: If y ∝ x and y = 10 when x = 2, then:
10 = k × 2 ⇒ k = 5
So, the relationship becomes y = 5x

5.2 Calculating Unknown Values:

Use the value of k to find missing values.

Example: From y = 5x, if x = 4, then y = 5 × 4 = 20

6. Importance of Units and Conversion

In variation problems, keeping units consistent is essential. Variables like time, distance, work, and speed must be used with proper units and converted when necessary.

Example: Convert time from hours to minutes by multiplying by 60.

7. Real-life Applications of Variation

7.1 Finance and Transactions:

Cost and quantity are directly proportional. Buying more items increases the total cost.

7.2 Speed and Travel Time:

Speed and time are inversely proportional when distance is constant — more speed, less time.

7.3 Work and Workforce:

Time taken to finish work is inversely proportional to the number of workers and their efficiency — a classic case of joint variation.

8. Graphical Interpretation

8.1 Direct Variation Graph:

A straight line that passes through the origin showing a linear relationship.

8.2 Inverse Variation Graph:

A downward-curving graph known as a hyperbola that shows the product of variables remains constant.

8.3 Importance of Graphs:

Graphs help visually identify the nature of the variation and understand the rate at which one variable changes in response to another.

9. Summary

The chapter “Variation” is a fundamental part of algebra in Class 10 Mathematics, helping students analyze how two or more quantities are related. Understanding the formulas and relationships is key to solving real-life mathematical problems.

• In Direct Variation, both variables increase or decrease together (y = kx). 
• In Inverse Variation, one increases while the other decreases (y = k/x). 
• In Joint Variation, a variable depends on multiple others (y = kxz, or similar forms).