Direct and Inverse Proportions

Chapter Revision Notes: Direct and Inverse Proportions

Class 8 Mathematics (MP Board)

Introduction / परिचय

Quantities can change in relation to each other. When an increase in one quantity leads to a corresponding increase in another, or a decrease leads to a decrease, they are related. This chapter deals with two types of proportions: Direct Proportion and Inverse Proportion.

### Concept 1: Direct Proportion (सीधा समानुपात)

Definition:

Two quantities $x$ and $y$ are said to be in direct proportion if they increase (or decrease) together in such a way that the ratio of their corresponding values remains constant.

• If $\frac{x}{y} = k$ (where $k$ is a positive constant, known as the constant of proportionality), then $x$ and $y$ vary directly.
• If $y_1$ and $y_2$ are the values of $y$ corresponding to the values $x_1$ and $x_2$ of $x$ respectively, then: $$\frac{x_1}{y_1} = \frac{x_2}{y_2}$$

Key Examples in Real Life:

• More items purchased $\rightarrow$ More cost (अधिक वस्तुएँ $\rightarrow$ अधिक मूल्य).
• More speed (at constant time) $\rightarrow$ More distance covered.
• More working hours $\rightarrow$ More work done.

### Concept 2: Inverse Proportion (व्युत्क्रमानुपात / प्रतिलोम समानुपात)

Definition:

Two quantities $x$ and $y$ are said to be in inverse proportion if an increase in one quantity leads to a decrease in the other quantity and vice versa, in such a way that their product remains constant.

• If $x \times y = k$ (where $k$ is a constant), then $x$ and $y$ vary inversely.
• If $y_1$ and $y_2$ are the values of $y$ corresponding to the values $x_1$ and $x_2$ of $x$ respectively, then: $$x_1 \times y_1 = x_2 \times y_2 \quad \text{or} \quad \frac{x_1}{x_2} = \frac{y_2}{y_1}$$

Key Examples in Real Life:

• More workers $\rightarrow$ Less time required to finish a job (अधिक मजदूर $\rightarrow$ कम समय).
• More speed $\rightarrow$ Less time taken to cover a fixed distance (अधिक चाल $\rightarrow$ कम समय).

### Summary of Formulas (महत्वपूर्ण सूत्र)

### Quick Tips for Problem Solving (हल करने के लिए महत्वपूर्ण टिप्स)

1. Identify the Type: Read the word problem carefully to check whether the quantities increase/decrease together (Direct) or one increases while the other decreases (Inverse).
2. Make a Table: Tabulate the given values with proper variables ($x_1, x_2, y_1, y_2$).
3. Apply the Formula:
   • Use $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ for Direct Proportion.
   • Use $x_1 \times y_1 = x_2 \times y_2$ for Inverse Proportion.
4. Solve for Unknown: Find the missing value using simple linear equations.