Chapter 10 Direct and Inverse Variations
RD Sharma Class 8 Maths Chapter 10 – Direct and Inverse Variations
RD Sharma Chapter 10 introduces the concept of variation, an essential topic in algebra that explains how two quantities are related to each other. In this chapter, students learn about direct variation and inverse variation, along with their applications in real-life situations. These RD Sharma class 8 Solutions and notes provide a simple and structured explanation of all important concepts.
Find the PDF Solutions of all the exercises in Chapter 10 – Direct and Inverse Variations Notes
What is Variation?
Variation describes the relationship between two quantities where a change in one quantity results in a change in another. There are mainly two types of variations:
- Direct Variation
- Inverse Variation
Understanding these relationships helps in solving problems involving proportional changes.
Direct Variation
Two quantities are said to be in direct variation if an increase in one leads to a proportional increase in the other, and a decrease in one leads to a proportional decrease in the other.
Definition
If x and y are two quantities such that:
y ∝ x
Then,
y = kx
where k is a constant called the constant of proportionality.
Example
If y varies directly as x and y = 10 when x = 2, then:
k = y/x = 10/2 = 5
So, the equation becomes:
y = 5x
Key Points of Direct Variation
- The ratio y/x remains constant
- Graph of direct variation is a straight line passing through the origin
- Both variables increase or decrease together
Real-Life Examples
- Cost of items varies directly with quantity
- Distance varies directly with time (at constant speed)
Inverse Variation
Two quantities are said to be in inverse variation if an increase in one leads to a decrease in the other, and vice versa.
Definition
If x and y are two quantities such that:
y ∝ 1/x
Then,
y = k/x
where k is a constant.
Example
If y varies inversely as x and y = 12 when x = 3, then:
k = xy = 12 × 3 = 36
So, the equation becomes:
y = 36/x
Key Points of Inverse Variation
- The product xy remains constant
- As one variable increases, the other decreases
- Graph is a curve (hyperbola), not a straight line
Real-Life Examples
- Speed and time (for a fixed distance)
- Number of workers and time taken to complete a task
Difference Between Direct and Inverse Variation
Solving Problems on Variation
To solve variation problems, follow these steps:
- Identify the type of variation (direct or inverse)
- Write the formula (y = kx or y = k/x)
- Use given values to find k
- Substitute k to find unknown values
Example Problem
If y varies directly as x and y = 15 when x = 5, find y when x = 8.
Solution:
k = 15/5 = 3
y = 3 × 8 = 24
Applications of Direct and Inverse Variation
Variation is widely used in real-life situations such as:
- Calculating wages based on hours worked
- Finding time required to complete tasks
- Understanding speed, distance, and time relationships
- Solving problems in physics and economics
Tips to Master the Chapter
- Clearly identify whether the relation is direct or inverse
- Memorize the basic formulas
- Practice solving numerical problems
- Understand real-life applications
- Check units and values carefully
Common Mistakes to Avoid
- Confusing direct variation with inverse variation
- Using incorrect formulas
- Not calculating the constant k correctly
- Ignoring units in practical problems